From b4d6b431adc64df33c5b3d907bb600c4880ac06b Mon Sep 17 00:00:00 2001 From: Enrico Tassi Date: Tue, 9 Jun 2020 17:25:11 +0200 Subject: [PATCH 01/25] [core] generalize context from term |-> type to term |-> 'a --- src/core/ctxt.ml | 2 +- src/core/term.ml | 4 ++-- src/core/term.mli | 3 ++- 3 files changed, 5 insertions(+), 4 deletions(-) diff --git a/src/core/ctxt.ml b/src/core/ctxt.ml index e21d620ce..abca46037 100644 --- a/src/core/ctxt.ml +++ b/src/core/ctxt.ml @@ -7,7 +7,7 @@ open Timed (** [type_of x ctx] returns the type of [x] in the context [ctx] when it appears in it, and @raise [Not_found] otherwise. *) -let type_of : tvar -> ctxt -> term = fun x ctx -> +let type_of : tvar -> 'a actxt -> 'a = fun x ctx -> let (_,a,_) = List.find (fun (y,_,_) -> Bindlib.eq_vars x y) ctx in a (** [def_of x ctx] returns the definition of [x] in the context [ctx] if it diff --git a/src/core/term.ml b/src/core/term.ml index 332f3781e..0fa4f9f86 100644 --- a/src/core/term.ml +++ b/src/core/term.ml @@ -275,8 +275,8 @@ end list [xn:An;..;x1:A1] in reverse order (last added variable comes first). Note that it cannot be implemented by a map as the order is important. *) -type ctxt = (tvar * term * term option) list - +type 'a actxt = (tvar * 'a * term option) list +type ctxt = term actx (** Typing context with lifted terms. Used to optimise type checking and avoid lifting terms several times. Definitions are not included because these contexts are used to create meta variables types, which do not use [let] diff --git a/src/core/term.mli b/src/core/term.mli index cbf5d1dfb..4f528cb30 100644 --- a/src/core/term.mli +++ b/src/core/term.mli @@ -256,7 +256,8 @@ end definition. The typing environment [x1:A1,..,xn:An] is represented by the list [xn:An;..;x1:A1] in reverse order (last added variable comes first). *) -type ctxt = (tvar * term * term option) list +type 'a actxt = (tvar * 'a * term option) list +type ctxt = term actx (** Typing context with lifted terms. Used to optimise type checking and avoid lifting terms several times. Definitions are not included because these From 0c56e85d85aaf850c434d7dc6f42d0789e492ae1 Mon Sep 17 00:00:00 2001 From: Enrico Tassi Date: Tue, 9 Jun 2020 19:59:28 +0200 Subject: [PATCH 02/25] [opam] depend on Elpi --- lambdapi.opam | 1 + 1 file changed, 1 insertion(+) diff --git a/lambdapi.opam b/lambdapi.opam index 829e849c9..4557e1b92 100644 --- a/lambdapi.opam +++ b/lambdapi.opam @@ -62,6 +62,7 @@ depends: [ "cmdliner" {>= "1.1.0"} "stdlib-shims" {>= "0.1.0"} "odoc" {with-doc} + "elpi" { >= "1.16.0" & < "1.17.0" } ] build: [ ["dune" "subst"] {dev} From 32bb53130015643a309ef98a55b59800e4314e24 Mon Sep 17 00:00:00 2001 From: Enrico Tassi Date: Tue, 9 Jun 2020 17:24:04 +0200 Subject: [PATCH 03/25] [elpi] minimalistic integration --- dune-workspace | 0 lambdap.elpi | 1298 + lambdapi.opam | 2 +- out/Coq__Arith__Arith__base.dk | 4 + out/Coq__Arith__Between.dk | 2513 + out/Coq__Arith__Compare__dec.dk | 5066 ++ out/Coq__Arith__EqNat.dk | 2688 + out/Coq__Arith__Factorial.dk | 342 + out/Coq__Arith__Gt.dk | 1422 + out/Coq__Arith__Le.dk | 319 + out/Coq__Arith__Lt.dk | 1835 + out/Coq__Arith__Minus.dk | 796 + out/Coq__Arith__Mult.dk | 2942 + out/Coq__Arith__Peano__dec.dk | 6849 +++ out/Coq__Arith__Plus.dk | 2831 + out/Coq__Arith__Wf__nat.dk | 3746 ++ out/Coq__Bool__Bool.dk | 48232 ++++++++++++++++ out/Coq__Bool__BoolEq.dk | 1120 + out/Coq__Bool__DecBool.dk | 216 + out/Coq__Bool__IfProp.dk | 1245 + out/Coq__Classes__CMorphisms.dk | 29593 ++++++++++ out/Coq__Classes__CRelationClasses.dk | 11122 ++++ out/Coq__Classes__Equivalence.dk | 5430 ++ out/Coq__Classes__Init.dk | 30 + out/Coq__Classes__Morphisms.dk | 20641 +++++++ out/Coq__Classes__Morphisms__Prop.dk | 5654 ++ out/Coq__Classes__RelationClasses.dk | 17089 ++++++ out/Coq__Classes__SetoidTactics.dk | 98 + out/Coq__Init__Datatypes.dk | 9568 +++ out/Coq__Init__Decimal.dk | 816 + out/Coq__Init__Logic.dk | 20146 +++++++ out/Coq__Init__Logic__Type.dk | 398 + out/Coq__Init__Nat.dk | 1457 + out/Coq__Init__Notations.dk | 4 + out/Coq__Init__Peano.dk | 5988 ++ out/Coq__Init__Prelude.dk | 4 + out/Coq__Init__Specif.dk | 48062 +++++++++++++++ out/Coq__Init__Tactics.dk | 87 + out/Coq__Init__Tauto.dk | 4 + out/Coq__Init__Wf.dk | 1621 + out/Coq__Logic__Decidable.dk | 9614 +++ out/Coq__Logic__EqdepFacts.dk | 12316 ++++ out/Coq__Logic__Eqdep__dec.dk | 13414 +++++ out/Coq__Logic__ProofIrrelevanceFacts.dk | 4 + out/Coq__Numbers__NatInt__NZAdd.dk | 4 + out/Coq__Numbers__NatInt__NZAddOrder.dk | 4 + out/Coq__Numbers__NatInt__NZAxioms.dk | 4 + out/Coq__Numbers__NatInt__NZBase.dk | 4 + out/Coq__Numbers__NatInt__NZBits.dk | 4 + out/Coq__Numbers__NatInt__NZDiv.dk | 4 + out/Coq__Numbers__NatInt__NZGcd.dk | 4 + out/Coq__Numbers__NatInt__NZLog.dk | 4 + out/Coq__Numbers__NatInt__NZMul.dk | 4 + out/Coq__Numbers__NatInt__NZMulOrder.dk | 4 + out/Coq__Numbers__NatInt__NZOrder.dk | 4 + out/Coq__Numbers__NatInt__NZParity.dk | 4 + out/Coq__Numbers__NatInt__NZPow.dk | 4 + out/Coq__Numbers__NatInt__NZProperties.dk | 4 + out/Coq__Numbers__NatInt__NZSqrt.dk | 4 + out/Coq__Numbers__Natural__Abstract__NAdd.dk | 4 + ...__Numbers__Natural__Abstract__NAddOrder.dk | 4 + ...oq__Numbers__Natural__Abstract__NAxioms.dk | 4 + out/Coq__Numbers__Natural__Abstract__NBase.dk | 4 + out/Coq__Numbers__Natural__Abstract__NBits.dk | 4 + out/Coq__Numbers__Natural__Abstract__NDiv.dk | 4 + out/Coq__Numbers__Natural__Abstract__NGcd.dk | 4 + out/Coq__Numbers__Natural__Abstract__NLcm.dk | 4 + out/Coq__Numbers__Natural__Abstract__NLog.dk | 4 + ...oq__Numbers__Natural__Abstract__NMaxMin.dk | 4 + ...__Numbers__Natural__Abstract__NMulOrder.dk | 4 + ...Coq__Numbers__Natural__Abstract__NOrder.dk | 4 + ...oq__Numbers__Natural__Abstract__NParity.dk | 4 + out/Coq__Numbers__Natural__Abstract__NPow.dk | 4 + ...Numbers__Natural__Abstract__NProperties.dk | 4 + out/Coq__Numbers__Natural__Abstract__NSqrt.dk | 4 + out/Coq__Numbers__Natural__Abstract__NSub.dk | 4 + out/Coq__Numbers__NumPrelude.dk | 4 + out/Coq__Program__Basics.dk | 109 + out/Coq__Program__Tactics.dk | 22 + out/Coq__Relations__Operators__Properties.dk | 2889 + out/Coq__Relations__Relation__Definitions.dk | 804 + out/Coq__Relations__Relation__Operators.dk | 4526 ++ out/Coq__Relations__Relations.dk | 898 + out/Coq__Setoids__Setoid.dk | 513 + out/Coq__Structures__Equalities.dk | 4 + out/Coq__Structures__GenericMinMax.dk | 48 + out/Coq__Structures__Orders.dk | 4 + out/Coq__Structures__OrdersFacts.dk | 4 + out/Coq__Structures__OrdersTac.dk | 131 + out/Makefile | 4 + out/cupicef.lp | 345 + out/lambdap.elpi | 1298 + out/lambdapi.pkg | 2 + src/cli/lambdapi.ml | 1 + src/common/error.ml | 2 + src/core/ctxt.ml | 11 + src/core/dune | 2 +- src/core/elpi_lambdapi.ml | 259 + src/core/sig_state.ml | 9 +- src/core/sign.ml | 16 + src/core/term.ml | 5 +- src/core/term.mli | 7 +- src/core/unif_rule.ml | 6 +- src/export/coq.ml | 3 + src/handle/command.ml | 38 +- src/handle/dune | 2 +- src/handle/elpi_handle.ml | 269 + src/handle/query.ml | 25 +- src/handle/query.mli | 2 +- src/handle/rewrite.ml | 2 +- src/handle/tactic.ml | 2 +- src/parsing/lpLexer.ml | 6 + src/parsing/lpParser.mly | 7 + src/parsing/scope.ml | 18 +- src/parsing/scope.mli | 5 + src/parsing/syntax.ml | 3 + tcsolver.elpi | 34 + tests/OK/elpitest.elpi | 29 + tests/OK/elpitest.lp | 44 + tests/rewriting.ml | 2 +- 120 files changed, 309129 insertions(+), 39 deletions(-) create mode 100644 dune-workspace create mode 100644 lambdap.elpi create mode 100644 out/Coq__Arith__Arith__base.dk create mode 100644 out/Coq__Arith__Between.dk create mode 100644 out/Coq__Arith__Compare__dec.dk create mode 100644 out/Coq__Arith__EqNat.dk create mode 100644 out/Coq__Arith__Factorial.dk create mode 100644 out/Coq__Arith__Gt.dk create mode 100644 out/Coq__Arith__Le.dk create mode 100644 out/Coq__Arith__Lt.dk create mode 100644 out/Coq__Arith__Minus.dk create mode 100644 out/Coq__Arith__Mult.dk create mode 100644 out/Coq__Arith__Peano__dec.dk create mode 100644 out/Coq__Arith__Plus.dk create mode 100644 out/Coq__Arith__Wf__nat.dk create mode 100644 out/Coq__Bool__Bool.dk create mode 100644 out/Coq__Bool__BoolEq.dk create mode 100644 out/Coq__Bool__DecBool.dk create mode 100644 out/Coq__Bool__IfProp.dk create mode 100644 out/Coq__Classes__CMorphisms.dk create mode 100644 out/Coq__Classes__CRelationClasses.dk create mode 100644 out/Coq__Classes__Equivalence.dk create mode 100644 out/Coq__Classes__Init.dk create mode 100644 out/Coq__Classes__Morphisms.dk create mode 100644 out/Coq__Classes__Morphisms__Prop.dk create mode 100644 out/Coq__Classes__RelationClasses.dk create mode 100644 out/Coq__Classes__SetoidTactics.dk create mode 100644 out/Coq__Init__Datatypes.dk create mode 100644 out/Coq__Init__Decimal.dk create mode 100644 out/Coq__Init__Logic.dk create mode 100644 out/Coq__Init__Logic__Type.dk create mode 100644 out/Coq__Init__Nat.dk create mode 100644 out/Coq__Init__Notations.dk create mode 100644 out/Coq__Init__Peano.dk create mode 100644 out/Coq__Init__Prelude.dk create mode 100644 out/Coq__Init__Specif.dk create mode 100644 out/Coq__Init__Tactics.dk create mode 100644 out/Coq__Init__Tauto.dk create mode 100644 out/Coq__Init__Wf.dk create mode 100644 out/Coq__Logic__Decidable.dk create mode 100644 out/Coq__Logic__EqdepFacts.dk create mode 100644 out/Coq__Logic__Eqdep__dec.dk create mode 100644 out/Coq__Logic__ProofIrrelevanceFacts.dk create mode 100644 out/Coq__Numbers__NatInt__NZAdd.dk create mode 100644 out/Coq__Numbers__NatInt__NZAddOrder.dk create mode 100644 out/Coq__Numbers__NatInt__NZAxioms.dk create mode 100644 out/Coq__Numbers__NatInt__NZBase.dk create mode 100644 out/Coq__Numbers__NatInt__NZBits.dk create mode 100644 out/Coq__Numbers__NatInt__NZDiv.dk create mode 100644 out/Coq__Numbers__NatInt__NZGcd.dk create mode 100644 out/Coq__Numbers__NatInt__NZLog.dk create mode 100644 out/Coq__Numbers__NatInt__NZMul.dk create mode 100644 out/Coq__Numbers__NatInt__NZMulOrder.dk create mode 100644 out/Coq__Numbers__NatInt__NZOrder.dk create mode 100644 out/Coq__Numbers__NatInt__NZParity.dk create mode 100644 out/Coq__Numbers__NatInt__NZPow.dk create mode 100644 out/Coq__Numbers__NatInt__NZProperties.dk create mode 100644 out/Coq__Numbers__NatInt__NZSqrt.dk create mode 100644 out/Coq__Numbers__Natural__Abstract__NAdd.dk create mode 100644 out/Coq__Numbers__Natural__Abstract__NAddOrder.dk create mode 100644 out/Coq__Numbers__Natural__Abstract__NAxioms.dk create mode 100644 out/Coq__Numbers__Natural__Abstract__NBase.dk create mode 100644 out/Coq__Numbers__Natural__Abstract__NBits.dk create mode 100644 out/Coq__Numbers__Natural__Abstract__NDiv.dk create mode 100644 out/Coq__Numbers__Natural__Abstract__NGcd.dk create mode 100644 out/Coq__Numbers__Natural__Abstract__NLcm.dk create mode 100644 out/Coq__Numbers__Natural__Abstract__NLog.dk create mode 100644 out/Coq__Numbers__Natural__Abstract__NMaxMin.dk create mode 100644 out/Coq__Numbers__Natural__Abstract__NMulOrder.dk create mode 100644 out/Coq__Numbers__Natural__Abstract__NOrder.dk create mode 100644 out/Coq__Numbers__Natural__Abstract__NParity.dk create mode 100644 out/Coq__Numbers__Natural__Abstract__NPow.dk create mode 100644 out/Coq__Numbers__Natural__Abstract__NProperties.dk create mode 100644 out/Coq__Numbers__Natural__Abstract__NSqrt.dk create mode 100644 out/Coq__Numbers__Natural__Abstract__NSub.dk create mode 100644 out/Coq__Numbers__NumPrelude.dk create mode 100644 out/Coq__Program__Basics.dk create mode 100644 out/Coq__Program__Tactics.dk create mode 100644 out/Coq__Relations__Operators__Properties.dk create mode 100644 out/Coq__Relations__Relation__Definitions.dk create mode 100644 out/Coq__Relations__Relation__Operators.dk create mode 100644 out/Coq__Relations__Relations.dk create mode 100644 out/Coq__Setoids__Setoid.dk create mode 100644 out/Coq__Structures__Equalities.dk create mode 100644 out/Coq__Structures__GenericMinMax.dk create mode 100644 out/Coq__Structures__Orders.dk create mode 100644 out/Coq__Structures__OrdersFacts.dk create mode 100644 out/Coq__Structures__OrdersTac.dk create mode 100644 out/Makefile create mode 100644 out/cupicef.lp create mode 100644 out/lambdap.elpi create mode 100644 out/lambdapi.pkg create mode 100644 src/core/elpi_lambdapi.ml create mode 100644 src/handle/elpi_handle.ml create mode 100644 tcsolver.elpi create mode 100644 tests/OK/elpitest.elpi create mode 100644 tests/OK/elpitest.lp diff --git a/dune-workspace b/dune-workspace new file mode 100644 index 000000000..e69de29bb diff --git a/lambdap.elpi b/lambdap.elpi new file mode 100644 index 000000000..0f2e420ab --- /dev/null +++ b/lambdap.elpi @@ -0,0 +1,1298 @@ +% automatically generated + +% ---- Lambdapi datatypes ---- + +% A symbol +typeabbrev symbol (ctype "symbol"). + + + +kind term type. +type typ term. +type kin term. +type symb symbol -> term. +type appl term -> term -> term. +type abst term -> (term -> term) -> term. +type prod term -> (term -> term) -> term. + + +kind sealed-goal type. +type nabla (term -> sealed-goal) -> sealed-goal. +type seal goal -> sealed-goal. + +kind goal type. +type goal list prop -> term -> term -> goal. + +type of term -> term -> prop. + +pred msolve o:list sealed-goal. + + + +% ---- Builtin predicates ---- + +% [lp.sig S T] Gives the type of a symbol +external pred lp.sig i:symbol, o:term. + +% [lp.tc-instances L] Gives the list of type class instances +external pred lp.tc-instances o:list symbol. + +% [lp.tc? S] Tells if S is a type class +external pred lp.tc? i:symbol. + +% [lp.term->string T S] Pretty prints a term T to the string S +external pred lp.term->string i:term, o:string. + +% ---- Elpi predicates ---- + +% == Core builtins ===================================== + +% -- Logic -- + +pred true. + +true. + +pred fail. + +pred false. + +external pred (=) o:A, o:A. % unification + +typeabbrev int (ctype "int"). + + +typeabbrev string (ctype "string"). + + +typeabbrev float (ctype "float"). + + +pred (;) o:prop, o:prop. + +(A ; _) :- A. + +(_ ; B) :- B. + +type (:-) prop -> prop -> prop. + +type (:-) prop -> list prop -> prop. + +type (,) variadic prop prop. + +type uvar A. + +type (as) A -> A -> A. + +type (=>) prop -> prop -> prop. + +type (=>) list prop -> prop -> prop. + +% -- Control -- + +external pred !. % The cut operator + +pred not i:prop. + +not X :- X, !, fail. + +not _. + +% [declare_constraint C Key1 Key2...] declares C blocked +% on Key1 Key2 ... (variables, or lists thereof). +external type declare_constraint variadic any prop. + +external pred print_constraints. % prints all constraints + +% [halt ...] halts the program and print the terms +external type halt variadic any prop. + +pred stop. + +stop :- halt. + +% -- Evaluation -- + +% [calc Expr Out] unifies Out with the value of Expr. It can be used in +% tandem with spilling, eg [f {calc (N + 1)}] +external pred calc i:A, o:A. + +pred (is) o:A, i:A. + +X is Y :- calc Y X. + +type (-) A -> A -> A. + +type (i-) int -> int -> int. + +type (r-) float -> float -> float. + +type (+) int -> int -> int. + +type (+) float -> float -> float. + +type (i+) int -> int -> int. + +type (r+) float -> float -> float. + +type (*) int -> int -> int. + +type (*) float -> float -> float. + +type (/) float -> float -> float. + +type (mod) int -> int -> int. + +type (div) int -> int -> int. + +type (^) string -> string -> string. + +type (~) int -> int. + +type (~) float -> float. + +type (i~) int -> int. + +type (r~) float -> float. + +type abs int -> int. + +type abs float -> float. + +type iabs int -> int. + +type rabs float -> float. + +type max int -> int -> int. + +type max float -> float -> float. + +type min int -> int -> int. + +type min float -> float -> float. + +type sqrt float -> float. + +type sin float -> float. + +type cos float -> float. + +type arctan float -> float. + +type ln float -> float. + +type int_to_real int -> float. + +type floor float -> int. + +type ceil float -> int. + +type truncate float -> int. + +type size string -> int. + +type chr int -> string. + +type rhc string -> int. + +type string_to_int string -> int. + +type int_to_string int -> string. + +type substring string -> int -> int -> string. + +type real_to_string float -> string. + +% -- Arithmetic tests -- + +% [lt_ X Y] checks if X < Y. Works for string, int and float +external pred lt_ i:A, i:A. + +% [gt_ X Y] checks if X > Y. Works for string, int and float +external pred gt_ i:A, i:A. + +% [le_ X Y] checks if X =< Y. Works for string, int and float +external pred le_ i:A, i:A. + +% [ge_ X Y] checks if X >= Y. Works for string, int and float +external pred ge_ i:A, i:A. + +type (<), (>), (=<), (>=) A -> A -> prop. + +X > Y :- gt_ X Y. + +X < Y :- lt_ X Y. + +X =< Y :- le_ X Y. + +X >= Y :- ge_ X Y. + +type (i<), (i>), (i=<), (i>=) int -> int -> prop. + +X i< Y :- lt_ X Y. + +X i> Y :- gt_ X Y. + +X i=< Y :- le_ X Y. + +X i>= Y :- ge_ X Y. + +type (r<), (r>), (r=<), (r>=) float -> float -> prop. + +X r< Y :- lt_ X Y. + +X r> Y :- gt_ X Y. + +X r=< Y :- le_ X Y. + +X r>= Y :- ge_ X Y. + +type (s<), (s>), (s=<), (s>=) string -> string -> prop. + +X s< Y :- lt_ X Y. + +X s> Y :- gt_ X Y. + +X s=< Y :- le_ X Y. + +X s>= Y :- ge_ X Y. + +% -- Standard data types (supported in the FFI) -- + +kind list type -> type. + +type (::) X -> list X -> list X. + +type ([]) list X. + +% Boolean values: tt and ff since true and false are predicates +kind bool type. +type tt bool. +type ff bool. + +% Pair: the constructor is pr, since ',' is for conjunction +kind pair type -> type -> type. +type pr A -> B -> pair A B. + +pred fst i:pair A B, o:A. + +fst (pr A _) A. + +pred snd i:pair A B, o:B. + +snd (pr _ B) B. + +% The option type (aka Maybe) +kind option type -> type. +type none option A. +type some A -> option A. + +% Result of a comparison +kind cmp type. +type eq cmp. +type lt cmp. +type gt cmp. + +% Used in builtin variants that return Coq's error rather than failing +kind diagnostic type. +type ok diagnostic. % Success +type error string -> diagnostic. % Failure + +% == I/O builtins ===================================== + +% -- I/O -- + +typeabbrev in_stream (ctype "in_stream"). + +type std_in in_stream. + +typeabbrev out_stream (ctype "out_stream"). + +type std_out out_stream. +type std_err out_stream. + +% [open_in FileName InStream] opens FileName for input +external pred open_in i:string, o:in_stream. + +% [open_out FileName OutStream] opens FileName for output +external pred open_out i:string, o:out_stream. + +% [open_append FileName OutStream] opens FileName for output in append mode +external pred open_append i:string, o:out_stream. + +% [close_in InStream] closes input stream InStream +external pred close_in i:in_stream. + +% [close_out OutStream] closes output stream OutStream +external pred close_out i:out_stream. + +% [output OutStream Data] writes Data to OutStream +external pred output i:out_stream, i:string. + +% [flush OutStream] flush all output not yet finalized to OutStream +external pred flush i:out_stream. + +% [input InStream Bytes Data] reads Bytes from InStream +external pred input i:in_stream, i:int, o:string. + +% [input_line InStream Line] reads a full line from InStream +external pred input_line i:in_stream, o:string. + +% [eof InStream] checks if no more data can be read from InStream +external pred eof i:in_stream. + +% -- System -- + +% [gettimeofday T] sets T to the number of seconds elapsed since 1/1/1970 +external pred gettimeofday o:float. + +% [getenv VarName Value] Like Sys.getenv +external pred getenv i:string, o:option string. + +% [system Command RetVal] executes Command and sets RetVal to the exit code +external pred system i:string, o:int. + +% -- Debugging -- + +% [term_to_string T S] prints T to S +external pred term_to_string i:any, o:string. + +% == Lambda Prolog builtins ===================================== + +% -- Extra I/O -- + +% [open_string DataIn InStream] opens DataIn as an input stream +external pred open_string i:string, o:in_stream. + +% [lookahead InStream NextChar] peeks one byte from InStream +external pred lookahead i:in_stream, o:string. + +% -- Hacks -- + +% [string_to_term S T] parses a term T from S +external pred string_to_term i:string, o:any. + +% [readterm InStream T] reads T from InStream, ends with \n +external pred readterm i:in_stream, o:any. + +pred printterm i:out_stream, i:A. + +printterm S T :- term_to_string T T1, output S T1. + +pred read o:A. + +read S :- flush std_out, input_line std_in X, string_to_term X S. + +% == Elpi builtins ===================================== + +% [dprint ...] prints raw terms (debugging) +external type dprint variadic any prop. + +% [print ...] prints terms +external type print variadic any prop. + +% Deprecated, use trace.counter +pred counter i:string, o:int. +counter C N :- trace.counter C N. + +% [quote_syntax FileName QueryText QuotedProgram QuotedQuery] quotes the +% program from FileName and the QueryText. See elpi-quoted_syntax.elpi for +% the syntax tree +external pred quote_syntax i:string, i:string, o:list A, o:A. + +typeabbrev loc (ctype "Loc.t"). + + +% [loc.fields Loc File StartChar StopChar Line LineStartsAtChar] Decomposes +% a loc into its fields +external pred loc.fields i:loc, o:string, o:int, o:int, o:int, o:int. + +% == Regular Expressions ===================================== + +% [rex.match Rex Subject] checks if Subject matches Rex. Matching is based +% on OCaml's Str library +external pred rex.match i:string, i:string. + +% [rex.replace Rex Replacement Subject Out] Out is obtained by replacing all +% occurrences of Rex with Replacement in Subject. See also OCaml's +% Str.global_replace +external pred rex.replace i:string, i:string, i:string, o:string. + +% [rex.split Rex Subject Out] Out is obtained by splitting Subject at all +% occurrences of Rex. See also OCaml's Str.split +external pred rex.split i:string, i:string, o:list string. + +% Deprecated, use rex.match +pred rex_match i:string, i:string. +rex_match Rx S :- rex.match Rx S. + +% Deprecated, use rex.replace +pred rex_replace i:string, i:string, i:string, o:string. +rex_replace Rx R S O :- rex.replace Rx R S O. + +% Deprecated, use rex.split +pred rex_split i:string, i:string, o:list string. +rex_split Rx S L :- rex.split Rx S L. + +% == Elpi nonlogical builtins ===================================== + +% Opaque ML data types +kind ctyp type. +type ctype string -> ctyp. + +% [var V ...] checks if the term V is a variable. When used with tree +% arguments it relates an applied variable with its head and argument list. +external type var any -> variadic any prop. + +% [prune V L] V is pruned to L (V is unified with a variable that only sees +% the list of names L) +external pred prune o:any, i:list any. + +% [distinct_names L] checks if L is a list of distinct names. If L is the +% scope of a unification variable (its arguments, as per var predicate) then +% distinct_names L checks that such variable is in the Miller pattern +% fragment (L_\lambda) +external pred distinct_names i:list any. + +% [same_var V1 V2] checks if the two terms V1 and V2 are the same variable, +% ignoring the arguments of the variables +external pred same_var i:A, i:A. + +% [same_term T1 T2] checks if the two terms T1 and T2 are syntactically +% equal (no unification). It behaves differently than same_var since it +% recursively compares the arguments of the variables +external pred same_term i:A, i:A. + + +% Infix notation for same_term +pred (==) i:A, i:A. +X == Y :- same_term X Y. + + +% [cmp_term A B Cmp] Compares A and B. Only works if A and B are ground. +external pred cmp_term i:any, i:any, o:cmp. + +% [name T ...] checks if T is a eigenvariable. When used with tree arguments +% it relates an applied name with its head and argument list. +external type name any -> variadic any prop. + +% [constant T ...] checks if T is a (global) constant. When used with tree +% arguments it relates an applied constant with its head and argument list. +external type constant any -> variadic any prop. + +external pred names % generates the list of eigenvariable + o:list any. % list of eigenvariables in order of age (young first) + +external pred occurs % checks if the atom occurs in the term + i:any, % an atom, that is a global constant or a bound name (aka eigenvariable) + i:any. % a term + +% [closed_term T] unify T with a variable that has no eigenvariables in +% scope +external pred closed_term o:any. + +% [ground_term T] Checks if T contains unification variables +external pred ground_term i:any. + +% [is_cdata T Ctype] checks if T is primitive of type Ctype, eg (ctype +% "int") +external pred is_cdata i:any, o:ctyp. + +pred primitive? i:A, i:string. + +primitive? X S :- is_cdata X (ctype S). + +% [new_int N] unifies N with a different int every time it is called. Values +% of N are guaranteed to be incresing. +external pred new_int o:int. + +% [findall_solution P L] finds all the solved instances of P and puts them +% in L +% in the order in which they are found. Instances can contain +% eigenvariables +% and unification variables. + +external pred findall_solutions i:prop, o:list prop. + +% Holds data across bracktracking; can only contain closed terms +typeabbrev safe (ctype "safe"). + + +% [new_safe Safe] creates a safe: a store that persists across backtracking +external pred new_safe o:safe. + +% [stash_in_safe Safe Data] stores Data in the Safe +external pred stash_in_safe i:safe, i:A. + +% [open_safe Safe Data] retrieves the Data stored in Safe +external pred open_safe i:safe, o:list A. + + +% [if C T E] picks the first success of C then runs T (never E). +% if C has no success it runs E. +pred if i:prop, i:prop, i:prop. +if B T _ :- B, !, T. +if _ _ E :- E. + +% [if2 C1 B1 C2 B2 E] like if but with 2 then branches (and one else branch). +pred if2 i:prop, i:prop, i:prop, i:prop, i:prop. +if2 G1 P1 _ _ _ :- G1, !, P1. +if2 _ _ G2 P2 _ :- G2, !, P2. +if2 _ _ _ _ E :- !, E. + +% [random.init Seed] Initialize OCaml's PRNG with the given Seed +external pred random.init i:int. + +% [random.self_init] Initialize OCaml's PRNG with some seed +external pred random.self_init . + +% [random.int Bound N] unifies N with a random int between 0 and Bound +% (excluded) +external pred random.int i:int, o:int. + +#line 0 "builtin_stdlib.elpi" +% == stdlib ======================================================= + +% Conventions: +% - all predicates declare a mode with some input arguments, unless... +% - predicates whose name ends with R are relations (work in any direction, +% that is all arguments are in output mode) +% - predicates whose name ends with ! do contain a cut and generate only the +% first result +% - all errors given by this library end up calling fatal-error[-w-data], +% override it in order to handle them differently +% - all debug prints by this library end up calling debug-print, override it +% in order to handle them differently + +namespace std { + +pred fatal-error i:string. +:name "default-fatal-error" +fatal-error Msg :- halt Msg. + +pred fatal-error-w-data i:string, i:A. +:name "default-fatal-error-w-data" +fatal-error-w-data Msg Data :- halt Msg ":" Data. + +pred debug-print i:string, i:A. +:name "default-debug-print" +debug-print Msg Data :- print Msg Data. + +% -- Errors, Debugging, Hacks -- + +pred ignore-failure! i:prop. +ignore-failure! P :- P, !. +ignore-failure! _. + +% [assert! C M] takes the first success of C or fails with message M +pred assert! i:prop, i:string. +assert! Cond Msg :- (Cond ; fatal-error-w-data Msg Cond), !. + +% [assert-ok! C M] like assert! but the last argument of the predicate must +% be a diagnostic that is printed after M in case it is not ok +pred assert-ok! i:(diagnostic -> prop), i:string. +assert-ok! Cond Msg :- Cond Diagnostic, !, (Diagnostic = ok ; Diagnostic = error S, fatal-error-w-data Msg S), !. +assert-ok! _ Msg :- fatal-error-w-data Msg "no diagnostic returned". + +% [spy P] traces the call to P, printing all success and the final failure +pred spy i:prop. +spy P :- trace.counter "run" NR, if (not(NR = 0)) (debug-print "run=" NR) true, + debug-print "----<<---- enter: " P, + P, + debug-print "---->>---- exit: " P. +spy P :- debug-print "---->>---- fail: " P, fail. + +% [spy! P] traces the first call to P without leaving a choice point +pred spy! i:prop. +spy! P :- trace.counter "run" NR, if (not(NR = 0)) (debug-print "run=" NR) true, + debug-print "----<<---- enter: " P, + P, + debug-print "---->>---- exit: " P, !. +spy! P :- debug-print "---->>---- fail: " P, fail. + +% to silence the type checker +pred unsafe-cast o:A, o:B. +unsafe-cast X X. + +% -- List processing -- + +pred length i:list A, o:int. +length [_|L] N :- length L N1, N is N1 + 1. +length [] 0. + +pred rev i:list A, o:list A. +rev L RL :- rev.aux L [] RL. +rev.aux [X|XS] ACC R :- rev.aux XS [X|ACC] R. +rev.aux [] L L. + +pred last i:list A, o:A. +last [] _ :- fatal-error "last on empty list". +last [X] X :- !. +last [_|XS] R :- last XS R. + +pred append i:list A, i:list A, o:list A. +append [X|XS] L [X|L1] :- append XS L L1 . +append [] L L . + +pred appendR o:list A, o:list A, o:list A. +appendR [X|XS] L [X|L1] :- appendR XS L L1 . +appendR [] L L . + +pred take i:int, i:list A, o:list A. +take 0 _ [] :- !. +take N [X|XS] [X|L] :- !, N1 is N - 1, take N1 XS L. +take _ _ _ :- fatal-error "take run out of list items". + +pred take-last i:int, i:list A, o:list A. +take-last N L R :- + length L M, + D is M - N, + drop D L R. + +pred drop i:int, i:list A, o:list A. +drop 0 L L :- !. +drop N [_|XS] L :- !, N1 is N - 1, drop N1 XS L. +drop _ _ _ :- fatal-error "drop run out of list items". + +pred drop-last i:int, i:list A, o:list A. +drop-last N L R :- + length L M, I is M - N, take I L R. + +pred split-at i:int, i:list A, o:list A, o:list A. +split-at 0 L [] L :- !. +split-at N [X|XS] [X|LN] LM :- !, N1 is N - 1, split-at N1 XS LN LM. +split-at _ _ _ _ :- fatal-error "split-at run out of list items". + +pred fold i:list B, i:A, i:(B -> A -> A -> prop), o:A. +fold [] A _ A. +fold [X|XS] A F R :- F X A A1, fold XS A1 F R. + +pred fold2 i:list C, i:list B, i:A, i:(C -> B -> A -> A -> prop), o:A. +fold2 [] [_|_] _ _ _ :- fatal-error "fold2 on lists of different length". +fold2 [_|_] [] _ _ _ :- fatal-error "fold2 on lists of different length". +fold2 [] [] A _ A. +fold2 [X|XS] [Y|YS] A F R :- F X Y A A1, fold2 XS YS A1 F R. + +pred map i:list A, i:(A -> B -> prop), o:list B. +map [] _ []. +map [X|XS] F [Y|YS] :- F X Y, map XS F YS. + +pred map-i i:list A, i:(int -> A -> B -> prop), o:list B. +map-i L F R :- map-i.aux L 0 F R. +map-i.aux [] _ _ []. +map-i.aux [X|XS] N F [Y|YS] :- F N X Y, M is N + 1, map-i.aux XS M F YS. + +pred map-filter i:list A, i:(A -> B -> prop), o:list B. +map-filter [] _ []. +map-filter [X|XS] F [Y|YS] :- F X Y, !, map-filter XS F YS. +map-filter [_|XS] F YS :- map-filter XS F YS. + +:index(1 1) +pred map2 i:list A, i:list B, i:(A -> B -> C -> prop), o:list C. +map2 [] [_|_] _ _ :- fatal-error "map2 on lists of different length". +map2 [_|_] [] _ _ :- fatal-error "map2 on lists of different length". +map2 [] [] _ []. +map2 [X|XS] [Y|YS] F [Z|ZS] :- F X Y Z, map2 XS YS F ZS. + +pred map2-filter i:list A, i:list B, i:(A -> B -> C -> prop), o:list C. +map2-filter [] [_|_] _ _ :- fatal-error "map2-filter on lists of different length". +map2-filter [_|_] [] _ _ :- fatal-error "map2-filter on lists of different length". +map2-filter [] [] _ []. +map2-filter [X|XS] [Y|YS] F [Z|ZS] :- F X Y Z, !, map2-filter XS YS F ZS. +map2-filter [_|XS] [_|YS] F ZS :- map2-filter XS YS F ZS. + +pred map-ok i:list A, i:(A -> B -> diagnostic -> prop), o:list A, o:diagnostic. +map-ok [X|L] P [Y|YS] S :- P X Y S0, if (S0 = ok) (map-ok L P YS S) (S = S0). +map-ok [] _ [] ok. + +pred fold-map i:list A, i:B, i:(A -> B -> C -> B -> prop), o:list C, o:B. +fold-map [] A _ [] A. +fold-map [X|XS] A F [Y|YS] A2 :- F X A Y A1, fold-map XS A1 F YS A2. + +pred omap i:option A, i:(A -> B -> prop), o:option B. +omap none _ none. +omap (some X) F (some Y) :- F X Y. + +% [nth N L X] picks in X the N-th element of L (L must be of length > N) +pred nth i:int, i:list A, o:A. +nth 0 [X|_ ] R :- !, X = R. +nth N [_|XS] R :- N > 0, !, N1 is N - 1, nth N1 XS R. +nth N _ _ :- N < 0, !, fatal-error "nth got a negative index". +nth _ _ _ :- fatal-error "nth run out of list items". + +% [lookup L K V] sees L as a map from K to V +pred lookup i:list (pair A B), i:A, o:B. +lookup [pr X Y|_] X Y. +lookup [_|LS] X Y :- lookup LS X Y. + +% [lookup! L K V] sees L as a map from K to V, stops at the first binding +pred lookup! i:list (pair A B), i:A, o:B. +lookup! [pr X Y|_] X Y :- !. +lookup! [_|LS] X Y :- lookup! LS X Y. + +% [mem! L X] succeeds once if X occurs inside L +pred mem! i:list A, o:A. +mem! [X|_] X :- !. +mem! [_|L] X :- mem! L X. + +% [mem L X] succeeds every time if X occurs inside L +pred mem i:list A, o:A. +mem [X|_] X. +mem [_|L] X :- mem L X. + +pred exists i:list A, i:(A -> prop). +exists [X|_] P :- P X. +exists [_|L] P :- exists L P. + +pred exists2 i:list A, i:list B, i:(A -> B -> prop). +exists2 [] [_|_] _ :- fatal-error "exists2 on lists of different length". +exists2 [_|_] [] _ :- fatal-error "exists2 on lists of different length". +exists2 [X|_] [Y|_] P :- P X Y. +exists2 [_|L] [_|M] P :- exists2 L M P. + +pred forall i:list A, i:(A -> prop). +forall [] _. +forall [X|L] P :- P X, forall L P. + +pred forall-ok i:list A, i:(A -> diagnostic -> prop), o:diagnostic. +forall-ok [X|L] P S :- P X S0, if (S0 = ok) (forall-ok L P S) (S = S0). +forall-ok [] _ ok. + +pred forall2 i:list A, i:list B, i:(A -> B -> prop). +forall2 [] [_|_] _ :- fatal-error "forall2 on lists of different length". +forall2 [_|_] [] _ :- fatal-error "forall2 on lists of different length". +forall2 [X|XS] [Y|YS] P :- P X Y, forall2 XS YS P. +forall2 [] [] _. + +pred filter i:list A, i:(A -> prop), o:list A. +filter [] _ []. +filter [X|L] P R :- if (P X) (R = X :: L1) (R = L1), filter L P L1. + +pred zip i:list A, i:list B, o:list (pair A B). +zip [_|_] [] _ :- fatal-error "zip on lists of different length". +zip [] [_|_] _ :- fatal-error "zip on lists of different length". +zip [X|LX] [Y|LY] [pr X Y|LR] :- zip LX LY LR. +zip [] [] []. + +pred unzip i:list (pair A B), o:list A, o:list B. +unzip [] [] []. +unzip [pr X Y|L] [X|LX] [Y|LY] :- unzip L LX LY. + +pred flatten i:list (list A), o:list A. +flatten [X|LS] R :- flatten LS LS', append X LS' R. +flatten [] []. + +pred null i:list A. +null []. + +pred iota i:int, o:list int. +iota N L :- iota.aux 0 N L. +iota.aux X X [] :- !. +iota.aux N X [N|R] :- M is N + 1, iota.aux M X R. + +% [intersperse X L R] R is [L0, X, ..., X, LN] +:index(_ 1) +pred intersperse i:A, i:list A, o:list A. +intersperse _ [] []. +intersperse _ [X] [X] :- !. +intersperse Sep [X|XS] [X,Sep|YS] :- intersperse Sep XS YS. + +% -- Misc -- + +pred flip i:(A -> B -> prop), i:B, i:A. +flip P X Y :- P Y X. + +pred time i:prop, o:float. +time P T :- gettimeofday Before, P, gettimeofday After, T is After - Before. + +pred do! i:list prop. +do! []. +do! [P|PS] :- P, !, do! PS. + +:index(_ 1) +pred do-ok! o:diagnostic, i:list (diagnostic -> prop). +do-ok! ok []. +do-ok! S [P|PS] :- P S0, !, if (S0 = ok) (do-ok! S PS) (S = S0). + +pred lift-ok i:prop, i:string, o:diagnostic. +lift-ok P Msg R :- (P, R = ok; R = error Msg). + +pred spy-do! i:list prop. +spy-do! L :- map L (x\y\y = spy x) L1, do! L1. + +pred while-ok-do! i:diagnostic, i:list (diagnostic -> prop), o:diagnostic. +while-ok-do! (error _ as E) _ E. +while-ok-do! ok [] ok. +while-ok-do! ok [P|PS] R :- P C, !, while-ok-do! C PS R. + +pred any->string i:A, o:string. +any->string X Y :- term_to_string X Y. + +pred max i:A, i:A, o:A. +max N M N :- N >= M, !. +max _ M M. + +% [findall P L] L is the list [P1,P2,P3..] where each Pi is a solution to P. +pred findall i:prop, o:list prop. +findall P L :- findall_solutions P L. + +} + +% [std.string.concat Separator Items Result] concatenates Items +% interspersing Separator +external pred std.string.concat i:string, i:list string, o:string. + +% CAVEAT: the type parameter of std.string.map must be a closed term + +kind std.string.map type -> type. + +% [std.string.map.empty M] The empty map +external pred std.string.map.empty o:std.string.map A. + +% [std.string.map.mem S M] Checks if S is bound in M +external pred std.string.map.mem i:string, i:std.string.map A. + +% [std.string.map.add S V M M1] M1 is M where V is bound to S +external pred std.string.map.add i:string, i:A, i:std.string.map A, + o:std.string.map A. + +% [std.string.map.remove S M M1] M1 is M where S is unbound +external pred std.string.map.remove i:string, i:std.string.map A, + o:std.string.map A. + +% [std.string.map.find S M V] V is the binding of S in M +external pred std.string.map.find i:string, i:std.string.map A, o:A. + +% [std.string.map.bindings M L] L is M transformed into an associative list +external pred std.string.map.bindings i:std.string.map A, + o:list (pair string A). + +% CAVEAT: the type parameter of std.int.map must be a closed term + +kind std.int.map type -> type. + +% [std.int.map.empty M] The empty map +external pred std.int.map.empty o:std.int.map A. + +% [std.int.map.mem S M] Checks if S is bound in M +external pred std.int.map.mem i:int, i:std.int.map A. + +% [std.int.map.add S V M M1] M1 is M where V is bound to S +external pred std.int.map.add i:int, i:A, i:std.int.map A, + o:std.int.map A. + +% [std.int.map.remove S M M1] M1 is M where S is unbound +external pred std.int.map.remove i:int, i:std.int.map A, o:std.int.map A. + +% [std.int.map.find S M V] V is the binding of S in M +external pred std.int.map.find i:int, i:std.int.map A, o:A. + +% [std.int.map.bindings M L] L is M transformed into an associative list +external pred std.int.map.bindings i:std.int.map A, o:list (pair int A). + +% CAVEAT: the type parameter of std.loc.map must be a closed term + +kind std.loc.map type -> type. + +% [std.loc.map.empty M] The empty map +external pred std.loc.map.empty o:std.loc.map A. + +% [std.loc.map.mem S M] Checks if S is bound in M +external pred std.loc.map.mem i:loc, i:std.loc.map A. + +% [std.loc.map.add S V M M1] M1 is M where V is bound to S +external pred std.loc.map.add i:loc, i:A, i:std.loc.map A, + o:std.loc.map A. + +% [std.loc.map.remove S M M1] M1 is M where S is unbound +external pred std.loc.map.remove i:loc, i:std.loc.map A, o:std.loc.map A. + +% [std.loc.map.find S M V] V is the binding of S in M +external pred std.loc.map.find i:loc, i:std.loc.map A, o:A. + +% [std.loc.map.bindings M L] L is M transformed into an associative list +external pred std.loc.map.bindings i:std.loc.map A, o:list (pair loc A). + +kind std.string.set type. + +% [std.string.set.empty A] The empty set +external pred std.string.set.empty o:std.string.set. + +% [std.string.set.mem Elem A] Checks if Elem is in a +external pred std.string.set.mem i:string, i:std.string.set. + +% [std.string.set.add Elem A B] B is A union {Elem} +external pred std.string.set.add i:string, i:std.string.set, + o:std.string.set. + +% [std.string.set.remove Elem A B] B is A \ {Elem} +external pred std.string.set.remove i:string, i:std.string.set, + o:std.string.set. + +% [std.string.set.union A B X] X is A union B +external pred std.string.set.union i:std.string.set, i:std.string.set, + o:std.string.set. + +% [std.string.set.inter A B X] X is A intersection B +external pred std.string.set.inter i:std.string.set, i:std.string.set, + o:std.string.set. + +% [std.string.set.diff A B X] X is A \ B +external pred std.string.set.diff i:std.string.set, i:std.string.set, + o:std.string.set. + +% [std.string.set.equal A B] tests A and B for equality +external pred std.string.set.equal i:std.string.set, i:std.string.set. + +% [std.string.set.subset A B] tests if A is a subset of B +external pred std.string.set.subset i:std.string.set, i:std.string.set. + +% [std.string.set.elements M L] L is M transformed into list +external pred std.string.set.elements i:std.string.set, o:list string. + +% [std.string.set.cardinal M N] N is the number of elements of M +external pred std.string.set.cardinal i:std.string.set, o:int. + +kind std.int.set type. + +% [std.int.set.empty A] The empty set +external pred std.int.set.empty o:std.int.set. + +% [std.int.set.mem Elem A] Checks if Elem is in a +external pred std.int.set.mem i:int, i:std.int.set. + +% [std.int.set.add Elem A B] B is A union {Elem} +external pred std.int.set.add i:int, i:std.int.set, o:std.int.set. + +% [std.int.set.remove Elem A B] B is A \ {Elem} +external pred std.int.set.remove i:int, i:std.int.set, o:std.int.set. + +% [std.int.set.union A B X] X is A union B +external pred std.int.set.union i:std.int.set, i:std.int.set, + o:std.int.set. + +% [std.int.set.inter A B X] X is A intersection B +external pred std.int.set.inter i:std.int.set, i:std.int.set, + o:std.int.set. + +% [std.int.set.diff A B X] X is A \ B +external pred std.int.set.diff i:std.int.set, i:std.int.set, + o:std.int.set. + +% [std.int.set.equal A B] tests A and B for equality +external pred std.int.set.equal i:std.int.set, i:std.int.set. + +% [std.int.set.subset A B] tests if A is a subset of B +external pred std.int.set.subset i:std.int.set, i:std.int.set. + +% [std.int.set.elements M L] L is M transformed into list +external pred std.int.set.elements i:std.int.set, o:list int. + +% [std.int.set.cardinal M N] N is the number of elements of M +external pred std.int.set.cardinal i:std.int.set, o:int. + +kind std.loc.set type. + +% [std.loc.set.empty A] The empty set +external pred std.loc.set.empty o:std.loc.set. + +% [std.loc.set.mem Elem A] Checks if Elem is in a +external pred std.loc.set.mem i:loc, i:std.loc.set. + +% [std.loc.set.add Elem A B] B is A union {Elem} +external pred std.loc.set.add i:loc, i:std.loc.set, o:std.loc.set. + +% [std.loc.set.remove Elem A B] B is A \ {Elem} +external pred std.loc.set.remove i:loc, i:std.loc.set, o:std.loc.set. + +% [std.loc.set.union A B X] X is A union B +external pred std.loc.set.union i:std.loc.set, i:std.loc.set, + o:std.loc.set. + +% [std.loc.set.inter A B X] X is A intersection B +external pred std.loc.set.inter i:std.loc.set, i:std.loc.set, + o:std.loc.set. + +% [std.loc.set.diff A B X] X is A \ B +external pred std.loc.set.diff i:std.loc.set, i:std.loc.set, + o:std.loc.set. + +% [std.loc.set.equal A B] tests A and B for equality +external pred std.loc.set.equal i:std.loc.set, i:std.loc.set. + +% [std.loc.set.subset A B] tests if A is a subset of B +external pred std.loc.set.subset i:std.loc.set, i:std.loc.set. + +% [std.loc.set.elements M L] L is M transformed into list +external pred std.loc.set.elements i:std.loc.set, o:list loc. + +% [std.loc.set.cardinal M N] N is the number of elements of M +external pred std.loc.set.cardinal i:std.loc.set, o:int. + +#line 0 "builtin_map.elpi" +kind std.map type -> type -> type. +type std.map std.map.private.map K V -> (K -> K -> cmp -> prop) -> std.map K V. + +namespace std.map { + +% [make Eq Ltn M] builds an empty map M where keys are compared using Eq and Ltn +pred make i:(K -> K -> cmp -> prop), o:std.map K V. +make Cmp (std.map private.empty Cmp). + +% [find K M V] looks in M for the value V associated to K +pred find i:K, i:std.map K V, o:V. +find K (std.map M Cmp) V :- private.find M Cmp K V. + +% [add K V M M1] M1 is M where K is bound to V +pred add i:K, i:V, i:std.map K V, o:std.map K V. +add K V (std.map M Cmp) (std.map M1 Cmp) :- private.add M Cmp K V M1. + +% [remove K M M1] M1 is M where K is unbound +pred remove i:K, i:std.map K V, o:std.map K V. +remove K (std.map M Cmp) (std.map M1 Cmp) :- private.remove M Cmp K M1. + +% [bindings M L] L is the key-value pairs in increasing order +pred bindings i:std.map K V, o:list (pair K V). +bindings (std.map M _) L :- private.bindings M [] L. + +namespace private { + +% Taken from OCaml's map.ml +kind map type -> type -> type. +type empty map K V. +type node map K V -> K -> V -> map K V -> int -> map K V. + +pred height i:map K V, o:int. +height empty 0. +height (node _ _ _ _ H) H. + +pred create i:map K V, i:K, i:V, i:map K V, o:map K V. +create L K V R (node L K V R H) :- H is {std.max {height L} {height R}} + 1. + +pred bal i:map K V, i:K, i:V, i:map K V, o:map K V. +bal L K V R T :- + height L HL, + height R HR, + HL2 is HL + 2, + HR2 is HR + 2, + bal.aux HL HR HL2 HR2 L K V R T. + +bal.aux HL _ _ HR2 (node LL LV LD LR _) X D R T :- + HL > HR2, {height LL} >= {height LR}, !, + create LL LV LD {create LR X D R} T. +bal.aux HL _ _ HR2 (node LL LV LD (node LRL LRV LRD LRR _) _) X D R T :- + HL > HR2, !, + create {create LL LV LD LRL} LRV LRD {create LRR X D R} T. +bal.aux _ HR HL2 _ L X D (node RL RV RD RR _) T :- + HR > HL2, {height RR} >= {height RL}, !, + create {create L X D RL} RV RD RR T. +bal.aux _ HR HL2 _ L X D (node (node RLL RLV RLD RLR _) RV RD RR _) T :- + HR > HL2, !, + create {create L X D RLL} RLV RLD {create RLR RV RD RR} T. +bal.aux _ _ _ _ L K V R T :- create L K V R T. + +pred add i:map K V, i:(K -> K -> cmp -> prop), i:K, i:V, o:map K V. +add empty _ K V T :- create empty K V empty T. +add (node _ X _ _ _ as M) Cmp X1 XD M1 :- Cmp X1 X E, add.aux E M Cmp X1 XD M1. +add.aux eq (node L _ _ R H) _ X XD T :- T = node L X XD R H. +add.aux lt (node L V D R _) Cmp X XD T :- bal {add L Cmp X XD} V D R T. +add.aux gt (node L V D R _) Cmp X XD T :- bal L V D {add R Cmp X XD} T. + +pred find i:map K V, i:(K -> K -> cmp -> prop), i:K, o:V. +find (node L K1 V1 R _) Cmp K V :- Cmp K K1 E, find.aux E Cmp L R V1 K V. +find.aux eq _ _ _ V _ V. +find.aux lt Cmp L _ _ K V :- find L Cmp K V. +find.aux gt Cmp _ R _ K V :- find R Cmp K V. + +pred remove-min-binding i:map K V, o:map K V. +remove-min-binding (node empty _ _ R _) R :- !. +remove-min-binding (node L V D R _) X :- bal {remove-min-binding L} V D R X. + +pred min-binding i:map K V, o:K, o:V. +min-binding (node empty V D _ _) V D :- !. +min-binding (node L _ _ _ _) V D :- min-binding L V D. + +pred merge i:map K V, i:map K V, o:map K V. +merge empty X X :- !. +merge X empty X :- !. +merge M1 M2 R :- + min-binding M2 X D, + bal M1 X D {remove-min-binding M2} R. + +pred remove i:map K V, i:(K -> K -> cmp -> prop), i:K, o:map K V. +remove empty _ _ empty :- !. +remove (node L V D R _) Cmp X M :- Cmp X V E, remove.aux E Cmp L R V D X M. +remove.aux eq _ L R _ _ _ M :- merge L R M. +remove.aux lt Cmp L R V D X M :- bal {remove L Cmp X} V D R M. +remove.aux gt Cmp L R V D X M :- bal L V D {remove R Cmp X} M. + +pred bindings i:map K V, i:list (pair K V), o:list (pair K V). +bindings empty X X. +bindings (node L V D R _) X X1 :- + bindings L [pr V D|{bindings R X}] X1. + + +} % std.map.private +} % std.map + + +#line 0 "builtin_set.elpi" +kind std.set type -> type. +type std.set std.set.private.set E -> (E -> E -> cmp -> prop) -> std.set E. + +namespace std.set { + +% [make Eq Ltn M] builds an empty set M where keys are compared using Eq and Ltn +pred make i:(E -> E -> cmp -> prop), o:std.set E. +make Cmp (std.set private.empty Cmp). + +% [mem E M] looks if E is in M +pred mem i:E, i:std.set E. +mem E (std.set M Cmp):- private.mem M Cmp E. + +% [add E M M1] M1 is M + {E} +pred add i:E, i:std.set E, o:std.set E. +add E (std.set M Cmp) (std.set M1 Cmp) :- private.add M Cmp E M1. + +% [remove E M M1] M1 is M - {E} +pred remove i:E, i:std.set E, o:std.set E. +remove E (std.set M Cmp) (std.set M1 Cmp) :- private.remove M Cmp E M1. + +% [cardinal S N] N is the number of elements of S +pred cardinal i:std.set E, o:int. +cardinal (std.set M _) N :- private.cardinal M N. + +pred elements i:std.set E, o:list E. +elements (std.set M _) L :- private.elements M [] L. + +namespace private { + +% Taken from OCaml's set.ml +kind set type -> type. +type empty set E. +type node set E -> E -> set E -> int -> set E. + +pred height i:set E, o:int. +height empty 0. +height (node _ _ _ H) H. + +pred create i:set E, i:E, i:set E, o:set E. +create L E R (node L E R H) :- H is {std.max {height L} {height R}} + 1. + +pred bal i:set E, i:E, i:set E, o:set E. +bal L E R T :- + height L HL, + height R HR, + HL2 is HL + 2, + HR2 is HR + 2, + bal.aux HL HR HL2 HR2 L E R T. + +bal.aux HL _ _ HR2 (node LL LV LR _) X R T :- + HL > HR2, {height LL} >= {height LR}, !, + create LL LV {create LR X R} T. +bal.aux HL _ _ HR2 (node LL LV (node LRL LRV LRR _) _) X R T :- + HL > HR2, !, + create {create LL LV LRL} LRV {create LRR X R} T. +bal.aux _ HR HL2 _ L X (node RL RV RR _) T :- + HR > HL2, {height RR} >= {height RL}, !, + create {create L X RL} RV RR T. +bal.aux _ HR HL2 _ L X (node (node RLL RLV RLR _) RV RR _) T :- + HR > HL2, !, + create {create L X RLL} RLV {create RLR RV RR} T. +bal.aux _ _ _ _ L E R T :- create L E R T. + +pred add i:set E, i:(E -> E -> cmp -> prop), i:E, o:set E. +add empty _ E T :- create empty E empty T. +add (node L X R H) Cmp X1 S :- Cmp X1 X E, add.aux E Cmp L R X X1 H S. +add.aux eq _ L R X _ H (node L X R H). +add.aux lt Cmp L R E X _ T :- bal {add L Cmp X} E R T. +add.aux gt Cmp L R E X _ T :- bal L E {add R Cmp X} T. + +pred mem i:set E, i:(E -> E -> cmp -> prop), i:E. +mem (node L K R _) Cmp E :- Cmp E K O, mem.aux O Cmp L R E. +mem.aux eq _ _ _ _. +mem.aux lt Cmp L _ E :- mem L Cmp E. +mem.aux gt Cmp _ R E :- mem R Cmp E. + +pred remove-min-binding i:set E, o:set E. +remove-min-binding (node empty _ R _) R :- !. +remove-min-binding (node L E R _) X :- bal {remove-min-binding L} E R X. + +pred min-binding i:set E, o:E. +min-binding (node empty E _ _) E :- !. +min-binding (node L _ _ _) E :- min-binding L E. + +pred merge i:set E, i:set E, o:set E. +merge empty X X :- !. +merge X empty X :- !. +merge M1 M2 R :- + min-binding M2 X, + bal M1 X {remove-min-binding M2} R. + +pred remove i:set E, i:(E -> E -> cmp -> prop), i:E, o:set E. +remove empty _ _ empty. +remove (node L E R _) Cmp X M :- Cmp X E O, remove.aux O Cmp L R E X M. +remove.aux eq _ L R _ _ M :- merge L R M. +remove.aux lt Cmp L R E X M :- bal {remove L Cmp X} E R M. +remove.aux gt Cmp L R E X M :- bal L E {remove R Cmp X} M. + +pred cardinal i:set E, o:int. +cardinal empty 0. +cardinal (node L _ R _) N :- N is {cardinal L} + 1 + {cardinal R}. + +pred elements i:set E, i:list E, o:list E. +elements empty X X. +elements (node L E R _) Acc X :- + elements L [E|{elements R Acc}] X. + +} % std.set.private +} % std.set + + +% == Elpi runtime builtins ===================================== + +% [trace.counter Name Value] reads the Value of a trace point Name +external pred trace.counter i:string, o:int. + +% [gc.get MinorHeapSize MajorHeapIncrement SpaceOverhead Verbose MaxOverhead +% StackLimit AllocationPolicy WindowSize] Reads the current settings of the +% garbage collector. See also OCaml's Gc.control type documentation. +external pred gc.get o:int, o:int, o:int, o:int, o:int, o:int, o:int, + o:int. + +% [gc.set MinorHeapSize MajorHeapIncrement SpaceOverhead Verbose MaxOverhead +% StackLimit AllocationPolicy WindowSize] Writes the current settings of the +% garbage collector. Any parameter left unspecificed (eg _) is not changed. +% See also OCaml's Gc.control type documentation. +external pred gc.set i:int, i:int, i:int, i:int, i:int, i:int, i:int, + i:int. + +% [gc.minor] See OCaml's Gc.minor documentation. +external pred gc.minor . + +% [gc.major] See OCaml's Gc.major documentation. +external pred gc.major . + +% [gc.full] See OCaml's Gc.full_major documentation. +external pred gc.full . + +% [gc.compact] See OCaml's Gc.compact documentation. +external pred gc.compact . + +% [gc.stat MinorWords PromotedWords MajorWords MinorCollections +% MajorCollections HeapWords HeapChunks LiveWords LiveBlocks FreeWords +% FreeBlocks LargestFree Fragments Compactions TopHeapWords StackSize] See +% OCaml's Gc.stat documentation. +external pred gc.stat o:float, o:float, o:float, o:int, o:int, o:int, + o:int, o:int, o:int, o:int, o:int, o:int, o:int, + o:int, o:int, o:int. + +% [gc.quick-stat MinorWords PromotedWords MajorWords MinorCollections +% MajorCollections HeapWords HeapChunks Compactions TopHeapWords StackSize] +% See OCaml's Gc.quick_stat documentation. +external pred gc.quick-stat o:float, o:float, o:float, o:int, o:int, + o:int, o:int, o:int, o:int, o:int. + + + + diff --git a/lambdapi.opam b/lambdapi.opam index 4557e1b92..0bb6353fa 100644 --- a/lambdapi.opam +++ b/lambdapi.opam @@ -62,7 +62,7 @@ depends: [ "cmdliner" {>= "1.1.0"} "stdlib-shims" {>= "0.1.0"} "odoc" {with-doc} - "elpi" { >= "1.16.0" & < "1.17.0" } + "elpi" { >= "1.16.6" & < "1.17.0" } ] build: [ ["dune" "subst"] {dev} diff --git a/out/Coq__Arith__Arith__base.dk b/out/Coq__Arith__Arith__base.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Arith__Arith__base.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Arith__Between.dk b/out/Coq__Arith__Between.dk new file mode 100644 index 000000000..f47ebbcb1 --- /dev/null +++ b/out/Coq__Arith__Between.dk @@ -0,0 +1,2513 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def between : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop. + +def bet__emp : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (between P k k). + +def bet__S : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (between P k l) -> + __0 : cupicef.Term cupicef.prop (P l) -> + cupicef.Term cupicef.prop (between P k (Coq__Init__Datatypes.S l)). + +[ ] cupicef.code_guarded? _ (bet__emp _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (bet__S _ _ _ _ _) --> cupicef.guarded. + +def match____between : + s : cupicef.Sort -> + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + P0 : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (between P k __) -> + cupicef.Univ s) -> + case__bet__emp : cupicef.Term s (P0 k (bet__emp P k)) -> + case__bet__S : + (l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (between P k l) -> + __0 : cupicef.Term cupicef.prop (P l) -> + cupicef.Term s (P0 (Coq__Init__Datatypes.S l) (bet__S P k l __ __0))) -> + __ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + x : cupicef.Term cupicef.prop (between P k __) -> + cupicef.Term s (P0 __ x). + +[ s, P, k, P0, case__bet__emp, case__bet__S] + match____between s P k P0 case__bet__emp case__bet__S {k} (bet__emp P k) + --> + case__bet__emp. + +[ s, P, k, P0, case__bet__emp, case__bet__S, l, __, __0] + match____between s P k P0 case__bet__emp case__bet__S + {Coq__Init__Datatypes.S l} (bet__S P k l __ __0) + --> + case__bet__S l __ __0. + +[ s, P, k, P0, s'] + match____between s P k + (__ => x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P0 __ x))) + --> + match____between s' P k (__ => x => P0 __ x). + +def between__ind : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + P0 : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + f : cupicef.Term cupicef.prop (P0 k) -> + f0 : + (l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (between P k l) -> + __0 : cupicef.Term cupicef.prop (P0 l) -> + __1 : cupicef.Term cupicef.prop (P l) -> + cupicef.Term cupicef.prop (P0 (Coq__Init__Datatypes.S l))) -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + b : cupicef.Term cupicef.prop (between P k n) -> + cupicef.Term cupicef.prop (P0 n) + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + P0 : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + f : cupicef.Term cupicef.prop (P0 k) => + f0 : + (l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (between P k l) -> + __0 : cupicef.Term cupicef.prop (P0 l) -> + __1 : cupicef.Term cupicef.prop (P l) -> + cupicef.Term cupicef.prop (P0 (Coq__Init__Datatypes.S l))) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 cupicef.prop + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (between P k n) + (b : cupicef.Term cupicef.prop (between P k n) => P0 n))))) + (c => + c + (F => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + b : cupicef.Term cupicef.prop (between P k n) => + match____between cupicef.prop P k + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + __ : cupicef.Term cupicef.prop (between P k n0) => + P0 n0) + f + (l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + b0 : cupicef.Term cupicef.prop (between P k l) => + p : cupicef.Term cupicef.prop (P l) => + f0 l b0 (F l b0) p) + n b)) + cupicef.0. + +thm bet__eq : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + l k) -> + cupicef.Term cupicef.prop (between P k l) + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + l k) => + Coq__Init__Logic.eq__ind__r + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + k + (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + between P k l0) + (bet__emp P k) l H. + +thm between__le : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (between P k l) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (between P k l) => + between__ind P k + (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Peano.le k l0) + (Coq__Init__Peano.le__n k) + (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (between P k l0) => + IHbetween : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l0) => + H1 : cupicef.Term cupicef.prop (P l0) => + Coq__Init__Peano.le__S k l0 IHbetween) + l H. + +thm between__Sk__l : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (between P k l) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) l) -> + cupicef.Term cupicef.prop (between P (Coq__Init__Datatypes.S k) l) + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (between P k l) => + between__ind P k + (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) l0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) l0) => + between P (Coq__Init__Datatypes.S k) l0)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) k) => + bet__eq P (Coq__Init__Datatypes.S k) k + (Coq__Arith__PeanoNat.Nat__le__antisymm k (Coq__Init__Datatypes.S k) + (Coq__Arith__PeanoNat.Nat__le__succ__diag__r k) H0)) + (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (between P k l0) => + match____between cupicef.prop P k + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + __ : cupicef.Term cupicef.prop (between P k n) => + cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) n) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) n) => + between P (Coq__Init__Datatypes.S k) n)) + (__0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) n) -> + cupicef.Term cupicef.prop + (between P (Coq__Init__Datatypes.S k) n)) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (P n) + (__1 : cupicef.Term cupicef.prop (P n) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) + (Coq__Init__Datatypes.S n)) + (__2 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) + (Coq__Init__Datatypes.S n)) => + between P (Coq__Init__Datatypes.S k) + (Coq__Init__Datatypes.S n))))) + (IHbetween : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) k) -> + cupicef.Term cupicef.prop + (between P (Coq__Init__Datatypes.S k) k)) => + H1 : cupicef.Term cupicef.prop (P k) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) + (Coq__Init__Datatypes.S k)) => + bet__emp P (Coq__Init__Datatypes.S k)) + (l1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H1 : cupicef.Term cupicef.prop (between P k l1) => + H2 : cupicef.Term cupicef.prop (P l1) => + IHbetween : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) + (Coq__Init__Datatypes.S l1)) -> + cupicef.Term cupicef.prop + (between P (Coq__Init__Datatypes.S k) + (Coq__Init__Datatypes.S l1))) => + H3 : cupicef.Term cupicef.prop (P (Coq__Init__Datatypes.S l1)) => + H4 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) + (Coq__Init__Datatypes.S (Coq__Init__Datatypes.S l1))) => + bet__S P (Coq__Init__Datatypes.S k) (Coq__Init__Datatypes.S l1) + (IHbetween (Coq__Arith__Le.le__n__S k l1 (between__le P k l1 H1))) + H3) + l0 H0) + l H. + +thm between__restr : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) -> + __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le l m) -> + __1 : cupicef.Term cupicef.prop (between P k m) -> + cupicef.Term cupicef.prop (between P l m) + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) => + Coq__Init__Peano.le__ind k + (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (Coq__Init__Peano.le l0 m) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le l0 m) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (between P k m) + (__0 : cupicef.Term cupicef.prop (between P k m) => between P l0 m))) + (H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k m) => + H1 : cupicef.Term cupicef.prop (between P k m) => + H1) + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k m0) => + IHle : + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le m0 m) -> + __0 : cupicef.Term cupicef.prop (between P k m) -> + cupicef.Term cupicef.prop (between P m0 m)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S m0) m) => + H2 : cupicef.Term cupicef.prop (between P k m) => + between__Sk__l P m0 m + (IHle (Coq__Arith__PeanoNat.Nat__lt__le__incl m0 m H1) H2) H1) + l H. + +def exists__between : + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop. + +def exists__S : + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (exists__between Q k l) -> + cupicef.Term cupicef.prop (exists__between Q k (Coq__Init__Datatypes.S l)). + +def exists__le : + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) -> + __0 : cupicef.Term cupicef.prop (Q l) -> + cupicef.Term cupicef.prop (exists__between Q k (Coq__Init__Datatypes.S l)). + +[ ] cupicef.code_guarded? _ (exists__S _ _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (exists__le _ _ _ _ _) --> cupicef.guarded. + +def match____exists__between : + s : cupicef.Sort -> + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (exists__between Q k __) -> + cupicef.Univ s) -> + case__exists__S : + (l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (exists__between Q k l) -> + cupicef.Term s (P (Coq__Init__Datatypes.S l) (exists__S Q k l __))) -> + case__exists__le : + (l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) -> + __0 : cupicef.Term cupicef.prop (Q l) -> + cupicef.Term s (P (Coq__Init__Datatypes.S l) (exists__le Q k l __ __0))) -> + __ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + x : cupicef.Term cupicef.prop (exists__between Q k __) -> + cupicef.Term s (P __ x). + +[ s, Q, k, P, case__exists__S, case__exists__le, l, __] + match____exists__between s Q k P case__exists__S case__exists__le + {Coq__Init__Datatypes.S l} (exists__S Q k l __) + --> + case__exists__S l __. + +[ s, Q, k, P, case__exists__S, case__exists__le, l, __, __0] + match____exists__between s Q k P case__exists__S case__exists__le + {Coq__Init__Datatypes.S l} (exists__le Q k l __ __0) + --> + case__exists__le l __ __0. + +[ s, Q, k, P, s'] + match____exists__between s Q k + (__ => x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P __ x))) + --> + match____exists__between s' Q k (__ => x => P __ x). + +def exists__between__ind : + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + f : + (l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (exists__between Q k l) -> + __0 : cupicef.Term cupicef.prop (P l) -> + cupicef.Term cupicef.prop (P (Coq__Init__Datatypes.S l))) -> + f0 : + (l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) -> + __0 : cupicef.Term cupicef.prop (Q l) -> + cupicef.Term cupicef.prop (P (Coq__Init__Datatypes.S l))) -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + e : cupicef.Term cupicef.prop (exists__between Q k n) -> + cupicef.Term cupicef.prop (P n) + +:= Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + f : + (l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (exists__between Q k l) -> + __0 : cupicef.Term cupicef.prop (P l) -> + cupicef.Term cupicef.prop (P (Coq__Init__Datatypes.S l))) => + f0 : + (l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) -> + __0 : cupicef.Term cupicef.prop (Q l) -> + cupicef.Term cupicef.prop (P (Coq__Init__Datatypes.S l))) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 cupicef.prop + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (exists__between Q k n) + (e : cupicef.Term cupicef.prop (exists__between Q k n) => + P n))))) + (c => + c + (F => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + e : cupicef.Term cupicef.prop (exists__between Q k n) => + match____exists__between cupicef.prop Q k + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + __ : cupicef.Term cupicef.prop (exists__between Q k n0) => + P n0) + (l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + e0 : cupicef.Term cupicef.prop (exists__between Q k l) => + f l e0 (F l e0)) + (l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) => + q : cupicef.Term cupicef.prop (Q l) => + f0 l l0 q) + n e)) + cupicef.0. + +thm exists__le__S : + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (exists__between Q k l) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) l) + +:= Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (exists__between Q k l) => + exists__between__ind Q k + (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Peano.le (Coq__Init__Datatypes.S k) l0) + (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (exists__between Q k l0) => + IHexists__between : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) l0) => + Coq__Init__Peano.le__S (Coq__Init__Datatypes.S k) l0 IHexists__between) + (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l0) => + H1 : cupicef.Term cupicef.prop (Q l0) => + Coq__Arith__Le.le__n__S k l0 H0) + l H. + +thm exists__lt : + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (exists__between Q k l) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt k l) + +:= Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + exists__le__S Q. + +thm exists__S__le : + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (exists__between Q k (Coq__Init__Datatypes.S l)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) + +:= Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (exists__between Q k (Coq__Init__Datatypes.S l)) => + Coq__Arith__Le.le__S__n k l + (exists__le__S Q k (Coq__Init__Datatypes.S l) H). + +def in__int : + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop + +:= p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and (Coq__Init__Peano.le p r) (Coq__Init__Peano.lt r q). + +thm in__int__intro : + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le p r) -> + __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt r q) -> + cupicef.Term cupicef.prop (in__int p q r) + +:= p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (Coq__Init__Peano.le p r) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt r q) => + Coq__Init__Logic.conj (Coq__Init__Peano.le p r) (Coq__Init__Peano.lt r q) + H H0. + +thm in__int__lt : + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int p q r) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt p q) + +:= p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (in__int p q r) => + Coq__Init__Logic.match____and cupicef.prop (Coq__Init__Peano.le p r) + (Coq__Init__Peano.lt r q) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Coq__Init__Peano.le p r) + (Coq__Init__Peano.lt r q)) => + Coq__Init__Peano.lt p q) + (H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le p r) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt r q) => + Coq__Arith__PeanoNat.Nat__le__lt__trans p r q H0 H1) + H. + +def let__o : + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : cupicef.Term cupicef.prop (in__int p (Coq__Init__Datatypes.S q) r) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le p r) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt r (Coq__Init__Datatypes.S q)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt r q) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + r q)) + +:= p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (in__int p (Coq__Init__Datatypes.S q) r) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le p r) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt r (Coq__Init__Datatypes.S q)) => + Coq__Arith__Lt.le__lt__or__eq r q (Coq__Arith__Lt.lt__n__Sm__le r q H1). + +thm in__int__p__Sq : + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int p (Coq__Init__Datatypes.S q) r) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (in__int p q r) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + r q)) + +:= p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (in__int p (Coq__Init__Datatypes.S q) r) => + Coq__Init__Logic.match____and cupicef.prop (Coq__Init__Peano.le p r) + (Coq__Init__Peano.lt r (Coq__Init__Datatypes.S q)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Coq__Init__Peano.le p r) + (Coq__Init__Peano.lt r (Coq__Init__Datatypes.S q))) => + Coq__Init__Logic.or (in__int p q r) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + r q)) + (H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le p r) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt r (Coq__Init__Datatypes.S q)) => + Coq__Init__Logic.match____or cupicef.prop (Coq__Init__Peano.lt r q) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + r q) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt r q) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + r q)) => + Coq__Init__Logic.or (in__int p q r) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + r q)) + (H2 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt r q) => + Coq__Init__Logic.or__introl (in__int p q r) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + r q) + (in__int__intro p q r H0 H2)) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + r q) => + Coq__Init__Logic.or__intror (in__int p q r) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + r q) + H2) + (let__o p q r H H0 H1)) + H. + +thm in__int__S : + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int p q r) -> + cupicef.Term cupicef.prop (in__int p (Coq__Init__Datatypes.S q) r) + +:= p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (in__int p q r) => + Coq__Init__Logic.match____and cupicef.prop (Coq__Init__Peano.le p r) + (Coq__Init__Peano.lt r q) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Coq__Init__Peano.le p r) + (Coq__Init__Peano.lt r q)) => + in__int p (Coq__Init__Datatypes.S q) r) + (H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le p r) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt r q) => + in__int__intro p (Coq__Init__Datatypes.S q) r H0 + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S r) + (Coq__Init__Datatypes.S q)) + (Coq__Init__Peano.lt r (Coq__Init__Datatypes.S q)) cupicef.I + (Coq__Init__Peano.le__S (Coq__Init__Datatypes.S r) q H1))) + H. + +thm in__int__Sp__q : + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int (Coq__Init__Datatypes.S p) q r) -> + cupicef.Term cupicef.prop (in__int p q r) + +:= p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (in__int (Coq__Init__Datatypes.S p) q r) => + Coq__Init__Logic.match____and cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S p) r) + (Coq__Init__Peano.lt r q) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Peano.le (Coq__Init__Datatypes.S p) r) + (Coq__Init__Peano.lt r q)) => + in__int p q r) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S p) r) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt r q) => + in__int__intro p q r (Coq__Arith__PeanoNat.Nat__lt__le__incl p r H0) H1) + H. + +def let__o0 : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : cupicef.Term cupicef.prop (between P k l) -> + l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H0 : cupicef.Term cupicef.prop (between P k l0) -> + IHbetween : + (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k l0 r) -> + cupicef.Term cupicef.prop (P r)) -> + H1 : cupicef.Term cupicef.prop (P l0) -> + r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H2 : cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S l0) r) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (in__int k l0 r) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + r l0)) + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (between P k l) => + l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (between P k l0) => + IHbetween : + (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k l0 r) -> + cupicef.Term cupicef.prop (P r)) => + H1 : cupicef.Term cupicef.prop (P l0) => + r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H2 : cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S l0) r) => + in__int__p__Sq k l0 r H2. + +thm between__in__int : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (between P k l) -> + r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __0 : cupicef.Term cupicef.prop (in__int k l r) -> + cupicef.Term cupicef.prop (P r) + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (between P k l) => + between__ind P k + (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (in__int k l0 r) + (__ : cupicef.Term cupicef.prop (in__int k l0 r) => P r))) + (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (in__int k k r) => + Coq__Init__Logic.False__ind (P r) + (Coq__Arith__PeanoNat.Nat__lt__irrefl k (in__int__lt k k r H0))) + (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (between P k l0) => + IHbetween : + (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k l0 r) -> + cupicef.Term cupicef.prop (P r)) => + H1 : cupicef.Term cupicef.prop (P l0) => + r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H2 : + cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S l0) r) => + Coq__Init__Logic.match____or cupicef.prop (in__int k l0 r) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + r l0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (in__int k l0 r) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + r l0)) => + P r) + (H3 : cupicef.Term cupicef.prop (in__int k l0 r) => IHbetween r H3) + (H3 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + r l0) => + Coq__Init__Logic.eq__ind__r + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + l0 + (r0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (in__int k (Coq__Init__Datatypes.S l0) r0) + (H4 : + cupicef.Term cupicef.prop + (in__int k (Coq__Init__Datatypes.S l0) r0) => + P r0)) + (H4 : + cupicef.Term cupicef.prop + (in__int k (Coq__Init__Datatypes.S l0) l0) => + H1) + r H3 H2) + (let__o0 P k l H l0 H0 IHbetween H1 r H2)) + l H. + +thm in__int__between : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) -> + __0 : + (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __0 : cupicef.Term cupicef.prop (in__int k l r) -> + cupicef.Term cupicef.prop (P r)) -> + cupicef.Term cupicef.prop (between P k l) + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) => + Coq__Init__Peano.le__ind k + (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.prop + (cupicef.rule + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (in__int k l0 r) + (__ : cupicef.Term cupicef.prop (in__int k l0 r) => P r))) + (__ : + (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k l0 r) -> + cupicef.Term cupicef.prop (P r)) => + between P k l0)) + (H0 : + (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k k r) -> + cupicef.Term cupicef.prop (P r)) => + bet__emp P k) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k m) => + IHle : + (__ : + (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k m r) -> + cupicef.Term cupicef.prop (P r)) -> + cupicef.Term cupicef.prop (between P k m)) => + H1 : + (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S m) r) -> + cupicef.Term cupicef.prop (P r)) => + bet__S P k m + (IHle + (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H2 : cupicef.Term cupicef.prop (in__int k m r) => + H1 r (in__int__S k m r H2))) + (H1 m + (in__int__intro k (Coq__Init__Datatypes.S m) m H0 + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S m) + (Coq__Init__Datatypes.S m)) + (Coq__Init__Peano.lt m (Coq__Init__Datatypes.S m)) cupicef.I + (Coq__Init__Peano.le__n (Coq__Init__Datatypes.S m)))))) + l H. + +thm exists__in__int : + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (exists__between Q k l) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex2 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + in__int k l m) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => Q m)) + +:= Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (exists__between Q k l) => + exists__between__ind Q k + (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.ex2 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + in__int k l0 m) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => Q m)) + (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (exists__between Q k l0) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex2 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + in__int k l0 m) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => Q m)) => + Coq__Init__Logic.match____ex2 cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + in__int k l0 m) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => Q m) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex2 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + in__int k l0 m) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => Q m)) => + Coq__Init__Logic.ex2 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + in__int k (Coq__Init__Datatypes.S l0) m) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => Q m)) + (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H2 : cupicef.Term cupicef.prop (in__int k l0 p) => + H3 : cupicef.Term cupicef.prop (Q p) => + Coq__Init__Logic.ex__intro2 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + in__int k (Coq__Init__Datatypes.S l0) m) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => Q m) p + (in__int__S k l0 p H2) H3) + H1) + (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l0) => + H1 : cupicef.Term cupicef.prop (Q l0) => + Coq__Init__Logic.ex__intro2 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + in__int k (Coq__Init__Datatypes.S l0) m) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => Q m) l0 + (in__int__intro k (Coq__Init__Datatypes.S l0) l0 H0 + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S l0) + (Coq__Init__Datatypes.S l0)) + (Coq__Init__Peano.lt l0 (Coq__Init__Datatypes.S l0)) cupicef.I + (Coq__Init__Peano.le__n (Coq__Init__Datatypes.S l0)))) + H1) + l H. + +thm in__int__exists : + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k l r) -> + __0 : cupicef.Term cupicef.prop (Q r) -> + cupicef.Term cupicef.prop (exists__between Q k l) + +:= Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (in__int k l r) => + Coq__Init__Logic.match____and cupicef.prop (Coq__Init__Peano.le k r) + (Coq__Init__Peano.lt r l) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Coq__Init__Peano.le k r) + (Coq__Init__Peano.lt r l)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (Q r) + (__0 : cupicef.Term cupicef.prop (Q r) => exists__between Q k l)) + (H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k r) => + lt__r__l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt r l) => + H1 : cupicef.Term cupicef.prop (Q r) => + Coq__Init__Peano.le__ind (Coq__Init__Datatypes.S r) + (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + exists__between Q k l0) + (exists__le Q k r H0 H1) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + lt__r__l0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S r) m) => + IHlt__r__l : cupicef.Term cupicef.prop (exists__between Q k m) => + exists__S Q k m IHlt__r__l) + l lt__r__l) + H. + +def let__o1 : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k m) -> + IHle : + (__ : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k m n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.or (P n) (Q n))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (between P k m) (exists__between Q k m))) -> + P__or__Q : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S m) n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.or (P n) (Q n))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (between P k m) (exists__between Q k m)) + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k m) => + IHle : + (__ : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k m n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.or (P n) (Q n))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (between P k m) (exists__between Q k m))) => + P__or__Q : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S m) n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.or (P n) (Q n))) => + IHle + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H1 : cupicef.Term cupicef.prop (in__int k m n) => + P__or__Q n (in__int__S k m n H1)). + +def let__o2 : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k m) -> + IHle : + (__ : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k m n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.or (P n) (Q n))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (between P k m) (exists__between Q k m))) -> + P__or__Q : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S m) n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.or (P n) (Q n))) -> + H1 : cupicef.Term cupicef.prop (between P k m) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.or (P m) (Q m)) + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k m) => + IHle : + (__ : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k m n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.or (P n) (Q n))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (between P k m) (exists__between Q k m))) => + P__or__Q : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S m) n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.or (P n) (Q n))) => + H1 : cupicef.Term cupicef.prop (between P k m) => + P__or__Q m + (in__int__intro k (Coq__Init__Datatypes.S m) m H0 + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S m) + (Coq__Init__Datatypes.S m)) + (Coq__Init__Peano.lt m (Coq__Init__Datatypes.S m)) cupicef.I + (Coq__Init__Peano.le__n (Coq__Init__Datatypes.S m)))). + +thm between__or__exists : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) -> + __0 : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __0 : cupicef.Term cupicef.prop (in__int k l n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.or (P n) (Q n))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (between P k l) (exists__between Q k l)) + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) => + Coq__Init__Peano.le__ind k + (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.prop + (cupicef.rule + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (in__int k l0 n) + (__ : cupicef.Term cupicef.prop (in__int k l0 n) => + Coq__Init__Logic.or (P n) (Q n)))) + (__ : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k l0 n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.or (P n) (Q n))) => + Coq__Init__Logic.or (between P k l0) (exists__between Q k l0))) + (H0 : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k k n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.or (P n) (Q n))) => + Coq__Init__Logic.or__introl (between P k k) (exists__between Q k k) + (bet__emp P k)) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k m) => + IHle : + (__ : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k m n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.or (P n) (Q n))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (between P k m) (exists__between Q k m))) => + P__or__Q : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S m) n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.or (P n) (Q n))) => + Coq__Init__Logic.match____or cupicef.prop (between P k m) + (exists__between Q k m) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (between P k m) (exists__between Q k m)) => + Coq__Init__Logic.or (between P k (Coq__Init__Datatypes.S m)) + (exists__between Q k (Coq__Init__Datatypes.S m))) + (H1 : cupicef.Term cupicef.prop (between P k m) => + Coq__Init__Logic.match____or cupicef.prop (P m) (Q m) + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.or (P m) (Q m)) => + Coq__Init__Logic.or (between P k (Coq__Init__Datatypes.S m)) + (exists__between Q k (Coq__Init__Datatypes.S m))) + (H2 : cupicef.Term cupicef.prop (P m) => + Coq__Init__Logic.or__introl + (between P k (Coq__Init__Datatypes.S m)) + (exists__between Q k (Coq__Init__Datatypes.S m)) + (bet__S P k m H1 H2)) + (H2 : cupicef.Term cupicef.prop (Q m) => + Coq__Init__Logic.or__intror + (between P k (Coq__Init__Datatypes.S m)) + (exists__between Q k (Coq__Init__Datatypes.S m)) + (exists__le Q k m H0 H2)) + (let__o2 P Q k l H m H0 IHle P__or__Q H1)) + (H1 : cupicef.Term cupicef.prop (exists__between Q k m) => + Coq__Init__Logic.or__intror (between P k (Coq__Init__Datatypes.S m)) + (exists__between Q k (Coq__Init__Datatypes.S m)) + (exists__S Q k m H1)) + (let__o1 P Q k l H m H0 IHle P__or__Q)) + l H. + +def let__e : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : cupicef.Term cupicef.prop (between P k l) -> + l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H0 : cupicef.Term cupicef.prop (between P k l0) -> + IHbetween : + (__ : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k l0 n) -> + __0 : cupicef.Term cupicef.prop (P n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (exists__between Q k l0))) -> + H1 : cupicef.Term cupicef.prop (P l0) -> + H2 : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S l0) n) -> + __0 : cupicef.Term cupicef.prop (P n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) -> + H3 : + cupicef.Term cupicef.prop + (exists__between Q k (Coq__Init__Datatypes.S l0)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex2 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + in__int k (Coq__Init__Datatypes.S l0) m) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => Q m)) + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (between P k l) => + l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (between P k l0) => + IHbetween : + (__ : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k l0 n) -> + __0 : cupicef.Term cupicef.prop (P n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (exists__between Q k l0))) => + H1 : cupicef.Term cupicef.prop (P l0) => + H2 : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S l0) n) -> + __0 : cupicef.Term cupicef.prop (P n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) => + H3 : + cupicef.Term cupicef.prop + (exists__between Q k (Coq__Init__Datatypes.S l0)) => + exists__in__int Q k (Coq__Init__Datatypes.S l0) H3. + +def let__o3 : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : cupicef.Term cupicef.prop (between P k l) -> + l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H0 : cupicef.Term cupicef.prop (between P k l0) -> + IHbetween : + (__ : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k l0 n) -> + __0 : cupicef.Term cupicef.prop (P n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (exists__between Q k l0))) -> + H1 : cupicef.Term cupicef.prop (P l0) -> + H2 : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S l0) n) -> + __0 : cupicef.Term cupicef.prop (P n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) -> + H3 : + cupicef.Term cupicef.prop + (exists__between Q k (Coq__Init__Datatypes.S l0)) -> + l_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H4 : + cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S l0) l_27_) -> + H5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l_27_) -> + H6 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt l_27_ (Coq__Init__Datatypes.S l0)) -> + H7 : cupicef.Term cupicef.prop (Q l_27_) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt l_27_ l0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + l_27_ l0)) + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (between P k l) => + l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (between P k l0) => + IHbetween : + (__ : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k l0 n) -> + __0 : cupicef.Term cupicef.prop (P n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (exists__between Q k l0))) => + H1 : cupicef.Term cupicef.prop (P l0) => + H2 : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S l0) n) -> + __0 : cupicef.Term cupicef.prop (P n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) => + H3 : + cupicef.Term cupicef.prop + (exists__between Q k (Coq__Init__Datatypes.S l0)) => + l_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H4 : + cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S l0) l_27_) => + H5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l_27_) => + H6 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt l_27_ (Coq__Init__Datatypes.S l0)) => + H7 : cupicef.Term cupicef.prop (Q l_27_) => + Coq__Arith__Lt.le__lt__or__eq l_27_ l0 + (Coq__Arith__Lt.lt__n__Sm__le l_27_ l0 H6). + +def let__H6 : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : cupicef.Term cupicef.prop (between P k l) -> + l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H0 : cupicef.Term cupicef.prop (between P k l0) -> + IHbetween : + (__ : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k l0 n) -> + __0 : cupicef.Term cupicef.prop (P n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (exists__between Q k l0))) -> + H1 : cupicef.Term cupicef.prop (P l0) -> + H2 : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S l0) n) -> + __0 : cupicef.Term cupicef.prop (P n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) -> + H3 : + cupicef.Term cupicef.prop + (exists__between Q k (Coq__Init__Datatypes.S l0)) -> + l_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H4 : + cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S l0) l_27_) -> + H5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l_27_) -> + H6 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt l_27_ (Coq__Init__Datatypes.S l0)) -> + H7 : cupicef.Term cupicef.prop (Q l_27_) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + l_27_ l0) + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (between P k l) => + l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (between P k l0) => + IHbetween : + (__ : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k l0 n) -> + __0 : cupicef.Term cupicef.prop (P n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (exists__between Q k l0))) => + H1 : cupicef.Term cupicef.prop (P l0) => + H2 : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S l0) n) -> + __0 : cupicef.Term cupicef.prop (P n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) => + H3 : + cupicef.Term cupicef.prop + (exists__between Q k (Coq__Init__Datatypes.S l0)) => + l_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H4 : + cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S l0) l_27_) => + H5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l_27_) => + H6 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt l_27_ (Coq__Init__Datatypes.S l0)) => + H7 : cupicef.Term cupicef.prop (Q l_27_) => + Coq__Init__Logic.match____or cupicef.prop (Coq__Init__Peano.lt l_27_ l0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + l_27_ l0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt l_27_ l0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + l_27_ l0)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + l_27_ l0) + (H8 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt l_27_ l0) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + l_27_ l0) + (IHbetween + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H9 : cupicef.Term cupicef.prop (in__int k l0 n) => + H10 : cupicef.Term cupicef.prop (P n) => + H2 n (in__int__S k l0 n H9) H10) + (in__int__exists Q k l0 l_27_ (in__int__intro k l0 l_27_ H5 H8) H7))) + (H8 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + l_27_ l0) => + H8) + (let__o3 P Q k l H l0 H0 IHbetween H1 H2 H3 l_27_ H4 H5 H6 H7). + +thm between__not__exists : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (between P k l) -> + __0 : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __0 : cupicef.Term cupicef.prop (in__int k l n) -> + __1 : cupicef.Term cupicef.prop (P n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (exists__between Q k l)) + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (between P k l) => + between__ind P k + (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.rule + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (in__int k l0 n) + (__ : cupicef.Term cupicef.prop (in__int k l0 n) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + P n) + (__0 : cupicef.Term cupicef.prop (P n) => + Coq__Init__Logic.not (Q n))))) + (__ : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k l0 n) -> + __0 : cupicef.Term cupicef.prop (P n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) => + Coq__Init__Logic.not (exists__between Q k l0))) + (cupicef.cast + (cupicef.rule + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop) + (cupicef.prod + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (in__int k k n) + (__ : cupicef.Term cupicef.prop (in__int k k n) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + P n) + (__0 : cupicef.Term cupicef.prop (P n) => + Coq__Init__Logic.not (Q n))))) + (H0 : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k k n) -> + __0 : cupicef.Term cupicef.prop (P n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (exists__between Q k k) + (H1 : cupicef.Term cupicef.prop (exists__between Q k k) => + Coq__Init__Logic.False))) + (cupicef.prod + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.rule + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (in__int k k n) + (__ : cupicef.Term cupicef.prop (in__int k k n) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + P n) + (__0 : cupicef.Term cupicef.prop (P n) => + Coq__Init__Logic.not (Q n))))) + (__ : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k k n) -> + __0 : cupicef.Term cupicef.prop (P n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) => + Coq__Init__Logic.not (exists__between Q k k))) + cupicef.I + (H0 : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k k n) -> + __0 : cupicef.Term cupicef.prop (P n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) => + H1 : cupicef.Term cupicef.prop (exists__between Q k k) => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (Coq__Arith__PeanoNat.Nat__lt__irrefl k + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) k) + (Coq__Init__Peano.lt k k) cupicef.I (exists__le__S Q k k H1))))) + (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (between P k l0) => + IHbetween : + (__ : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (in__int k l0 n) -> + __0 : cupicef.Term cupicef.prop (P n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (exists__between Q k l0))) => + H1 : cupicef.Term cupicef.prop (P l0) => + cupicef.cast + (cupicef.rule + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop) + (cupicef.prod + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (in__int k (Coq__Init__Datatypes.S l0) n) + (__ : + cupicef.Term cupicef.prop + (in__int k (Coq__Init__Datatypes.S l0) n) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + P n) + (__0 : cupicef.Term cupicef.prop (P n) => + Coq__Init__Logic.not (Q n))))) + (H2 : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (in__int k (Coq__Init__Datatypes.S l0) n) -> + __0 : cupicef.Term cupicef.prop (P n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (exists__between Q k (Coq__Init__Datatypes.S l0)) + (H3 : + cupicef.Term cupicef.prop + (exists__between Q k (Coq__Init__Datatypes.S l0)) => + Coq__Init__Logic.False))) + (cupicef.prod + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.rule + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (in__int k (Coq__Init__Datatypes.S l0) n) + (__ : + cupicef.Term cupicef.prop + (in__int k (Coq__Init__Datatypes.S l0) n) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + P n) + (__0 : cupicef.Term cupicef.prop (P n) => + Coq__Init__Logic.not (Q n))))) + (__ : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (in__int k (Coq__Init__Datatypes.S l0) n) -> + __0 : cupicef.Term cupicef.prop (P n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) => + Coq__Init__Logic.not + (exists__between Q k (Coq__Init__Datatypes.S l0)))) + cupicef.I + (H2 : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (in__int k (Coq__Init__Datatypes.S l0) n) -> + __0 : cupicef.Term cupicef.prop (P n) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) => + H3 : + cupicef.Term cupicef.prop + (exists__between Q k (Coq__Init__Datatypes.S l0)) => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (H2 l0 + (in__int__intro k (Coq__Init__Datatypes.S l0) l0 + (exists__S__le Q k l0 H3) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S l0) + (Coq__Init__Datatypes.S l0)) + (Coq__Init__Peano.lt l0 (Coq__Init__Datatypes.S l0)) + cupicef.I + (Coq__Init__Peano.le__n (Coq__Init__Datatypes.S l0)))) + H1 + (Coq__Init__Logic.match____ex2 cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + in__int k (Coq__Init__Datatypes.S l0) m) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Q m) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex2 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (m : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + in__int k (Coq__Init__Datatypes.S l0) m) + (m : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Q m)) => + Q l0) + (l_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H4 : + cupicef.Term cupicef.prop + (in__int k (Coq__Init__Datatypes.S l0) l_27_) => + Coq__Init__Logic.match____and cupicef.prop + (Coq__Init__Peano.le k l_27_) + (Coq__Init__Peano.lt l_27_ (Coq__Init__Datatypes.S l0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Coq__Init__Peano.le k l_27_) + (Coq__Init__Peano.lt l_27_ + (Coq__Init__Datatypes.S l0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Q l_27_) + (__0 : cupicef.Term cupicef.prop (Q l_27_) => Q l0)) + (H5 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le k l_27_) => + H6 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt l_27_ + (Coq__Init__Datatypes.S l0)) => + H7 : cupicef.Term cupicef.prop (Q l_27_) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + l_27_ + (l1 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Q l1) + H7 l0 + (let__H6 P Q k l H l0 H0 IHbetween H1 H2 H3 l_27_ H4 + H5 H6 H7)) + H4) + (let__e P Q k l H l0 H0 IHbetween H1 H2 H3))))) + l H. + +def P__nth : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + init : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop. + +def nth__O : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + init : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (P__nth P Q init init Coq__Init__Datatypes.O). + +def nth__S : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + init : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (P__nth P Q init k n) -> + __0 : cupicef.Term cupicef.prop (between P (Coq__Init__Datatypes.S k) l) -> + __1 : cupicef.Term cupicef.prop (Q l) -> + cupicef.Term cupicef.prop (P__nth P Q init l (Coq__Init__Datatypes.S n)). + +[ ] cupicef.code_guarded? _ (nth__O _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (nth__S _ _ _ _ _ _ _ _ _) --> cupicef.guarded. + +def match____P__nth : + s : cupicef.Sort -> + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + init : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + P0 : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (P__nth P Q init __ __0) -> + cupicef.Univ s) -> + case__nth__O : + cupicef.Term s (P0 init Coq__Init__Datatypes.O (nth__O P Q init)) -> + case__nth__S : + (k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (P__nth P Q init k n) -> + __0 : cupicef.Term cupicef.prop (between P (Coq__Init__Datatypes.S k) l) -> + __1 : cupicef.Term cupicef.prop (Q l) -> + cupicef.Term s + (P0 l (Coq__Init__Datatypes.S n) (nth__S P Q init k l n __ __0 __1))) -> + __ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + x : cupicef.Term cupicef.prop (P__nth P Q init __ __0) -> + cupicef.Term s (P0 __ __0 x). + +[ s, P, Q, init, P0, case__nth__O, case__nth__S] + match____P__nth s P Q init P0 case__nth__O case__nth__S {init} + {Coq__Init__Datatypes.O} (nth__O P Q init) + --> + case__nth__O. + +[ s, P, Q, init, P0, case__nth__O, case__nth__S, k, l, n, __, __0, __1] + match____P__nth s P Q init P0 case__nth__O case__nth__S {l} + {Coq__Init__Datatypes.S n} (nth__S P Q init k l n __ __0 __1) + --> + case__nth__S k l n __ __0 __1. + +[ s, P, Q, init, P0, s'] + match____P__nth s P Q init + (__ => + __0 => + x => + cupicef.uncode _ (cupicef.code (cupicef.cU s') (P0 __ __0 x))) + --> + match____P__nth s' P Q init (__ => __0 => x => P0 __ __0 x). + +def P__nth__ind : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + init : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + P0 : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + f : cupicef.Term cupicef.prop (P0 init Coq__Init__Datatypes.O) -> + f0 : + (k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (P__nth P Q init k n) -> + __0 : cupicef.Term cupicef.prop (P0 k n) -> + __1 : cupicef.Term cupicef.prop (between P (Coq__Init__Datatypes.S k) l) -> + __2 : cupicef.Term cupicef.prop (Q l) -> + cupicef.Term cupicef.prop (P0 l (Coq__Init__Datatypes.S n))) -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.prop (P__nth P Q init n n0) -> + cupicef.Term cupicef.prop (P0 n n0) + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + init : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + P0 : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + f : cupicef.Term cupicef.prop (P0 init Coq__Init__Datatypes.O) => + f0 : + (k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (P__nth P Q init k n) -> + __0 : cupicef.Term cupicef.prop (P0 k n) -> + __1 : + cupicef.Term cupicef.prop (between P (Coq__Init__Datatypes.S k) l) -> + __2 : cupicef.Term cupicef.prop (Q l) -> + cupicef.Term cupicef.prop (P0 l (Coq__Init__Datatypes.S n))) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.2 cupicef.prop + (cupicef.prod cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P__nth P Q init n n0) + (p : cupicef.Term cupicef.prop (P__nth P Q init n n0) => + P0 n n0)))))) + (c => + c + (F => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.prop (P__nth P Q init n n0) => + match____P__nth cupicef.prop P Q init + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + __ : cupicef.Term cupicef.prop (P__nth P Q init n1 n2) => + P0 n1 n2) + f + (k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p0 : cupicef.Term cupicef.prop (P__nth P Q init k n1) => + b : + cupicef.Term cupicef.prop + (between P (Coq__Init__Datatypes.S k) l) => + q : cupicef.Term cupicef.prop (Q l) => + f0 k l n1 p0 (F k n1 p0) b q) + n n0 p)) + cupicef.0. + +thm nth__le : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + init : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (P__nth P Q init l n) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le init l) + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + init : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (P__nth P Q init l n) => + P__nth__ind P Q init + (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Peano.le init l0) + (Coq__Init__Peano.le__n init) + (k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (P__nth P Q init k n0) => + IHP__nth : cupicef.Term cupicef.prop (Coq__Init__Peano.le init k) => + H1 : + cupicef.Term cupicef.prop (between P (Coq__Init__Datatypes.S k) l0) => + H2 : cupicef.Term cupicef.prop (Q l0) => + Coq__Arith__PeanoNat.Nat__le__trans init k l0 IHP__nth + (Coq__Arith__PeanoNat.Nat__le__trans k k l0 + (Coq__Init__Peano.le__n k) + (Coq__Arith__PeanoNat.Nat__le__trans k k l0 + (Coq__Init__Peano.le__n k) + (Coq__Arith__PeanoNat.Nat__le__trans k k l0 + (Coq__Init__Peano.le__n k) + (Coq__Arith__PeanoNat.Nat__le__trans k + (Coq__Init__Datatypes.S k) l0 + (Coq__Arith__PeanoNat.Nat__le__succ__diag__r k) + (between__le P (Coq__Init__Datatypes.S k) l0 H1)))))) + l n H. + +def eventually : + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop + +:= Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.ex2 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Peano.le k n) + (k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => Q k). + +def let__H1 : + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + H : cupicef.Term cupicef.prop (eventually Q Coq__Init__Datatypes.O) -> + x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H0 : + cupicef.Term cupicef.prop (Coq__Init__Peano.le x Coq__Init__Datatypes.O) -> + H1 : cupicef.Term cupicef.prop (Q x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + x Coq__Init__Datatypes.O) + +:= Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + H : cupicef.Term cupicef.prop (eventually Q Coq__Init__Datatypes.O) => + x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : + cupicef.Term cupicef.prop (Coq__Init__Peano.le x Coq__Init__Datatypes.O) => + H1 : cupicef.Term cupicef.prop (Q x) => + Coq__Arith__PeanoNat.Nat__le__antisymm x Coq__Init__Datatypes.O H0 + (Coq__Arith__PeanoNat.Nat__le__0__l x). + +thm event__O : + Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + __ : cupicef.Term cupicef.prop (eventually Q Coq__Init__Datatypes.O) -> + cupicef.Term cupicef.prop (Q Coq__Init__Datatypes.O) + +:= Q : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + H : cupicef.Term cupicef.prop (eventually Q Coq__Init__Datatypes.O) => + Coq__Init__Logic.match____ex2 cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Peano.le k Coq__Init__Datatypes.O) + (k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => Q k) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex2 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Peano.le k Coq__Init__Datatypes.O) + (k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => Q k)) => + Q Coq__Init__Datatypes.O) + (x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le x Coq__Init__Datatypes.O) => + H1 : cupicef.Term cupicef.prop (Q x) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + x (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => Q n) H1 + Coq__Init__Datatypes.O (let__H1 Q H x H0 H1)) + H. + +(; End of translation. ;) diff --git a/out/Coq__Arith__Compare__dec.dk b/out/Coq__Arith__Compare__dec.dk new file mode 100644 index 000000000..561ce601f --- /dev/null +++ b/out/Coq__Arith__Compare__dec.dk @@ -0,0 +1,5066 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def zerop : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n Coq__Init__Datatypes.O) + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O) + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) + (Coq__Init__Specif.left + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O) + (Coq__Init__Peano.lt Coq__Init__Datatypes.O Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O)) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Specif.right + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) + (Coq__Init__Peano.lt Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S n0)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.le + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O) + (Coq__Init__Datatypes.S n0)) + (Coq__Init__Peano.lt Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S n0)) + cupicef.I + (Coq__Arith__Gt.gt__le__S Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S n0) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S n0)) + (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n0) + Coq__Init__Datatypes.O) + cupicef.I (Coq__Arith__PeanoNat.Nat__lt__0__succ n0))))) + n. + +def let__s : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn : + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumor cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (Coq__Init__Peano.lt m0 n0))) -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumor cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n0 m1) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m1)) + (Coq__Init__Peano.lt m1 n0)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumor cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (Coq__Init__Peano.lt m0 n0))) => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn m1. + +def lt__eq__lt__dec : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumor cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) + (Coq__Init__Peano.lt m n)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__rec + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Specif.sumor cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (Coq__Init__Peano.lt m0 n0))) + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Specif.sumor cupicef.set + (Coq__Init__Specif.sumbool + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O n0)) + (Coq__Init__Peano.lt n0 Coq__Init__Datatypes.O)) + (Coq__Init__Specif.inleft + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sumbool + (Coq__Init__Peano.lt Coq__Init__Datatypes.O + Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O))) + (Coq__Init__Peano.lt Coq__Init__Datatypes.O Coq__Init__Datatypes.O) + (Coq__Init__Specif.right + (Coq__Init__Peano.lt Coq__Init__Datatypes.O + Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O))) + (m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Specif.inleft + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sumbool + (Coq__Init__Peano.lt Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S m1)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1)))) + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S m1) + Coq__Init__Datatypes.O) + (Coq__Init__Specif.left + (Coq__Init__Peano.lt Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S m1)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.le + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O) + (Coq__Init__Datatypes.S m1)) + (Coq__Init__Peano.lt Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S m1)) + cupicef.I + (Coq__Arith__Gt.gt__le__S Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S m1) (Coq__Arith__Gt.gt__Sn__O m1))))) + m0) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumor cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (Coq__Init__Peano.lt m0 n0))) => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Specif.sumor cupicef.set + (Coq__Init__Specif.sumbool + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) n1) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) n1)) + (Coq__Init__Peano.lt n1 (Coq__Init__Datatypes.S n0))) + (Coq__Init__Specif.inright + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sumbool + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O))) + (Coq__Init__Peano.lt Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S n0)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.le + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O) + (Coq__Init__Datatypes.S n0)) + (Coq__Init__Peano.lt Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S n0)) + cupicef.I + (Coq__Arith__Gt.gt__le__S Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S n0) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S n0)) + (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n0) + Coq__Init__Datatypes.O) + cupicef.I (Coq__Arith__PeanoNat.Nat__lt__0__succ n0))))) + (m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Specif.match____sumor cupicef.set + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n0 m1) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m1))) + (Coq__Init__Peano.lt m1 n0) + (s0 : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumor cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n0 m1) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m1)) + (Coq__Init__Peano.lt m1 n0)) => + Coq__Init__Specif.sumor cupicef.set + (Coq__Init__Specif.sumbool + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) (Coq__Init__Datatypes.S m1))) + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S m1) + (Coq__Init__Datatypes.S n0))) + (H : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n0 m1) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m1)) => + Coq__Init__Specif.match____sumbool cupicef.set + (Coq__Init__Peano.lt n0 m1) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m1) + (s0 : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n0 m1) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m1)) => + Coq__Init__Specif.sumor cupicef.set + (Coq__Init__Specif.sumbool + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1))) + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S m1) + (Coq__Init__Datatypes.S n0))) + (l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m1) => + Coq__Init__Specif.inleft + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sumbool + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)))) + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S m1) + (Coq__Init__Datatypes.S n0)) + (Coq__Init__Specif.left + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.le + (Coq__Init__Datatypes.S (Coq__Init__Datatypes.S n0)) + (Coq__Init__Datatypes.S m1)) + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)) + cupicef.I + (Coq__Arith__Le.le__n__S (Coq__Init__Datatypes.S n0) + m1 l)))) + (e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m1) => + Coq__Init__Specif.inleft + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sumbool + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)))) + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S m1) + (Coq__Init__Datatypes.S n0)) + (Coq__Init__Specif.right + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)) + (Coq__Init__Peano.f__equal__nat + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.S n0 m1 e))) + H) + (H : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m1 n0) => + Coq__Init__Specif.inright + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sumbool + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)))) + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S m1) + (Coq__Init__Datatypes.S n0)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.le + (Coq__Init__Datatypes.S (Coq__Init__Datatypes.S m1)) + (Coq__Init__Datatypes.S n0)) + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S m1) + (Coq__Init__Datatypes.S n0)) + cupicef.I + (Coq__Arith__Gt.gt__le__S (Coq__Init__Datatypes.S m1) + (Coq__Init__Datatypes.S n0) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S m1) + (Coq__Init__Datatypes.S n0)) + (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)) + cupicef.I + (Coq__Arith__Lt.le__lt__n__Sm + (Coq__Init__Datatypes.S m1) n0 H))))) + (let__s n m n0 IHn m0 m1)) + m0) + n m. + +def gt__eq__gt__dec : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumor cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.gt m n) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) + (Coq__Init__Peano.gt n m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + lt__eq__lt__dec n m. + +def le__lt__dec : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.le n m) + (Coq__Init__Peano.lt m n)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__rec + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Specif.sumbool (Coq__Init__Peano.le n0 m0) + (Coq__Init__Peano.lt m0 n0))) + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Specif.left (Coq__Init__Peano.le Coq__Init__Datatypes.O m0) + (Coq__Init__Peano.lt m0 Coq__Init__Datatypes.O) + (Coq__Arith__PeanoNat.Nat__le__0__l m0)) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.le n0 m0) + (Coq__Init__Peano.lt m0 n0))) => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Specif.sumbool + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) n1) + (Coq__Init__Peano.lt n1 (Coq__Init__Datatypes.S n0))) + (Coq__Init__Specif.right + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + Coq__Init__Datatypes.O) + (Coq__Init__Peano.lt Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S n0)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.le + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O) + (Coq__Init__Datatypes.S n0)) + (Coq__Init__Peano.lt Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S n0)) + cupicef.I + (Coq__Arith__Gt.gt__le__S Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S n0) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S n0)) + (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n0) + Coq__Init__Datatypes.O) + cupicef.I + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.le + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O) + (Coq__Init__Datatypes.S n0)) + (Coq__Init__Peano.lt Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S n0)) + cupicef.I + (Coq__Arith__Lt.lt__le__S Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S n0) + (Coq__Arith__PeanoNat.Nat__lt__0__succ n0))))))) + (m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Specif.sumbool__rec (Coq__Init__Peano.le n0 m1) + (Coq__Init__Peano.lt m1 n0) + (s : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.le n0 m1) + (Coq__Init__Peano.lt m1 n0)) => + Coq__Init__Specif.sumbool + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)) + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S m1) + (Coq__Init__Datatypes.S n0))) + (a : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m1) => + Coq__Init__Specif.left + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)) + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S m1) + (Coq__Init__Datatypes.S n0)) + (Coq__Arith__Gt.gt__le__S n0 (Coq__Init__Datatypes.S m1) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m1)) + (Coq__Init__Peano.gt (Coq__Init__Datatypes.S m1) n0) + cupicef.I + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)) + (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m1)) + cupicef.I (Coq__Arith__Le.le__n__S n0 m1 a))))) + (b : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m1 n0) => + Coq__Init__Specif.right + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)) + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S m1) + (Coq__Init__Datatypes.S n0)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.le + (Coq__Init__Datatypes.S (Coq__Init__Datatypes.S m1)) + (Coq__Init__Datatypes.S n0)) + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S m1) + (Coq__Init__Datatypes.S n0)) + cupicef.I + (Coq__Arith__Gt.gt__le__S (Coq__Init__Datatypes.S m1) + (Coq__Init__Datatypes.S n0) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S m1) + (Coq__Init__Datatypes.S n0)) + (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)) + cupicef.I (Coq__Arith__Lt.lt__n__S m1 n0 b))))) + (IHn m1)) + m0) + n m. + +def le__le__S__dec : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.le n m) + (Coq__Init__Peano.le (Coq__Init__Datatypes.S m) n)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__lt__dec n m. + +def le__ge__dec : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.le n m) + (Coq__Init__Peano.ge n m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Specif.sumbool__rec (Coq__Init__Peano.le n m) + (Coq__Init__Peano.lt m n) + (s : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.le n m) + (Coq__Init__Peano.lt m n)) => + Coq__Init__Specif.sumbool (Coq__Init__Peano.le n m) + (Coq__Init__Peano.ge n m)) + (a : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) => + Coq__Init__Specif.left (Coq__Init__Peano.le n m) + (Coq__Init__Peano.ge n m) a) + (b : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) => + Coq__Init__Specif.right (Coq__Init__Peano.le n m) + (Coq__Init__Peano.ge n m) + (cupicef.cast cupicef.prop cupicef.prop (Coq__Init__Peano.le m n) + (Coq__Init__Peano.ge n m) cupicef.I + (Coq__Arith__PeanoNat.Nat__lt__le__incl m n b))) + (le__lt__dec n m). + +def le__gt__dec : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.le n m) + (Coq__Init__Peano.gt n m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__lt__dec n m. + +def let__s0 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumor cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) + (Coq__Init__Peano.lt m n)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) => + lt__eq__lt__dec n m. + +def le__lt__eq__dec : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) => + Coq__Init__Specif.match____sumor cupicef.set + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m))) + (Coq__Init__Peano.lt m n) + (s0 : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumor cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) + (Coq__Init__Peano.lt m n)) => + Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) + (s0 : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) => + s0) + (l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) => + Coq__Init__Logic.False__rec + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) + (Coq__Arith__Lt.le__not__lt n m H l)) + (let__s0 n m H). + +def let__s1 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.le n m) + (Coq__Init__Peano.gt n m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__gt__dec n m. + +def le__dec : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.le n m) + (Coq__Init__Logic.not (Coq__Init__Peano.le n m))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Specif.match____sumbool cupicef.set (Coq__Init__Peano.le n m) + (Coq__Init__Peano.gt n m) + (s0 : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.le n m) + (Coq__Init__Peano.gt n m)) => + Coq__Init__Specif.sumbool (Coq__Init__Peano.le n m) + (Coq__Init__Logic.not (Coq__Init__Peano.le n m))) + (l : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) => + Coq__Init__Specif.left (Coq__Init__Peano.le n m) + (Coq__Init__Logic.not (Coq__Init__Peano.le n m)) l) + (g : cupicef.Term cupicef.prop (Coq__Init__Peano.gt n m) => + Coq__Init__Specif.right (Coq__Init__Peano.le n m) + (Coq__Init__Logic.not (Coq__Init__Peano.le n m)) + (Coq__Arith__Gt.gt__not__le n m g)) + (let__s1 n m). + +def lt__dec : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.not (Coq__Init__Peano.lt n m))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__dec (Coq__Init__Datatypes.S n) m. + +def gt__dec : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.gt n m) + (Coq__Init__Logic.not (Coq__Init__Peano.gt n m))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + lt__dec m n. + +def ge__dec : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.ge n m) + (Coq__Init__Logic.not (Coq__Init__Peano.ge n m))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__dec m n. + +thm dec__le : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Logic__Decidable.decidable (Coq__Init__Peano.le n m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Arith__PeanoNat.Nat__le__decidable n m. + +thm dec__lt : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Logic__Decidable.decidable (Coq__Init__Peano.lt n m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Arith__PeanoNat.Nat__lt__decidable n m. + +thm dec__gt : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Logic__Decidable.decidable (Coq__Init__Peano.gt n m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Arith__PeanoNat.Nat__lt__decidable m n. + +thm dec__ge : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Logic__Decidable.decidable (Coq__Init__Peano.ge n m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Arith__PeanoNat.Nat__le__decidable m n. + +def let__H : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Peano.lt m0 n0)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) => + Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Peano.lt m0 n0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Peano.lt m0 n0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Peano.lt m0 n0)) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) => + Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Peano.lt m0 n0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Peano.lt m0 n0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Peano.lt m0 n0)) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) => + Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Peano.lt m0 n0))) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Peano.lt m0 n0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Peano.lt m0 n0)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0))) => + __) + (Coq__Arith__PeanoNat.Nat__lt__gt__cases n0 m0). + +thm not__eq : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt n m) (Coq__Init__Peano.lt m n)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H n m n m. + +def let__H0 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) => + Coq__Init__Peano.lt m0 n0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt m0 n0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) => + Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) => + Coq__Init__Peano.lt m0 n0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt m0 n0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) => + Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) => + Coq__Init__Peano.lt m0 n0)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0)) => + __0 : + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0))) => + __) + (Coq__Arith__PeanoNat.Nat__nle__gt n0 m0). + +thm not__le : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le n m)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.gt n m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H0 n m n m. + +def let__H1 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le m0 n0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) => + Coq__Init__Peano.le m0 n0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le m0 n0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le m0 n0) => + Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) => + Coq__Init__Peano.le m0 n0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le m0 n0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le m0 n0) => + Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) => + Coq__Init__Peano.le m0 n0)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le m0 n0)) => + __0 : + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m0 n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0))) => + __) + (Coq__Arith__PeanoNat.Nat__nlt__ge n0 m0). + +thm not__gt : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.gt n m)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H1 n m m n. + +def let__H2 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) => + Coq__Init__Peano.lt m0 n0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt m0 n0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) => + Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) => + Coq__Init__Peano.lt m0 n0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt m0 n0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) => + Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) => + Coq__Init__Peano.lt m0 n0)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0)) => + __0 : + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0))) => + __) + (Coq__Arith__PeanoNat.Nat__nle__gt n0 m0). + +thm not__ge : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.ge n m)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H2 n m m n. + +def let__H3 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le m0 n0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) => + Coq__Init__Peano.le m0 n0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le m0 n0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le m0 n0) => + Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) => + Coq__Init__Peano.le m0 n0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le m0 n0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le m0 n0) => + Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) => + Coq__Init__Peano.le m0 n0)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le m0 n0)) => + __0 : + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m0 n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0))) => + __) + (Coq__Arith__PeanoNat.Nat__nlt__ge n0 m0). + +thm not__lt : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.lt n m)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.ge n m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H3 n m n m. + +thm nat__compare__lt : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n m) Coq__Init__Datatypes.Lt)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Classes__RelationClasses.symmetry + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff Coq__Classes__RelationClasses.iff__Symmetric + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n m) Coq__Init__Datatypes.Lt) + (Coq__Init__Peano.lt n m) + (Coq__Arith__PeanoNat.Nat__compare__lt__iff n m). + +thm nat__compare__gt : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Coq__Init__Peano.gt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n m) Coq__Init__Datatypes.Gt)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Classes__RelationClasses.symmetry + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff Coq__Classes__RelationClasses.iff__Symmetric + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n m) Coq__Init__Datatypes.Gt) + (Coq__Init__Peano.gt n m) + (Coq__Arith__PeanoNat.Nat__compare__gt__iff n m). + +thm nat__compare__le : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Coq__Init__Peano.le n m) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n m) Coq__Init__Datatypes.Gt))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Classes__RelationClasses.symmetry + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff Coq__Classes__RelationClasses.iff__Symmetric + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n m) Coq__Init__Datatypes.Gt)) + (Coq__Init__Peano.le n m) + (Coq__Arith__PeanoNat.Nat__compare__le__iff n m). + +thm nat__compare__ge : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Coq__Init__Peano.ge n m) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n m) Coq__Init__Datatypes.Lt))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Classes__RelationClasses.symmetry + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff Coq__Classes__RelationClasses.iff__Symmetric + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n m) Coq__Init__Datatypes.Lt)) + (Coq__Init__Peano.ge n m) + (Coq__Arith__PeanoNat.Nat__compare__ge__iff n m). + +def let__H4 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Eq) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Eq) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) + Coq__Init__Datatypes.Eq) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Eq)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) + Coq__Init__Datatypes.Eq) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) + Coq__Init__Datatypes.Eq) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) + Coq__Init__Datatypes.Eq))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Eq) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) + Coq__Init__Datatypes.Eq) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) + Coq__Init__Datatypes.Eq) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) + Coq__Init__Datatypes.Eq)) => + __) + (Coq__Arith__PeanoNat.Nat__compare__eq__iff n0 m0). + +thm nat__compare__eq : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n m) Coq__Init__Datatypes.Eq) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H4 n m n m. + +def let__H5 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Lt) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Lt) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) + Coq__Init__Datatypes.Lt) => + Coq__Init__Peano.lt n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Lt)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) + Coq__Init__Datatypes.Lt) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) + Coq__Init__Datatypes.Lt) => + Coq__Init__Peano.lt n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) + Coq__Init__Datatypes.Lt))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Lt) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) + Coq__Init__Datatypes.Lt) => + Coq__Init__Peano.lt n0 m0)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) + Coq__Init__Datatypes.Lt) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => + __0 : + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) + Coq__Init__Datatypes.Lt)) => + __) + (Coq__Arith__PeanoNat.Nat__compare__lt__iff n0 m0). + +thm nat__compare__Lt__lt : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n m) Coq__Init__Datatypes.Lt) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H5 n m n m. + +def let__H6 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare x y) Coq__Init__Datatypes.Gt) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt y x) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare x y) Coq__Init__Datatypes.Gt) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare x y) + Coq__Init__Datatypes.Gt) => + Coq__Init__Peano.lt y x)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt y x) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt y x) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare x y) Coq__Init__Datatypes.Gt)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare x y) + Coq__Init__Datatypes.Gt) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare x y) + Coq__Init__Datatypes.Gt) => + Coq__Init__Peano.lt y x)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt y x) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt y x) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare x y) + Coq__Init__Datatypes.Gt))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare x y) Coq__Init__Datatypes.Gt) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare x y) + Coq__Init__Datatypes.Gt) => + Coq__Init__Peano.lt y x)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare x y) + Coq__Init__Datatypes.Gt) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt y x)) => + __0 : + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt y x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare x y) Coq__Init__Datatypes.Gt)) => + __) + (Coq__Arith__PeanoNat.Nat__compare__gt__iff x y). + +thm nat__compare__Gt__gt : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n m) Coq__Init__Datatypes.Gt) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.gt n m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H6 n m n m. + +def nat__compare__alt : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.comparison + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Specif.match____sumor cupicef.set + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m))) + (Coq__Init__Peano.lt m n) + (x : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumor cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) + (Coq__Init__Peano.lt m n)) => + Coq__Init__Datatypes.comparison) + (s : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) => + Coq__Init__Specif.match____sumbool cupicef.set + (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) + (s0 : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) => + Coq__Init__Datatypes.comparison) + (l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) => + Coq__Init__Datatypes.Lt) + (e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) => + Coq__Init__Datatypes.Eq) + s) + (l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) => + Coq__Init__Datatypes.Gt) + (lt__eq__lt__dec n m). + +def let__s2 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumor cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) + (Coq__Init__Peano.lt m n)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + lt__eq__lt__dec n m. + +def let__H7 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + s0 : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) -> + l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Lt) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + s0 : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) => + l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Lt) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) + Coq__Init__Datatypes.Lt) => + Coq__Init__Peano.lt n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Lt)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) + Coq__Init__Datatypes.Lt) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) + Coq__Init__Datatypes.Lt) => + Coq__Init__Peano.lt n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) + Coq__Init__Datatypes.Lt))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Lt)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) + Coq__Init__Datatypes.Lt) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => + __0 : + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) + Coq__Init__Datatypes.Lt)) => + __0) + (Coq__Arith__PeanoNat.Nat__compare__lt__iff n0 m0). + +def let__H8 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + s0 : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) -> + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Eq) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + s0 : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Eq) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) + Coq__Init__Datatypes.Eq) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Eq)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) + Coq__Init__Datatypes.Eq) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) + Coq__Init__Datatypes.Eq) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) + Coq__Init__Datatypes.Eq))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Eq)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) + Coq__Init__Datatypes.Eq) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n0 m0) + Coq__Init__Datatypes.Eq)) => + __0) + (Coq__Arith__PeanoNat.Nat__compare__eq__iff n0 m0). + +def let__H9 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) -> + x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt y x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare x y) Coq__Init__Datatypes.Gt) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) => + x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare x y) Coq__Init__Datatypes.Gt) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare x y) + Coq__Init__Datatypes.Gt) => + Coq__Init__Peano.lt y x)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt y x) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt y x) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare x y) Coq__Init__Datatypes.Gt)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare x y) + Coq__Init__Datatypes.Gt) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare x y) + Coq__Init__Datatypes.Gt) => + Coq__Init__Peano.lt y x)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt y x) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt y x) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare x y) + Coq__Init__Datatypes.Gt))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt y x) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt y x) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare x y) Coq__Init__Datatypes.Gt)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare x y) + Coq__Init__Datatypes.Gt) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt y x)) => + __0 : + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt y x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare x y) Coq__Init__Datatypes.Gt)) => + __0) + (Coq__Arith__PeanoNat.Nat__compare__gt__iff x y). + +thm nat__compare__equiv : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n m) (nat__compare__alt n m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n m) + (Coq__Init__Specif.match____sumor cupicef.set + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m))) + (Coq__Init__Peano.lt m n) + (x : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumor cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) + (Coq__Init__Peano.lt m n)) => + Coq__Init__Datatypes.comparison) + (s : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) => + Coq__Init__Specif.match____sumbool cupicef.set + (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) + (s0 : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) => + Coq__Init__Datatypes.comparison) + (l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) => + Coq__Init__Datatypes.Lt) + (e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) => + Coq__Init__Datatypes.Eq) + s) + (l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) => + Coq__Init__Datatypes.Gt) + (lt__eq__lt__dec n m))) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n m) (nat__compare__alt n m)) + cupicef.I + (Coq__Init__Specif.match____sumor cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m))) + (Coq__Init__Peano.lt m n) + (s0 : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumor cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) + (Coq__Init__Peano.lt m n)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n m) + (Coq__Init__Specif.match____sumor cupicef.set + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m))) + (Coq__Init__Peano.lt m n) + (x : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumor cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) + (Coq__Init__Peano.lt m n)) => + Coq__Init__Datatypes.comparison) + (s1 : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) => + Coq__Init__Specif.match____sumbool cupicef.set + (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) + (s2 : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) => + Coq__Init__Datatypes.comparison) + (l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) => + Coq__Init__Datatypes.Lt) + (e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) => + Coq__Init__Datatypes.Eq) + s1) + (l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) => + Coq__Init__Datatypes.Gt) + s0)) + (s0 : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) => + Coq__Init__Specif.match____sumbool cupicef.prop + (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) + (s1 : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n m) + (Coq__Init__Specif.match____sumbool cupicef.set + (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) + (s2 : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) => + Coq__Init__Datatypes.comparison) + (l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) => + Coq__Init__Datatypes.Lt) + (e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) => + Coq__Init__Datatypes.Eq) + s1)) + (l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) => + let__H7 n m s0 l n m l) + (e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) => + let__H8 n m s0 e n m e) + s0) + (l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) => + let__H9 n m l n m l) + (let__s2 n m)). + +thm leb__iff__conv : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n m) Coq__Init__Datatypes.false) + (Coq__Init__Peano.lt m n)) + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + (lemma : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n m) Coq__Init__Datatypes.false) + (Coq__Init__Logic.not (Coq__Init__Peano.le n m))) => + Coq__Classes__Morphisms.trans__co__eq__inv__impl__morphism + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff Coq__Classes__RelationClasses.iff__Transitive + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n m) Coq__Init__Datatypes.false) + (Coq__Init__Logic.not (Coq__Init__Peano.le n m)) lemma + (Coq__Init__Peano.lt m n) (Coq__Init__Peano.lt m n) + (Coq__Classes__Morphisms.eq__proper__proxy + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Init__Peano.lt m n))) + (Coq__Arith__PeanoNat.Nat__leb__nle n m) + (Coq__Arith__PeanoNat.Nat__nle__gt n m). + +def let__H10 : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.true) + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) + Coq__Init__Datatypes.true) => + Coq__Init__Peano.le n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) + Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) + Coq__Init__Datatypes.true) => + Coq__Init__Peano.le n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) + Coq__Init__Datatypes.true))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.true)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) + Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => + __0 : + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.true)) => + __0) + (Coq__Arith__PeanoNat.Nat__leb__le n0 m0). + +thm leb__correct : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb m n) Coq__Init__Datatypes.true) + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H10 m n m n. + +def let__H11 : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) + Coq__Init__Datatypes.true) => + Coq__Init__Peano.le n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) + Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) + Coq__Init__Datatypes.true) => + Coq__Init__Peano.le n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) + Coq__Init__Datatypes.true))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.true) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) + Coq__Init__Datatypes.true) => + Coq__Init__Peano.le n0 m0)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) + Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => + __0 : + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.true)) => + __) + (Coq__Arith__PeanoNat.Nat__leb__le n0 m0). + +thm leb__complete : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb m n) Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le m n) + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H11 m n m n. + +def let__H12 : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.false) + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) + Coq__Init__Datatypes.false) => + Coq__Init__Peano.lt m0 n0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt m0 n0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) + Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) + Coq__Init__Datatypes.false) => + Coq__Init__Peano.lt m0 n0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt m0 n0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) + Coq__Init__Datatypes.false))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt m0 n0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.false)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) + Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0)) => + __0 : + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) + Coq__Init__Datatypes.false)) => + __0) + (leb__iff__conv m0 n0). + +thm leb__correct__conv : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n m) Coq__Init__Datatypes.false) + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H12 m n m n. + +def let__H13 : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) + Coq__Init__Datatypes.false) => + Coq__Init__Peano.lt m0 n0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt m0 n0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) + Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) + Coq__Init__Datatypes.false) => + Coq__Init__Peano.lt m0 n0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt m0 n0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) + Coq__Init__Datatypes.false))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.false) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) + Coq__Init__Datatypes.false) => + Coq__Init__Peano.lt m0 n0)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) + Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0)) => + __0 : + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n0 m0) + Coq__Init__Datatypes.false)) => + __) + (leb__iff__conv m0 n0). + +thm leb__complete__conv : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n m) Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H13 m n m n. + +thm leb__compare : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n m) Coq__Init__Datatypes.true) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n m) Coq__Init__Datatypes.Gt))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + (lemma : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n m) + Coq__Init__Datatypes.Gt)) + (Coq__Init__Peano.le n m)) => + Coq__Classes__Morphisms.trans__sym__co__inv__impl__morphism + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff + (Coq__Classes__RelationClasses.Equivalence__PER + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff Coq__Classes__RelationClasses.iff__equivalence) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__leb n m) Coq__Init__Datatypes.true) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.comparison) + (Coq__Arith__PeanoNat.Nat__compare n m) Coq__Init__Datatypes.Gt)) + (Coq__Init__Peano.le n m) lemma) + (Coq__Arith__PeanoNat.Nat__compare__le__iff n m) + (Coq__Arith__PeanoNat.Nat__leb__le n m). + +(; End of translation. ;) diff --git a/out/Coq__Arith__EqNat.dk b/out/Coq__Arith__EqNat.dk new file mode 100644 index 000000000..df90cb1c9 --- /dev/null +++ b/out/Coq__Arith__EqNat.dk @@ -0,0 +1,2688 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def eq__nat : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))))) + (c => + c + (eq__nat0 => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat + (cupicef.type (cupicef.s cupicef.z)) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (Coq__Init__Datatypes.match____nat + (cupicef.type (cupicef.s cupicef.z)) + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.False) + m) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat + (cupicef.type (cupicef.s cupicef.z)) + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False + (m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + eq__nat0 n1 m1) + m) + n)) + cupicef.0. + +thm eq__nat__refl : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (eq__nat n n) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__ind + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => eq__nat n0 n0) + (cupicef.cast cupicef.prop cupicef.prop Coq__Init__Logic.True + (eq__nat Coq__Init__Datatypes.O Coq__Init__Datatypes.O) cupicef.I + Coq__Init__Logic.I) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : cupicef.Term cupicef.prop (eq__nat n0 n0) => + cupicef.cast cupicef.prop cupicef.prop (eq__nat n0 n0) + (eq__nat (Coq__Init__Datatypes.S n0) (Coq__Init__Datatypes.S n0)) + cupicef.I IHn) + n. + +thm eq__nat__is__eq : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (eq__nat n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eq__nat n m) + (__ : cupicef.Term cupicef.prop (eq__nat n m) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) => + eq__nat n m)) + (Coq__Init__Datatypes.nat__ind + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eq__nat n0 m0) + (__ : cupicef.Term cupicef.prop (eq__nat n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0))) + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.prop + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eq__nat Coq__Init__Datatypes.O n0) + (__ : + cupicef.Term cupicef.prop + (eq__nat Coq__Init__Datatypes.O n0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O n0)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.True + (H : cupicef.Term cupicef.prop Coq__Init__Logic.True => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eq__nat Coq__Init__Datatypes.O Coq__Init__Datatypes.O) + (__ : + cupicef.Term cupicef.prop + (eq__nat Coq__Init__Datatypes.O Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O)) + cupicef.I + (H : cupicef.Term cupicef.prop Coq__Init__Logic.True => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O)) + (m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.False + (H : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eq__nat Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1)) + (__ : + cupicef.Term cupicef.prop + (eq__nat Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S m1)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1))) + cupicef.I + (H : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1)) + H)) + m0) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (eq__nat n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.prop + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eq__nat (Coq__Init__Datatypes.S n0) n1) + (__ : + cupicef.Term cupicef.prop + (eq__nat (Coq__Init__Datatypes.S n0) n1) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) n1)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.False + (H : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eq__nat (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) + (__ : + cupicef.Term cupicef.prop + (eq__nat (Coq__Init__Datatypes.S n0) + Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O)) + cupicef.I + (H : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) + H)) + (m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eq__nat n0 m1) + (H : cupicef.Term cupicef.prop (eq__nat n0 m1) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) (Coq__Init__Datatypes.S m1))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eq__nat (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)) + (__ : + cupicef.Term cupicef.prop + (eq__nat (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) (Coq__Init__Datatypes.S m1))) + cupicef.I + (H : cupicef.Term cupicef.prop (eq__nat n0 m1) => + Coq__Init__Peano.f__equal__nat + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.S n0 m1 (IHn m1 H))) + m0) + n m) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + eq__nat n m0) + (eq__nat__refl n) m H). + +def let__H : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) -> + cupicef.Term cupicef.prop (eq__nat n0 m0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eq__nat n0 m0) + (__ : cupicef.Term cupicef.prop (eq__nat n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) => + eq__nat n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eq__nat n0 m0) + (__ : cupicef.Term cupicef.prop (eq__nat n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) => + eq__nat n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) => + eq__nat n0 m0)) + (__ : + (__ : cupicef.Term cupicef.prop (eq__nat n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) -> + cupicef.Term cupicef.prop (eq__nat n0 m0)) => + __0) + (eq__nat__is__eq n0 m0). + +thm eq__eq__nat : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) -> + cupicef.Term cupicef.prop (eq__nat n m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H n m n m. + +def let__H0 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (eq__nat n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eq__nat n0 m0) + (__ : cupicef.Term cupicef.prop (eq__nat n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) => + eq__nat n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eq__nat n0 m0) + (__ : cupicef.Term cupicef.prop (eq__nat n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) => + eq__nat n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eq__nat n0 m0) + (__0 : cupicef.Term cupicef.prop (eq__nat n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (__ : + (__ : cupicef.Term cupicef.prop (eq__nat n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) -> + cupicef.Term cupicef.prop (eq__nat n0 m0)) => + __) + (eq__nat__is__eq n0 m0). + +thm eq__nat__eq : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (eq__nat n m) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H0 n m n m. + +def let__H1 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + H : cupicef.Term cupicef.prop (P n) -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H0 : cupicef.Term cupicef.prop (eq__nat n m) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + H : cupicef.Term cupicef.prop (P n) => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (eq__nat n m) => + eq__nat__eq n m H0. + +thm eq__nat__elim : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + __ : cupicef.Term cupicef.prop (P n) -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __0 : cupicef.Term cupicef.prop (eq__nat n m) -> + cupicef.Term cupicef.prop (P m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + H : cupicef.Term cupicef.prop (P n) => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (eq__nat n m) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => P m0) H m + (let__H1 n P H m H0). + +def eq__nat__decide : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (eq__nat n m) + (Coq__Init__Logic.not (eq__nat n m))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__rec + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Specif.sumbool (eq__nat n0 m) + (Coq__Init__Logic.not (eq__nat n0 m)))) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Specif.sumbool (eq__nat Coq__Init__Datatypes.O n0) + (Coq__Init__Logic.not (eq__nat Coq__Init__Datatypes.O n0))) + (cupicef.cast cupicef.set cupicef.set + (Coq__Init__Specif.sumbool Coq__Init__Logic.True + (Coq__Init__Logic.not Coq__Init__Logic.True)) + (Coq__Init__Specif.sumbool + (eq__nat Coq__Init__Datatypes.O Coq__Init__Datatypes.O) + (Coq__Init__Logic.not + (eq__nat Coq__Init__Datatypes.O Coq__Init__Datatypes.O))) + cupicef.I + (Coq__Init__Specif.left Coq__Init__Logic.True + (Coq__Init__Logic.not Coq__Init__Logic.True) Coq__Init__Logic.I)) + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.cast cupicef.set cupicef.set + (Coq__Init__Specif.sumbool Coq__Init__Logic.False + (Coq__Init__Logic.not Coq__Init__Logic.False)) + (Coq__Init__Specif.sumbool + (eq__nat Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m0)) + (Coq__Init__Logic.not + (eq__nat Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m0)))) + cupicef.I + (Coq__Init__Specif.right Coq__Init__Logic.False + (Coq__Init__Logic.not Coq__Init__Logic.False) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.False + (H : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not Coq__Init__Logic.False) cupicef.I + (H : cupicef.Term cupicef.prop Coq__Init__Logic.False => H)))) + m) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool (eq__nat n0 m) + (Coq__Init__Logic.not (eq__nat n0 m)))) => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Specif.sumbool (eq__nat (Coq__Init__Datatypes.S n0) n1) + (Coq__Init__Logic.not (eq__nat (Coq__Init__Datatypes.S n0) n1))) + (cupicef.cast cupicef.set cupicef.set + (Coq__Init__Specif.sumbool Coq__Init__Logic.False + (Coq__Init__Logic.not Coq__Init__Logic.False)) + (Coq__Init__Specif.sumbool + (eq__nat (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) + (Coq__Init__Logic.not + (eq__nat (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O))) + cupicef.I + (Coq__Init__Specif.right Coq__Init__Logic.False + (Coq__Init__Logic.not Coq__Init__Logic.False) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.False + (H : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not Coq__Init__Logic.False) cupicef.I + (H : cupicef.Term cupicef.prop Coq__Init__Logic.False => H)))) + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.cast cupicef.set cupicef.set + (Coq__Init__Specif.sumbool (eq__nat n0 m0) + (Coq__Init__Logic.not (eq__nat n0 m0))) + (Coq__Init__Specif.sumbool + (eq__nat (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) + (Coq__Init__Logic.not + (eq__nat (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)))) + cupicef.I (IHn m0)) + m) + n. + +thm beq__nat__refl : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (Coq__Arith__PeanoNat.Nat__eqb n n)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__eqb n n) Coq__Init__Datatypes.true + (Coq__Arith__PeanoNat.Nat__eqb__refl n). + +def let__H2 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__eqb n0 m0) Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__eqb n0 m0) Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__eqb n0 m0) + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__eqb n0 m0) Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__eqb n0 m0) + Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__eqb n0 m0) + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__eqb n0 m0) + Coq__Init__Datatypes.true))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__eqb n0 m0) Coq__Init__Datatypes.true) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__eqb n0 m0) + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__eqb n0 m0) + Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__eqb n0 m0) Coq__Init__Datatypes.true)) => + __) + (Coq__Arith__PeanoNat.Nat__eqb__eq n0 m0). + +thm beq__nat__true : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__eqb n m) Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H2 n m n m. + +def let__H3 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__eqb x y) Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + x y)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__eqb x y) Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__eqb x y) + Coq__Init__Datatypes.false) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + x y))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + x y)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + x y)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__eqb x y) Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__eqb x y) + Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__eqb x y) + Coq__Init__Datatypes.false) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + x y))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + x y)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + x y)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__eqb x y) + Coq__Init__Datatypes.false))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__eqb x y) Coq__Init__Datatypes.false) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__eqb x y) + Coq__Init__Datatypes.false) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + x y))) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__eqb x y) + Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + x y))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + x y)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__eqb x y) Coq__Init__Datatypes.false)) => + __) + (Coq__Arith__PeanoNat.Nat__eqb__neq x y). + +thm beq__nat__false : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Arith__PeanoNat.Nat__eqb n m) Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H3 n m n m. + +def let__H4 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True Coq__Init__Logic.False e) + Coq__Init__Logic.I Coq__Init__Datatypes.false H. + +def let__H5 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (Coq__Arith__PeanoNat.Nat__eqb n0 m)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m)) -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (Coq__Arith__PeanoNat.Nat__eqb n0 m)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m)) => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True Coq__Init__Logic.False e) + Coq__Init__Logic.I Coq__Init__Datatypes.false H. + +def beq__nat__eq : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (Coq__Arith__PeanoNat.Nat__eqb n m)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__ind + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (Coq__Arith__PeanoNat.Nat__eqb n0 m)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Arith__PeanoNat.Nat__eqb n0 m)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m))) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.prop + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Arith__PeanoNat.Nat__eqb Coq__Init__Datatypes.O n0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Arith__PeanoNat.Nat__eqb Coq__Init__Datatypes.O n0)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O n0)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Arith__PeanoNat.Nat__eqb Coq__Init__Datatypes.O + Coq__Init__Datatypes.O)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Arith__PeanoNat.Nat__eqb Coq__Init__Datatypes.O + Coq__Init__Datatypes.O)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O)) + cupicef.I + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O)) + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Arith__PeanoNat.Nat__eqb Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S m0))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Arith__PeanoNat.Nat__eqb Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S m0))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m0))) + cupicef.I + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m0)) + (let__H4 n m m0 H))) + m) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Arith__PeanoNat.Nat__eqb n0 m)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m)) => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.prop + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Arith__PeanoNat.Nat__eqb (Coq__Init__Datatypes.S n0) n1)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Arith__PeanoNat.Nat__eqb (Coq__Init__Datatypes.S n0) + n1)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) n1)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Arith__PeanoNat.Nat__eqb (Coq__Init__Datatypes.S n0) + Coq__Init__Datatypes.O)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Arith__PeanoNat.Nat__eqb + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O)) + cupicef.I + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) + (let__H5 n n0 IHn m H))) + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Arith__PeanoNat.Nat__eqb n0 m0)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Arith__PeanoNat.Nat__eqb n0 m0)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) (Coq__Init__Datatypes.S m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Arith__PeanoNat.Nat__eqb (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Arith__PeanoNat.Nat__eqb + (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) (Coq__Init__Datatypes.S m0))) + cupicef.I + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Arith__PeanoNat.Nat__eqb n0 m0)) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 + (y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 y) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) (Coq__Init__Datatypes.S y)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0)) + m0 (IHn m0 H))) + m) + n. + +(; End of translation. ;) diff --git a/out/Coq__Arith__Factorial.dk b/out/Coq__Arith__Factorial.dk new file mode 100644 index 000000000..ccb237589 --- /dev/null +++ b/out/Coq__Arith__Factorial.dk @@ -0,0 +1,342 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def fact : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat)))) + (c => + c + (fact0 => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Nat.mul (Coq__Init__Datatypes.S n0) (fact0 n0)) + n)) + cupicef.0. + +thm lt__O__fact : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O (fact n)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__ind + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Peano.lt Coq__Init__Datatypes.O (fact n0)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Peano.lt Coq__Init__Datatypes.O + (fact Coq__Init__Datatypes.O)) + cupicef.I + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.le + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Peano.lt Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + cupicef.I + (Coq__Init__Peano.le__n + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)))) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O (fact n0)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O + (Coq__Init__Nat.add (fact n0) (Coq__Init__Nat.mul n0 (fact n0)))) + (Coq__Init__Peano.lt Coq__Init__Datatypes.O + (fact (Coq__Init__Datatypes.S n0))) + cupicef.I + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.le + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O) + (Coq__Init__Nat.add (fact n0) (Coq__Init__Nat.mul n0 (fact n0)))) + (Coq__Init__Peano.lt Coq__Init__Datatypes.O + (Coq__Init__Nat.add (fact n0) (Coq__Init__Nat.mul n0 (fact n0)))) + cupicef.I + (Coq__Arith__Plus.le__plus__trans + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O) (fact n0) + (Coq__Init__Nat.mul n0 (fact n0)) IHn))) + n. + +def let__H : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) => + Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) => + Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O))) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O))) => + __0) + (Coq__Arith__PeanoNat.Nat__neq__0__lt__0 n0). + +thm fact__neq__0 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (fact n) Coq__Init__Datatypes.O)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H n (fact n) (lt__O__fact n). + +thm fact__le : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le (fact n) (fact m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) => + Coq__Init__Peano.le__ind n + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Peano.le (fact n) (fact m0)) + (Coq__Init__Peano.le__n (fact n)) + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m0) => + IHle : + cupicef.Term cupicef.prop (Coq__Init__Peano.le (fact n) (fact m0)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.le (fact n) + (Coq__Init__Nat.add (fact m0) (Coq__Init__Nat.mul m0 (fact m0)))) + (Coq__Init__Peano.le (fact n) (fact (Coq__Init__Datatypes.S m0))) + cupicef.I + (Coq__Classes__RelationClasses.transitivity + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Peano.le + (Coq__Classes__RelationClasses.PreOrder__Transitive + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Peano.le Coq__Arith__PeanoNat.Nat__le__preorder) + (fact n) (fact m0) + (Coq__Init__Nat.add (fact m0) (Coq__Init__Nat.mul m0 (fact m0))) + IHle + (Coq__Arith__PeanoNat.Nat__le__add__r (fact m0) + (Coq__Init__Nat.mul m0 (fact m0))))) + m H. + +(; End of translation. ;) diff --git a/out/Coq__Arith__Gt.dk b/out/Coq__Arith__Gt.dk new file mode 100644 index 000000000..e2bcb94e8 --- /dev/null +++ b/out/Coq__Arith__Gt.dk @@ -0,0 +1,1422 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +thm gt__Sn__O : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n) Coq__Init__Datatypes.O) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Arith__PeanoNat.Nat__lt__0__succ n. + +thm gt__Sn__n : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n) n) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Arith__PeanoNat.Nat__lt__succ__diag__r n. + +def let__H : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) => + Coq__Init__Peano.lt n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) => + Coq__Init__Peano.lt n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) + (__ : + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => + __) + (Coq__Arith__PeanoNat.Nat__succ__lt__mono n0 m0). + +thm gt__n__S : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.gt n m) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n) + (Coq__Init__Datatypes.S m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H n m m n. + +def let__H0 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) => + Coq__Init__Peano.lt n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) => + Coq__Init__Peano.lt n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) => + Coq__Init__Peano.lt n0 m0)) + (__ : + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => + __0) + (Coq__Arith__PeanoNat.Nat__succ__lt__mono n0 m0). + +thm gt__S__n : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.gt (Coq__Init__Datatypes.S m) + (Coq__Init__Datatypes.S n)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.gt m n) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H0 n m n m. + +def let__H1 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n) m) -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n) m) => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) => + Coq__Init__Peano.le n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) => + Coq__Init__Peano.le n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0))) + (__ : + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => + __) + (Coq__Arith__PeanoNat.Nat__lt__eq__cases n0 m0). + +def let__H2 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n) m) -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n) m) => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) => + Coq__Init__Peano.le n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) => + Coq__Init__Peano.le n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) => + Coq__Init__Peano.le n0 m0)) + (__ : + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => + __0) + (Coq__Arith__PeanoNat.Nat__succ__le__mono n0 m0). + +thm gt__S : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n) m) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.gt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m n)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n) m) => + let__H1 n m H m n (let__H2 n m H m n H). + +def let__H3 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) => + Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0)) => + Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) => + Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 + (Coq__Arith__PeanoNat.Nat__pred m0)) => + Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) => + Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0))) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0)) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0)) => + __) + (Coq__Arith__PeanoNat.Nat__lt__succ__lt__pred n0 m0). + +thm gt__pred : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.gt m (Coq__Init__Datatypes.S n)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.gt (Coq__Init__Nat.pred m) n) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H3 n m n m. + +thm gt__irrefl : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Coq__Init__Peano.gt n n)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Arith__PeanoNat.Nat__lt__irrefl n. + +thm gt__asym : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.gt n m) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Coq__Init__Peano.gt m n)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Arith__PeanoNat.Nat__lt__asymm m n. + +def let__H4 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0)) => + Coq__Init__Peano.le n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0)) => + Coq__Init__Peano.le n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0))) + (__ : + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => + __) + (Coq__Arith__PeanoNat.Nat__le__ngt n0 m0). + +thm le__not__gt : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Coq__Init__Peano.gt n m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H4 n m n m. + +def let__H5 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) => + Coq__Init__Peano.lt n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) => + Coq__Init__Peano.lt n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0))) + (__ : + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => + __) + (Coq__Arith__PeanoNat.Nat__lt__nge n0 m0). + +thm gt__not__le : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.gt n m) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Coq__Init__Peano.le n m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H5 n m m n. + +def let__H6 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) => + Coq__Init__Peano.lt n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) => + Coq__Init__Peano.lt n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => + __0 : + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0)) => + __0) + (Coq__Arith__PeanoNat.Nat__le__succ__l n0 m0). + +thm le__S__gt : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n) m) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.gt m n) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H6 n m n m. + +def let__H7 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) => + Coq__Init__Peano.le n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) => + Coq__Init__Peano.le n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) => + Coq__Init__Peano.le n0 m0)) + (__ : + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => + __0) + (Coq__Arith__PeanoNat.Nat__succ__le__mono n0 m0). + +thm gt__S__le : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.gt (Coq__Init__Datatypes.S m) n) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H7 n m n m. + +def let__H8 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) => + Coq__Init__Peano.lt n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) => + Coq__Init__Peano.lt n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => + __0 : + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0)) => + __0) + (Coq__Arith__PeanoNat.Nat__le__succ__l n0 m0). + +thm gt__le__S : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.gt m n) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n) m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H8 n m n m. + +def let__H9 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) => + Coq__Init__Peano.le n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) => + Coq__Init__Peano.le n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) + (__ : + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => + __) + (Coq__Arith__PeanoNat.Nat__succ__le__mono n0 m0). + +thm le__gt__S : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.gt (Coq__Init__Datatypes.S m) n) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H9 n m n m. + +thm le__gt__trans : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n) -> + __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.gt m p) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.gt n p) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.gt m p) => + Coq__Arith__PeanoNat.Nat__lt__le__trans p m n H0 H. + +thm gt__le__trans : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.gt n m) -> + __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le p m) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.gt n p) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (Coq__Init__Peano.gt n m) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le p m) => + Coq__Arith__PeanoNat.Nat__le__lt__trans p m n H0 H. + +thm gt__trans : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.gt n m) -> + __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.gt m p) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.gt n p) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (Coq__Init__Peano.gt n m) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.gt m p) => + Coq__Arith__PeanoNat.Nat__lt__trans p m n H0 H. + +def let__H10 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n) m) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.gt m p) -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n) m) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.gt m p) => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) => + Coq__Init__Peano.le n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) => + Coq__Init__Peano.le n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) => + Coq__Init__Peano.le n0 m0)) + (__ : + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => + __0) + (Coq__Arith__PeanoNat.Nat__succ__le__mono n0 m0). + +thm gt__trans__S : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n) m) -> + __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.gt m p) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.gt n p) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n) m) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.gt m p) => + Coq__Arith__PeanoNat.Nat__lt__le__trans p m n H0 + (let__H10 n m p H H0 m n H). + +thm gt__0__eq : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.gt n Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O n)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.prop + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.or (Coq__Init__Peano.gt n0 Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O n0)) + (Coq__Init__Logic.or__intror + (Coq__Init__Peano.gt Coq__Init__Datatypes.O Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O)) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.or__introl + (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n0) + Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n0)) + (Coq__Arith__PeanoNat.Nat__lt__0__succ n0)) + n. + +def let__H11 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) => + Coq__Init__Peano.lt n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt + (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) => + Coq__Init__Peano.lt n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) => + Coq__Init__Peano.lt n0 m0)) + (__ : + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => + __0) + (Coq__Arith__PeanoNat.Nat__add__lt__mono__l n0 m0 p0). + +thm plus__gt__reg__l : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.gt (Coq__Init__Nat.add p n) (Coq__Init__Nat.add p m)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.gt n m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H11 n m p m n p. + +def let__H12 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) => + Coq__Init__Peano.lt n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt + (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) => + Coq__Init__Peano.lt n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0))) + (__ : + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => + __) + (Coq__Arith__PeanoNat.Nat__add__lt__mono__l n0 m0 p0). + +thm plus__gt__compat__l : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.gt n m) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.gt (Coq__Init__Nat.add p n) (Coq__Init__Nat.add p m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H12 n m p m n p. + +(; End of translation. ;) diff --git a/out/Coq__Arith__Le.dk b/out/Coq__Arith__Le.dk new file mode 100644 index 000000000..cbf0eb0ca --- /dev/null +++ b/out/Coq__Arith__Le.dk @@ -0,0 +1,319 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def let__H0 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop (Coq__Init__Peano.le n Coq__Init__Datatypes.O) -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 Coq__Init__Datatypes.O) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop (Coq__Init__Peano.le n Coq__Init__Datatypes.O) => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 Coq__Init__Datatypes.O) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le n0 Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O) => + Coq__Init__Peano.le n0 Coq__Init__Datatypes.O)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 Coq__Init__Datatypes.O) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le n0 Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O) => + Coq__Init__Peano.le n0 Coq__Init__Datatypes.O))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 Coq__Init__Datatypes.O) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le n0 Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le n0 Coq__Init__Datatypes.O) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le n0 Coq__Init__Datatypes.O)) => + __) + (Coq__Arith__PeanoNat.Nat__le__0__r n0). + +thm le__n__0__eq : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop (Coq__Init__Peano.le n Coq__Init__Datatypes.O) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O n) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop (Coq__Init__Peano.le n Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n Coq__Init__Datatypes.O (let__H0 n H n H). + +thm le__n__S : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n) + (Coq__Init__Datatypes.S m)) + +:= Coq__Init__Peano.le__n__S. + +thm le__S__n : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n) + (Coq__Init__Datatypes.S m)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) + +:= Coq__Init__Peano.le__S__n. + +thm le__Sn__le : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n) m) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) + +:= Coq__Arith__PeanoNat.Nat__lt__le__incl. + +thm le__elim__rel : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + __ : + (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (P Coq__Init__Datatypes.O p)) -> + __0 : + (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le p q) -> + __1 : cupicef.Term cupicef.prop (P p q) -> + cupicef.Term cupicef.prop + (P (Coq__Init__Datatypes.S p) (Coq__Init__Datatypes.S q))) -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> + cupicef.Term cupicef.prop (P n m) + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + H0 : + (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (P Coq__Init__Datatypes.O p)) => + HS : + (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le p q) -> + __0 : cupicef.Term cupicef.prop (P p q) -> + cupicef.Term cupicef.prop + (P (Coq__Init__Datatypes.S p) (Coq__Init__Datatypes.S q))) => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__ind + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m) => + P n0 m))) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le Coq__Init__Datatypes.O m) => + H0 m) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m) -> + cupicef.Term cupicef.prop (P n0 m)) => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Le : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m) => + Coq__Init__Peano.le__ind (Coq__Init__Datatypes.S n0) + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + P (Coq__Init__Datatypes.S n0) m0) + (HS n0 n0 (Coq__Init__Peano.le__n n0) + (IHn n0 (Coq__Init__Peano.le__n n0))) + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) => + H1 : cupicef.Term cupicef.prop (P (Coq__Init__Datatypes.S n0) m0) => + HS n0 m0 (le__Sn__le n0 m0 H) (IHn m0 (le__Sn__le n0 m0 H))) + m Le) + n. + +(; End of translation. ;) diff --git a/out/Coq__Arith__Lt.dk b/out/Coq__Arith__Lt.dk new file mode 100644 index 000000000..3af116c31 --- /dev/null +++ b/out/Coq__Arith__Lt.dk @@ -0,0 +1,1835 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def let__H : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) => + Coq__Init__Peano.lt n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) => + Coq__Init__Peano.lt n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => + __0 : + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0)) => + __0) + (Coq__Arith__PeanoNat.Nat__le__succ__l n0 m0). + +thm lt__le__S : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n) m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H n m n m. + +def let__H0 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) => + Coq__Init__Peano.le n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) => + Coq__Init__Peano.le n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) => + Coq__Init__Peano.le n0 m0)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => + __0 : + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0))) => + __) + (Coq__Arith__PeanoNat.Nat__lt__succ__r n0 m0). + +thm lt__n__Sm__le : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n (Coq__Init__Datatypes.S m)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H0 n m n m. + +def let__H1 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) => + Coq__Init__Peano.le n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) => + Coq__Init__Peano.le n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0))) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => + __0 : + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0))) => + __0) + (Coq__Arith__PeanoNat.Nat__lt__succ__r n0 m0). + +thm le__lt__n__Sm : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n (Coq__Init__Datatypes.S m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H1 n m n m. + +def let__H2 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0)) => + Coq__Init__Peano.le n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0)) => + Coq__Init__Peano.le n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0))) + (__ : + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => + __) + (Coq__Arith__PeanoNat.Nat__le__ngt n0 m0). + +thm le__not__lt : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Coq__Init__Peano.lt m n)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H2 n m n m. + +def let__H3 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) => + Coq__Init__Peano.lt n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) => + Coq__Init__Peano.lt n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0))) + (__ : + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => + __) + (Coq__Arith__PeanoNat.Nat__lt__nge n0 m0). + +thm lt__not__le : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Coq__Init__Peano.le m n)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H3 n m n m. + +def let__H4 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O n)) -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O n)) => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) => + Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) => + Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) => + Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O))) => + __) + (Coq__Arith__PeanoNat.Nat__neq__0__lt__0 n0). + +thm neq__0__lt : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O n)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O n)) => + let__H4 n H n + (Coq__Arith__PeanoNat.Nat__neq__sym Coq__Init__Datatypes.O n H). + +def let__H5 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n) -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n) => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) => + Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) => + Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O))) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O))) => + __0) + (Coq__Arith__PeanoNat.Nat__neq__0__lt__0 n0). + +thm lt__0__neq : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O n)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n) => + Coq__Arith__PeanoNat.Nat__neq__sym n Coq__Init__Datatypes.O + (let__H5 n H n H). + +def let__H6 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) => + Coq__Init__Peano.lt n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) => + Coq__Init__Peano.lt n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) + (__ : + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => + __) + (Coq__Arith__PeanoNat.Nat__succ__lt__mono n0 m0). + +thm lt__n__S : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n) + (Coq__Init__Datatypes.S m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H6 n m n m. + +def let__H7 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) => + Coq__Init__Peano.lt n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) => + Coq__Init__Peano.lt n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) => + Coq__Init__Peano.lt n0 m0)) + (__ : + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => + __0) + (Coq__Arith__PeanoNat.Nat__succ__lt__mono n0 m0). + +thm lt__S__n : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n) + (Coq__Init__Datatypes.S m)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H7 n m n m. + +thm S__pred : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n (Coq__Init__Datatypes.S (Coq__Init__Nat.pred n))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) => + Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S (Coq__Init__Nat.pred n)) n + (Coq__Arith__PeanoNat.Nat__lt__succ__pred m n H). + +thm S__pred__pos : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n (Coq__Init__Datatypes.S (Coq__Init__Nat.pred n))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + S__pred n Coq__Init__Datatypes.O. + +def let__H8 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) => + Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0)) => + Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) => + Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 + (Coq__Arith__PeanoNat.Nat__pred m0)) => + Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) => + Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0))) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0)) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0)) => + __) + (Coq__Arith__PeanoNat.Nat__lt__succ__lt__pred n0 m0). + +thm lt__pred : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n) m) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n (Coq__Init__Nat.pred m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H8 n m n m. + +def let__H9 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n) -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n) => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) => + Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) => + Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O))) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O)) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O))) => + __0) + (Coq__Arith__PeanoNat.Nat__neq__0__lt__0 n0). + +thm lt__pred__n__n : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt (Coq__Init__Nat.pred n) n) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n) => + Coq__Arith__PeanoNat.Nat__lt__pred__l n (let__H9 n H n H). + +def let__H10 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) => + Coq__Init__Peano.le n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) => + Coq__Init__Peano.le n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0))) + (__ : + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => + __) + (Coq__Arith__PeanoNat.Nat__lt__eq__cases n0 m0). + +thm le__lt__or__eq : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt n m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H10 n m n m. + +def let__H11 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Peano.lt m0 n0)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) => + Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Peano.lt m0 n0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Peano.lt m0 n0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Peano.lt m0 n0)) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) => + Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Peano.lt m0 n0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Peano.lt m0 n0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Peano.lt m0 n0)) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) => + Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Peano.lt m0 n0))) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Peano.lt m0 n0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) + (Coq__Init__Peano.lt m0 n0)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0))) => + __) + (Coq__Arith__PeanoNat.Nat__lt__gt__cases n0 m0). + +thm nat__total__order : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Peano.lt n m) (Coq__Init__Peano.lt m n)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H11 n m n m. + +(; End of translation. ;) diff --git a/out/Coq__Arith__Minus.dk b/out/Coq__Arith__Minus.dk new file mode 100644 index 000000000..c5dee9532 --- /dev/null +++ b/out/Coq__Arith__Minus.dk @@ -0,0 +1,796 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +thm minus__n__O : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n (Coq__Init__Nat.sub n Coq__Init__Datatypes.O)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.sub n Coq__Init__Datatypes.O) n + (Coq__Arith__PeanoNat.Nat__sub__0__r n). + +thm minus__Sn__m : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S (Coq__Init__Nat.sub n m)) + (Coq__Init__Nat.sub (Coq__Init__Datatypes.S n) m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n) => + Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.sub (Coq__Init__Datatypes.S n) m) + (Coq__Init__Datatypes.S (Coq__Init__Nat.sub n m)) + (Coq__Arith__PeanoNat.Nat__sub__succ__l m n H). + +thm pred__of__minus : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.pred n) + (Coq__Init__Nat.sub n (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.sub n (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Nat.pred n) (Coq__Arith__PeanoNat.Nat__sub__1__r n). + +thm minus__diag__reverse : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Nat.sub n n)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.sub n n) Coq__Init__Datatypes.O + (Coq__Arith__PeanoNat.Nat__sub__diag n). + +thm minus__plus__simpl__l__reverse : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.sub n m) + (Coq__Init__Nat.sub (Coq__Init__Nat.add p n) (Coq__Init__Nat.add p m))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq__ind__r + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__sub + (Coq__Arith__PeanoNat.Nat__sub (Coq__Init__Nat.add p n) p) m) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.sub n m) n0) + (Coq__Init__Logic.eq__ind__r + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__add n p) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.sub n m) + (Coq__Arith__PeanoNat.Nat__sub + (Coq__Arith__PeanoNat.Nat__sub n0 p) m)) + (Coq__Init__Logic.eq__ind__r + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.sub n m) (Coq__Arith__PeanoNat.Nat__sub n0 m)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__sub n m)) + (Coq__Arith__PeanoNat.Nat__sub (Coq__Arith__PeanoNat.Nat__add n p) + p) + (Coq__Arith__PeanoNat.Nat__add__sub n p)) + (Coq__Arith__PeanoNat.Nat__add p n) + (Coq__Arith__PeanoNat.Nat__add__comm p n)) + (Coq__Arith__PeanoNat.Nat__sub (Coq__Init__Nat.add p n) + (Coq__Arith__PeanoNat.Nat__add p m)) + (Coq__Arith__PeanoNat.Nat__sub__add__distr (Coq__Init__Nat.add p n) p m). + +thm plus__minus : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n (Coq__Init__Nat.add m p)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + p (Coq__Init__Nat.sub n m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n (Coq__Init__Nat.add m p)) => + Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.sub n m) p + (Coq__Arith__PeanoNat.Nat__add__sub__eq__l n m p + (Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n (Coq__Arith__PeanoNat.Nat__add m p) H)). + +thm minus__plus : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.sub (Coq__Init__Nat.add n m) n) m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq__ind__r + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__add m n) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.sub n0 n) m) + (Coq__Arith__PeanoNat.Nat__add__sub m n) + (Coq__Arith__PeanoNat.Nat__add n m) + (Coq__Arith__PeanoNat.Nat__add__comm n m). + +thm le__plus__minus__r : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add n (Coq__Init__Nat.sub m n)) m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq__ind__r + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__add (Coq__Init__Nat.sub m n) n) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n m) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m)) + (Coq__Arith__PeanoNat.Nat__sub__add n m) + (Coq__Arith__PeanoNat.Nat__add n (Coq__Init__Nat.sub m n)) + (Coq__Arith__PeanoNat.Nat__add__comm n (Coq__Init__Nat.sub m n)). + +thm le__plus__minus : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m (Coq__Init__Nat.add n (Coq__Init__Nat.sub m n))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) => + Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add n (Coq__Init__Nat.sub m n)) m + (Coq__Init__Logic.eq__ind__r + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__add (Coq__Init__Nat.sub m n) n) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m) + (Coq__Arith__PeanoNat.Nat__sub__add n m H) + (Coq__Arith__PeanoNat.Nat__add n (Coq__Init__Nat.sub m n)) + (Coq__Arith__PeanoNat.Nat__add__comm n (Coq__Init__Nat.sub m n))). + +def let__H : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__sub m0 p)) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p) m0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p) m0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p) m0) => + Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__sub m0 p))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__sub m0 p)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__sub m0 p)) => + Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p) m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p) m0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt + (Coq__Arith__PeanoNat.Nat__add n0 p) m0) => + Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__sub m0 p))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__sub m0 p)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 + (Coq__Arith__PeanoNat.Nat__sub m0 p)) => + Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p) m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__sub m0 p)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__sub m0 p)) => + Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p) m0)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p) m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__sub m0 p))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__sub m0 p)) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p) m0)) => + __0) + (Coq__Arith__PeanoNat.Nat__lt__add__lt__sub__r n0 m0 p). + +thm lt__O__minus__lt : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O (Coq__Init__Nat.sub n m)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H n m Coq__Init__Datatypes.O n m. + +def let__H0 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le m n)) -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__sub n0 m0) Coq__Init__Datatypes.O) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le m n)) => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__sub n0 m0) Coq__Init__Datatypes.O) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__sub n0 m0) Coq__Init__Datatypes.O) => + Coq__Init__Peano.le n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__sub n0 m0) Coq__Init__Datatypes.O)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__sub n0 m0) + Coq__Init__Datatypes.O) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__sub n0 m0) + Coq__Init__Datatypes.O) => + Coq__Init__Peano.le n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__sub n0 m0) + Coq__Init__Datatypes.O))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__sub n0 m0) Coq__Init__Datatypes.O)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__sub n0 m0) Coq__Init__Datatypes.O) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => + __0 : + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__sub n0 m0) Coq__Init__Datatypes.O)) => + __0) + (Coq__Arith__PeanoNat.Nat__sub__0__le n0 m0). + +def let__H1 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le m n)) -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le m n)) => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) => + Coq__Init__Peano.lt n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) => + Coq__Init__Peano.lt n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) => + Coq__Init__Peano.lt n0 m0)) + (__ : + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => + __0) + (Coq__Arith__PeanoNat.Nat__lt__nge n0 m0). + +thm not__le__minus__0 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le m n)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.sub n m) Coq__Init__Datatypes.O) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Peano.le m n)) => + let__H0 n m H n m + (Coq__Arith__PeanoNat.Nat__lt__le__incl n m (let__H1 n m H n m H)). + +(; End of translation. ;) diff --git a/out/Coq__Arith__Mult.dk b/out/Coq__Arith__Mult.dk new file mode 100644 index 000000000..16146079c --- /dev/null +++ b/out/Coq__Arith__Mult.dk @@ -0,0 +1,2942 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +thm mult__assoc__reverse : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.mul (Coq__Init__Nat.mul n m) p) + (Coq__Init__Nat.mul n (Coq__Init__Nat.mul m p))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.mul n (Coq__Init__Nat.mul m p)) + (Coq__Init__Nat.mul (Coq__Init__Nat.mul n m) p) + (Coq__Arith__PeanoNat.Nat__mul__assoc n m p). + +def let__H : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__mul n0 m0) Coq__Init__Datatypes.O) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m0 Coq__Init__Datatypes.O)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__mul n0 m0) Coq__Init__Datatypes.O) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__mul n0 m0) Coq__Init__Datatypes.O) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m0 Coq__Init__Datatypes.O))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m0 Coq__Init__Datatypes.O)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m0 Coq__Init__Datatypes.O)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__mul n0 m0) Coq__Init__Datatypes.O)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__mul n0 m0) + Coq__Init__Datatypes.O) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__mul n0 m0) + Coq__Init__Datatypes.O) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m0 Coq__Init__Datatypes.O))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m0 Coq__Init__Datatypes.O)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m0 Coq__Init__Datatypes.O)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__mul n0 m0) + Coq__Init__Datatypes.O))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__mul n0 m0) Coq__Init__Datatypes.O) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__mul n0 m0) Coq__Init__Datatypes.O) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m0 Coq__Init__Datatypes.O))) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__mul n0 m0) Coq__Init__Datatypes.O) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m0 Coq__Init__Datatypes.O))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m0 Coq__Init__Datatypes.O)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__mul n0 m0) Coq__Init__Datatypes.O)) => + __) + (Coq__Arith__PeanoNat.Nat__eq__mul__0 n0 m0). + +thm mult__is__O : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.mul n m) Coq__Init__Datatypes.O) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m Coq__Init__Datatypes.O)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H n m n m. + +def let__H0 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__mul n0 m0) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__mul n0 m0) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__mul n0 m0) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__mul n0 m0) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__mul n0 m0) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__mul n0 m0) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__mul n0 m0) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__mul n0 m0) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__mul n0 m0) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)))) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__mul n0 m0) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__mul n0 m0) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) => + __) + (Coq__Arith__PeanoNat.Nat__eq__mul__1 n0 m0). + +thm mult__is__one : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.mul n m) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H0 n m n m. + +thm mult__O__le : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m Coq__Init__Datatypes.O) + (Coq__Init__Peano.le n (Coq__Init__Nat.mul m n))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.prop + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O) + (Coq__Init__Peano.le n (Coq__Init__Nat.mul n0 n))) + (Coq__Init__Logic.or__introl + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O) + (Coq__Init__Peano.le n (Coq__Init__Nat.mul Coq__Init__Datatypes.O n)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O))) + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.or__intror + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) Coq__Init__Datatypes.O) + (Coq__Init__Peano.le n + (Coq__Init__Nat.mul (Coq__Init__Datatypes.S m0) n)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.le n + (Coq__Arith__PeanoNat.Nat__add n (Coq__Init__Nat.mul m0 n))) + (Coq__Init__Peano.le n + (Coq__Init__Nat.mul (Coq__Init__Datatypes.S m0) n)) + cupicef.I + (Coq__Arith__PeanoNat.Nat__le__add__r n (Coq__Init__Nat.mul m0 n)))) + m. + +thm mult__le__compat__l : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Nat.mul p n) (Coq__Init__Nat.mul p m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Arith__PeanoNat.Nat__mul__le__mono__nonneg__l n m p + (Coq__Arith__PeanoNat.Nat__le__0__l p). + +thm mult__le__compat__r : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Nat.mul n p) (Coq__Init__Nat.mul m p)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Arith__PeanoNat.Nat__mul__le__mono__nonneg__r n m p + (Coq__Arith__PeanoNat.Nat__le__0__l p). + +thm mult__le__compat : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> + __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le p q) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Nat.mul n p) (Coq__Init__Nat.mul m q)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le p q) => + Coq__Arith__PeanoNat.Nat__mul__le__mono__nonneg n m p q + (Coq__Arith__PeanoNat.Nat__le__0__l n) H + (Coq__Arith__PeanoNat.Nat__le__0__l p) H0. + +def let__H1 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O p0) -> + __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O p0) => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0)) => + Coq__Init__Peano.lt n0 m0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt + (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0)) => + Coq__Init__Peano.lt n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__1 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0))) + (__0 : + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0))) => + __1 : + (__1 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => + __0) + (Coq__Arith__PeanoNat.Nat__mul__lt__mono__pos__l p0 n0 m0 __). + +thm mult__S__lt__compat__l : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m p) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Nat.mul (Coq__Init__Datatypes.S n) m) + (Coq__Init__Nat.mul (Coq__Init__Datatypes.S n) p)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H1 n m p (Coq__Init__Datatypes.S n) m p + (Coq__Arith__PeanoNat.Nat__lt__0__succ n). + +def let__H2 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) -> + H0 : + cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O p) -> + p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O p0) -> + __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) => + H0 : + cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O p) => + p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O p0) => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0)) => + Coq__Init__Peano.lt n0 m0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt + (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0)) => + Coq__Init__Peano.lt n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__1 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0))) + (__0 : + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0))) => + __1 : + (__1 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => + __0) + (Coq__Arith__PeanoNat.Nat__mul__lt__mono__pos__l p0 n0 m0 __). + +thm mult__lt__compat__l : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) -> + __0 : + cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O p) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Nat.mul p n) (Coq__Init__Nat.mul p m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) => + H0 : + cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O p) => + let__H2 n m p H H0 p n m H0 H. + +def let__H3 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) -> + H0 : + cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O p) -> + p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O p0) -> + __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul n0 p0) + (Coq__Arith__PeanoNat.Nat__mul m0 p0)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) => + H0 : + cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O p) => + p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O p0) => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul n0 p0) + (Coq__Arith__PeanoNat.Nat__mul m0 p0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul n0 p0) + (Coq__Arith__PeanoNat.Nat__mul m0 p0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul n0 p0) + (Coq__Arith__PeanoNat.Nat__mul m0 p0)) => + Coq__Init__Peano.lt n0 m0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul n0 p0) + (Coq__Arith__PeanoNat.Nat__mul m0 p0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul n0 p0) + (Coq__Arith__PeanoNat.Nat__mul m0 p0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt + (Coq__Arith__PeanoNat.Nat__mul n0 p0) + (Coq__Arith__PeanoNat.Nat__mul m0 p0)) => + Coq__Init__Peano.lt n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__1 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul n0 p0) + (Coq__Arith__PeanoNat.Nat__mul m0 p0))) + (__0 : + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul n0 p0) + (Coq__Arith__PeanoNat.Nat__mul m0 p0))) => + __1 : + (__1 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul n0 p0) + (Coq__Arith__PeanoNat.Nat__mul m0 p0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => + __0) + (Coq__Arith__PeanoNat.Nat__mul__lt__mono__pos__r p0 n0 m0 __). + +thm mult__lt__compat__r : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) -> + __0 : + cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O p) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Nat.mul n p) (Coq__Init__Nat.mul m p)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) => + H0 : + cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O p) => + let__H3 n m p H H0 p n m H0 H. + +def let__H4 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O p0) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt Coq__Init__Datatypes.O p0) => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0)) => + Coq__Init__Peano.le n0 m0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le + (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0)) => + Coq__Init__Peano.le n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0)) + (__1 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0)) => + Coq__Init__Peano.le n0 m0)) + (__0 : + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0))) => + __1 : + (__1 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__mul p0 n0) + (Coq__Arith__PeanoNat.Nat__mul p0 m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => + __1) + (Coq__Arith__PeanoNat.Nat__mul__le__mono__pos__l n0 m0 p0 __). + +thm mult__S__le__reg__l : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Nat.mul (Coq__Init__Datatypes.S n) m) + (Coq__Init__Nat.mul (Coq__Init__Datatypes.S n) p)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le m p) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H4 n m p m p (Coq__Init__Datatypes.S n) + (Coq__Arith__PeanoNat.Nat__lt__0__succ n). + +thm odd__even__lem : + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add + (Coq__Init__Nat.mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + p) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Nat.mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + q))) + +:= p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add + (Coq__Init__Nat.mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + p) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Nat.mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + q)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add + (Coq__Init__Nat.mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + p) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Nat.mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + q)) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add + (Coq__Init__Nat.mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + p) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Nat.mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + q))) + cupicef.I + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add + (Coq__Init__Nat.mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + p) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Nat.mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + q)) => + Coq__Arith__PeanoNat.Nat__Even__Odd__False + (Coq__Init__Nat.mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + q) + (Coq__Init__Logic.ex__intro + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + q) + (Coq__Init__Nat.mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + m)) + q + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + q))) + (Coq__Init__Logic.ex__intro + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + q) + (Coq__Init__Nat.add + (Coq__Init__Nat.mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + m) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) + p + (Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add + (Coq__Init__Nat.mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + p) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Nat.mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + q) + H))). + +def mult__acc : + s : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.2 cupicef.set + (cupicef.prod cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set))) + cupicef.I Coq__Init__Datatypes.nat + (s : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set + (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + cupicef.I Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat)))))) + (c => + c + (mult__acc0 => + s : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + s + (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + mult__acc0 (Coq__Arith__Plus.tail__plus m s) m p) + n)) + cupicef.0. + +def let__H5 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add m (Coq__Init__Nat.mul n0 p)) (mult__acc m p n0))) -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 n0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add m (Coq__Init__Nat.mul n0 p)) + (mult__acc m p n0))) => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0. + +def let__H6 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add m (Coq__Init__Nat.mul n0 p)) (mult__acc m p n0))) -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + p p) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add m (Coq__Init__Nat.mul n0 p)) + (mult__acc m p n0))) => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + p. + +def let__H7 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add m (Coq__Init__Nat.mul n0 p)) (mult__acc m p n0))) -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__add m p) (Coq__Arith__Plus.tail__plus p m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add m (Coq__Init__Nat.mul n0 p)) + (mult__acc m p n0))) => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq__ind__r + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__add p m) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Arith__Plus.tail__plus p m)) + (Coq__Arith__Plus.plus__tail__plus p m) + (Coq__Arith__PeanoNat.Nat__add m p) + (Coq__Arith__PeanoNat.Nat__add__comm m p). + +thm mult__acc__aux : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add m (Coq__Init__Nat.mul n p)) (mult__acc m p n)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__ind + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set (cupicef.rule cupicef.set cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.set cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I + Coq__Init__Datatypes.nat + (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add m (Coq__Init__Nat.mul n0 p)) + (mult__acc m p n0)))) + (cupicef.cast + (cupicef.rule cupicef.set (cupicef.rule cupicef.set cupicef.prop)) + (cupicef.rule cupicef.set (cupicef.rule cupicef.set cupicef.prop)) + (cupicef.prod cupicef.set (cupicef.rule cupicef.set cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.set cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I + Coq__Init__Datatypes.nat + (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add m Coq__Init__Datatypes.O) m))) + (cupicef.prod cupicef.set (cupicef.rule cupicef.set cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.set cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I + Coq__Init__Datatypes.nat + (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add m + (Coq__Init__Nat.mul Coq__Init__Datatypes.O p)) + (mult__acc m p Coq__Init__Datatypes.O)))) + cupicef.I + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m (Coq__Init__Nat.add m Coq__Init__Datatypes.O) + (Coq__Init__Peano.plus__n__O m))) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add m (Coq__Init__Nat.mul n0 p)) + (mult__acc m p n0))) => + cupicef.cast + (cupicef.rule cupicef.set (cupicef.rule cupicef.set cupicef.prop)) + (cupicef.rule cupicef.set (cupicef.rule cupicef.set cupicef.prop)) + (cupicef.prod cupicef.set (cupicef.rule cupicef.set cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.set cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I + Coq__Init__Datatypes.nat + (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (mult__acc (Coq__Arith__Plus.tail__plus p m) p n0)) + (Coq__Arith__PeanoNat.Nat__add m + (Coq__Arith__PeanoNat.Nat__add p + (Coq__Init__Nat.mul n0 p)))))) + (cupicef.prod cupicef.set (cupicef.rule cupicef.set cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.set cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I + Coq__Init__Datatypes.nat + (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add m + (Coq__Init__Nat.mul (Coq__Init__Datatypes.S n0) p)) + (mult__acc m p (Coq__Init__Datatypes.S n0))))) + cupicef.I + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq__ind__r + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__add (Coq__Arith__PeanoNat.Nat__add m p) + (Coq__Init__Nat.mul n0 p)) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (mult__acc (Coq__Arith__Plus.tail__plus p m) p n0)) + (Coq__Init__Logic.eq__ind__r + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (mult__acc (Coq__Arith__PeanoNat.Nat__add m p) p n0) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (mult__acc (Coq__Arith__Plus.tail__plus p m) p n0)) + ((H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__add m p) + (Coq__Arith__Plus.tail__plus p m)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + p p) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 n0) => + Coq__Init__Logic.eq__trans + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (mult__acc (Coq__Init__Nat.add m p) p n0) + (mult__acc (Coq__Arith__Plus.tail__plus p m) p n0) + (mult__acc (Coq__Arith__Plus.tail__plus p m) p n0) + (Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (n1 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat))) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (f : + (n1 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat) => + f n0) + (mult__acc (Coq__Init__Nat.add m p) p) + (mult__acc (Coq__Arith__Plus.tail__plus p m) p) + (Coq__Init__Logic.eq__trans + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) + cupicef.I Coq__Init__Datatypes.nat + (n1 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat))) + (mult__acc (Coq__Init__Nat.add m p) p) + (mult__acc (Coq__Arith__Plus.tail__plus p m) p) + (mult__acc (Coq__Arith__Plus.tail__plus p m) p) + (Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (cupicef.prod cupicef.set + (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + cupicef.I Coq__Init__Datatypes.nat + (m0 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) + cupicef.I Coq__Init__Datatypes.nat + (n1 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat)))) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) + cupicef.I Coq__Init__Datatypes.nat + (n1 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat))) + (f : + (m0 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat -> + n1 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat) => + f p) + (mult__acc (Coq__Init__Nat.add m p)) + (mult__acc (Coq__Arith__Plus.tail__plus p m)) + (Coq__Init__Logic.eq__trans + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (cupicef.prod cupicef.set + (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + cupicef.I Coq__Init__Datatypes.nat + (m0 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) + cupicef.I Coq__Init__Datatypes.nat + (n1 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat)))) + (mult__acc (Coq__Init__Nat.add m p)) + (mult__acc (Coq__Init__Nat.add m p)) + (mult__acc (Coq__Arith__Plus.tail__plus p m)) + (Coq__Init__Logic.f__equal + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (cupicef.prod cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + cupicef.set)) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + cupicef.set))) + cupicef.I Coq__Init__Datatypes.nat + (s : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set + (cupicef.rule cupicef.set + cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + cupicef.set)) + cupicef.I Coq__Init__Datatypes.nat + (m0 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set + cupicef.set + (cupicef.rule cupicef.set + cupicef.set) + cupicef.I + Coq__Init__Datatypes.nat + (n1 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat))))) + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (cupicef.prod cupicef.set + (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + cupicef.set)) + cupicef.I Coq__Init__Datatypes.nat + (m0 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set + cupicef.set) + cupicef.I Coq__Init__Datatypes.nat + (n1 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat)))) + (f : + (s : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat -> + m0 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat -> + n1 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat) => + f (Coq__Init__Nat.add m p)) + mult__acc mult__acc + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (cupicef.prod cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + cupicef.set)) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + cupicef.set))) + cupicef.I Coq__Init__Datatypes.nat + (s : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set + (cupicef.rule cupicef.set + cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + cupicef.set)) + cupicef.I + Coq__Init__Datatypes.nat + (m0 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set + cupicef.set + (cupicef.rule cupicef.set + cupicef.set) + cupicef.I + Coq__Init__Datatypes.nat + (n1 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat))))) + mult__acc)) + (Coq__Init__Logic.f__equal + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (cupicef.prod cupicef.set + (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + cupicef.set)) + cupicef.I Coq__Init__Datatypes.nat + (m0 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set + cupicef.set) + cupicef.I Coq__Init__Datatypes.nat + (n1 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat)))) + mult__acc (Coq__Init__Nat.add m p) + (Coq__Arith__Plus.tail__plus p m) H0))) + (Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) + cupicef.I Coq__Init__Datatypes.nat + (n1 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat))) + (mult__acc (Coq__Arith__Plus.tail__plus p m)) p p + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + p)))) + (Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (mult__acc (Coq__Arith__Plus.tail__plus p m) p) n0 n0 + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0))) + (let__H7 n n0 IHn m p) (let__H6 n n0 IHn m p) + (let__H5 n n0 IHn m p)) + (Coq__Init__Nat.add (Coq__Arith__PeanoNat.Nat__add m p) + (Coq__Init__Nat.mul n0 p)) + (IHn (Coq__Arith__PeanoNat.Nat__add m p) p)) + (Coq__Arith__PeanoNat.Nat__add m + (Coq__Arith__PeanoNat.Nat__add p (Coq__Init__Nat.mul n0 p))) + (Coq__Arith__PeanoNat.Nat__add__assoc m p + (Coq__Init__Nat.mul n0 p)))) + n. + +def tail__mult : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + mult__acc Coq__Init__Datatypes.O m n. + +thm mult__tail__mult : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.mul n m) (tail__mult n m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.cast cupicef.prop cupicef.prop + ((n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.mul n m) n0) + (mult__acc Coq__Init__Datatypes.O m n)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.mul n m) (tail__mult n m)) + cupicef.I + (Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add Coq__Init__Datatypes.O (Coq__Init__Nat.mul n m)) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.mul n m) n0) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add Coq__Init__Datatypes.O + (Coq__Init__Nat.mul n m))) + (mult__acc Coq__Init__Datatypes.O m n) + (mult__acc__aux n Coq__Init__Datatypes.O m)). + +(; End of translation. ;) diff --git a/out/Coq__Arith__Peano__dec.dk b/out/Coq__Arith__Peano__dec.dk new file mode 100644 index 000000000..c6f7978f7 --- /dev/null +++ b/out/Coq__Arith__Peano__dec.dk @@ -0,0 +1,6849 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def O__or__S : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumor cupicef.set + (Coq__Init__Specif.sig cupicef.set Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m) n)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O n)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__rec + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Specif.sumor cupicef.set + (Coq__Init__Specif.sig cupicef.set Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m) n0)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O n0)) + (Coq__Init__Specif.inright + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (Coq__Init__Specif.sig cupicef.set Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m) Coq__Init__Datatypes.O))) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O)) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumor cupicef.set + (Coq__Init__Specif.sig cupicef.set Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m) n0)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O n0)) => + Coq__Init__Specif.inleft + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (Coq__Init__Specif.sig cupicef.set Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m) (Coq__Init__Datatypes.S n0)))) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n0)) + (Coq__Init__Specif.exist + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m) (Coq__Init__Datatypes.S n0)) + n0 + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0)))) + n. + +def dec__eq__nat : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Logic__Decidable.decidable + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Specif.sumbool__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) + (s : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m))) => + Coq__Logic__Decidable.decidable + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) + (a : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) => + Coq__Init__Logic.or__introl + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) + a) + (b : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) => + Coq__Init__Logic.or__intror + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) + b) + (Coq__Arith__PeanoNat.Nat__eq__dec n m). + +def UIP__nat : + x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + x y) -> + p2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + x y)) + p1 p2) + +:= Coq__Logic__Eqdep__dec.UIP__dec + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Arith__PeanoNat.Nat__eq__dec. + +def let__Coq__Arith__Peano__dec__internal__eq__rew__dep : + A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) -> + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + P : + (a : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + e : cupicef.Term cupicef.prop (Coq__Init__Logic.eq A x a) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x (Coq__Init__Logic.eq__refl A x)) -> + y : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + e : cupicef.Term cupicef.prop (Coq__Init__Logic.eq A x y) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y e) + +:= A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) => + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + P : + (a : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + e : cupicef.Term cupicef.prop (Coq__Init__Logic.eq A x a) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x (Coq__Init__Logic.eq__refl A x)) => + y : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + e : cupicef.Term cupicef.prop (Coq__Init__Logic.eq A x y) => + Coq__Init__Logic.match____eq (cupicef.type (cupicef.s cupicef.z)) A x + (y0 : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + e0 : cupicef.Term cupicef.prop (Coq__Init__Logic.eq A x y0) => + P y0 e0) + f y e. + +def let__n0 : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.S n. + +def let__H0 : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n1)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n1)) => + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat (cupicef.type (cupicef.s cupicef.z)) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.False) + e) + Coq__Init__Logic.I (Coq__Init__Datatypes.S n1) H. + +def let__H1 : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) -> + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) => + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) => + Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + n1 (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => n0) e) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2) H. + +def let__def__n2 : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) -> + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n1) + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) => + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) => + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1. + +def let__def__n3 : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) -> + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + n3 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn3 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) -> + le__mn4 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + n4 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n4) -> + IHn1 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n4 (Coq__Init__Datatypes.S n0)) -> + le__mn5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn6 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn5 le__mn6)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n4 n4) + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) => + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) => + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + n3 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn3 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) => + le__mn4 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + n4 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n4) => + IHn1 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n4 (Coq__Init__Datatypes.S n0)) -> + le__mn5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn6 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn5 le__mn6)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n4. + +def let__def__n4 : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) -> + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + n3 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn3 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) -> + le__mn4 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + IHn1 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m (Coq__Init__Datatypes.S n0)) -> + le__mn5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn6 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn5 le__mn6)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m m) + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) => + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) => + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + n3 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn3 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) => + le__mn4 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + IHn1 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m (Coq__Init__Datatypes.S n0)) -> + le__mn5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn6 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn5 le__mn6)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m. + +def let__def__n5 : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) -> + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + n3 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn3 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) -> + le__mn4 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m0) -> + IHn1 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) (Coq__Init__Datatypes.S n0)) -> + le__mn6 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn6 le__mn7)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) (Coq__Init__Datatypes.S m0)) + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) => + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) => + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + n3 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn3 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) => + le__mn4 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m0) => + IHn1 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) (Coq__Init__Datatypes.S n0)) -> + le__mn6 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn6 le__mn7)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0). + +def let__f : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) -> + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + n3 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn3 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) -> + le__mn4 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m0) -> + IHn1 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) (Coq__Init__Datatypes.S n0)) -> + le__mn6 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn6 le__mn7)) -> + def__n0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) m) -> + le__mn0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S m0) m0) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) => + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) => + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + n3 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn3 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) => + le__mn4 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m0) => + IHn1 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) (Coq__Init__Datatypes.S n0)) -> + le__mn6 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn6 le__mn7)) => + def__n0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) m) => + le__mn0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S m0) m0) => + Coq__Arith__PeanoNat.Nat__nle__succ__diag__l m0 le__mn0. + +def let__def__n6 : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) -> + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + n3 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn3 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn4 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m0) -> + le__mn5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + n4 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n4) -> + IHn1 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n4 (Coq__Init__Datatypes.S n0)) -> + le__mn6 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn6 le__mn7)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n4 n4) + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) => + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) => + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + n3 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn3 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn4 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m0) => + le__mn5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + n4 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n4) => + IHn1 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n4 (Coq__Init__Datatypes.S n0)) -> + le__mn6 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn6 le__mn7)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n4. + +def let__def__n7 : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) -> + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + n3 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn3 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn4 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m0) -> + le__mn5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + IHn1 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m (Coq__Init__Datatypes.S n0)) -> + le__mn6 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn6 le__mn7)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m m) + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) => + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) => + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + n3 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn3 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn4 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m0) => + le__mn5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + IHn1 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m (Coq__Init__Datatypes.S n0)) -> + le__mn6 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn6 le__mn7)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m. + +def let__f0 : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) -> + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + n3 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn3 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn4 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m0) -> + le__mn5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + IHn1 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m (Coq__Init__Datatypes.S n0)) -> + le__mn6 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn6 le__mn7)) -> + def__n0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m (Coq__Init__Datatypes.S m0)) -> + le__mn0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S m0) m0) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) => + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) => + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + n3 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn3 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn4 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m0) => + le__mn5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + IHn1 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m (Coq__Init__Datatypes.S n0)) -> + le__mn6 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn6 le__mn7)) => + def__n0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m (Coq__Init__Datatypes.S m0)) => + le__mn0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S m0) m0) => + Coq__Arith__PeanoNat.Nat__nle__succ__diag__l m0 le__mn0. + +def let__def__n8 : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) -> + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + n3 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn3 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn4 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m0) -> + le__mn5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn6 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m1) -> + IHn1 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S n0)) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn8 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn7 le__mn8)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S m1)) + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) => + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) => + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + n3 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn3 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn4 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m0) => + le__mn5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn6 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m1) => + IHn1 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S n0)) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn8 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn7 le__mn8)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1). + +def let__H : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) -> + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + n3 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn3 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn4 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m0) -> + le__mn5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn6 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m1) -> + IHn1 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S n0)) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn8 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn7 le__mn8)) -> + def__n0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S m0)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m1 m0) + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) => + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) => + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + n3 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn3 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn4 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m0) => + le__mn5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn6 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m1) => + IHn1 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S n0)) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn8 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn7 le__mn8)) => + def__n0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S m0)) => + Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + m1 (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => n0) e) + (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S m0) def__n0. + +def let__def__n9 : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) -> + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + n3 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn3 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn4 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m0) -> + le__mn5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn6 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m1) -> + IHn1 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S n0)) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn8 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn7 le__mn8)) -> + def__n0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S m0)) -> + H3 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m1 m0) -> + m2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn2 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m2) (Coq__Init__Datatypes.S n0)) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn8 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn7 le__mn8)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m2) (Coq__Init__Datatypes.S m2)) + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) => + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) => + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + n3 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn3 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn4 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m0) => + le__mn5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn6 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m1) => + IHn1 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S n0)) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn8 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn7 le__mn8)) => + def__n0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S m0)) => + H3 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m1 m0) => + m2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn2 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m2) (Coq__Init__Datatypes.S n0)) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn8 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn7 le__mn8)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m2). + +def let__def__n10 : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) -> + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + n3 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn3 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn4 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m0) -> + le__mn5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn6 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m1) -> + IHn1 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S n0)) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn8 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn7 le__mn8)) -> + def__n0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S m0)) -> + H3 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m1 m0) -> + IHn2 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) (Coq__Init__Datatypes.S n0)) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn8 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn7 le__mn8)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) (Coq__Init__Datatypes.S m0)) + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) => + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) => + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + n3 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn3 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn4 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m0) => + le__mn5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn6 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m1) => + IHn1 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S n0)) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn8 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn7 le__mn8)) => + def__n0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S m0)) => + H3 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m1 m0) => + IHn2 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) (Coq__Init__Datatypes.S n0)) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn8 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn7 le__mn8)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0). + +def let__H2 : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) -> + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + n3 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn3 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn4 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m0) -> + le__mn5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) -> + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn6 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m1) -> + IHn1 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S n0)) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn8 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn7 le__mn8)) -> + def__n0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S m0)) -> + H3 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m1 m0) -> + IHn2 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) (Coq__Init__Datatypes.S n0)) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn8 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn7 le__mn8)) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m0) -> + def__n1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) (Coq__Init__Datatypes.S m0)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m m0)) + le__mn4 le__mn7) + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn0 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n0)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn1 le__mn2)) => + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) => + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + n3 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn3 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn4 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m0) => + le__mn5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn6 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m1) => + IHn1 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S n0)) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn8 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn7 le__mn8)) => + def__n0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S m0)) => + H3 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m1 m0) => + IHn2 : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) (Coq__Init__Datatypes.S n0)) -> + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + le__mn8 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n0)) + le__mn7 le__mn8)) => + le__mn7 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m m0) => + def__n1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) (Coq__Init__Datatypes.S m0)) => + IHn2 m0 def__n1 le__mn4 le__mn7. + +thm le__unique : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n)) + le__mn1 le__mn2) + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__ind + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I Coq__Init__Datatypes.nat + (n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n2)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n2)) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (Coq__Init__Peano.le m n2) + (le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n2) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le m n2) + (le__mn2 : + cupicef.Term cupicef.prop (Coq__Init__Peano.le m n2) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n2)) + le__mn1 le__mn2))))) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n1)) => + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n1)) + le__mn1 le__mn2) + (let__H0 m n n1 H le__mn1 le__mn2)) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn0 : + (n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Datatypes.S n2)) -> + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n2) -> + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n2) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n2)) + le__mn1 le__mn2)) => + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n2)) => + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n2) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 + (n3 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n3)) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n3)) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (Coq__Init__Peano.le m n3) + (le__mn1 : + cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le m n3) + (le__mn2 : + cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n3)) + le__mn1 le__mn2)))) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n1) (Coq__Init__Datatypes.S n1)) => + le__mn1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + le__mn2 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + Coq__Init__Logic.eq__trans + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n1)) + le__mn1 + (Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Peano.le m) le__mn2 n1 + (let__def__n2 m n n1 IHn0 n2 H H1 H2 le__mn1 le__mn2)) + le__mn2 + ((n3 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn3 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n3) => + Coq__Init__Peano.match____le cupicef.prop m + (n4 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n4) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (Coq__Init__Peano.le m n1) + (le__mn4 : + cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n4) + (def__n3 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n4) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n4)) + l + (Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 (Coq__Init__Peano.le m) le__mn4 n4 def__n3)))) + (le__mn4 : + cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + Coq__Init__Peano.match____le cupicef.prop m + (n4 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n4) => + cupicef.prod + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I Coq__Init__Datatypes.nat + (n5 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n4 (Coq__Init__Datatypes.S n5)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n4 (Coq__Init__Datatypes.S n5)) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (Coq__Init__Peano.le m n5) + (le__mn5 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le m n5) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (Coq__Init__Peano.le m n5) + (le__mn6 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le m n5) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n5)) + le__mn5 le__mn6))))) + (IHn1 : + (n5 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n4 (Coq__Init__Datatypes.S n5)) -> + le__mn5 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le m n5) -> + le__mn6 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le m n5) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n5)) + le__mn5 le__mn6)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n4 m) + (def__n3 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n4 m) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m m)) + (Coq__Init__Peano.le__n m) + (Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n4 (Coq__Init__Peano.le m) l m def__n3)))) + (IHn1 : + (n4 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m (Coq__Init__Datatypes.S n4)) -> + le__mn5 : + cupicef.Term cupicef.prop (Coq__Init__Peano.le m n4) -> + le__mn6 : + cupicef.Term cupicef.prop (Coq__Init__Peano.le m n4) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n4)) + le__mn5 le__mn6)) => + def__n0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m m) => + Coq__Init__Logic.eq__ind__r + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m m)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m) + (def__n1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m m) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m m)) + (Coq__Init__Peano.le__n m) + (Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m (Coq__Init__Peano.le m) + (Coq__Init__Peano.le__n m) m def__n1)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m m)) + (Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m (Coq__Init__Peano.le m) + (Coq__Init__Peano.le__n m) m + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m))) + def__n0 + (UIP__nat m m def__n0 + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m))) + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn5 : + cupicef.Term cupicef.prop (Coq__Init__Peano.le m m0) => + IHn1 : + (n4 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) + (Coq__Init__Datatypes.S n4)) -> + le__mn6 : + cupicef.Term cupicef.prop (Coq__Init__Peano.le m n4) -> + le__mn7 : + cupicef.Term cupicef.prop (Coq__Init__Peano.le m n4) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n4)) + le__mn6 le__mn7)) => + def__n0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) m) => + let__Coq__Arith__Peano__dec__internal__eq__rew__dep + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) + (cupicef.cast + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I Coq__Init__Datatypes.nat + (m1 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) m1) + (def__n1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) m1) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I Coq__Init__Datatypes.nat + (a : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) a) + (e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) a) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I))) + cupicef.I + (m1 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + def__n1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) m1) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (Coq__Init__Peano.le m1 m0) + (le__mn6 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le m1 m0) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m1 m1)) + (Coq__Init__Peano.le__n m1) + (Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) + (Coq__Init__Peano.le m1) + (Coq__Init__Peano.le__S m1 m0 le__mn6) m1 + def__n1)))) + (le__mn0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S m0) + m0) => + Coq__Init__Logic.match____False cupicef.prop + (__ : + cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Peano.le + (Coq__Init__Datatypes.S m0) + (Coq__Init__Datatypes.S m0))) + (Coq__Init__Peano.le__n + (Coq__Init__Datatypes.S m0)) + (Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) + (Coq__Init__Peano.le + (Coq__Init__Datatypes.S m0)) + (Coq__Init__Peano.le__S + (Coq__Init__Datatypes.S m0) m0 le__mn0) + (Coq__Init__Datatypes.S m0) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0)))) + (let__f m n n1 IHn0 n2 H H1 H2 le__mn1 le__mn2 n3 + le__mn3 le__mn4 m0 le__mn5 IHn1 def__n0 le__mn0)) + m def__n0 le__mn5) + n1 le__mn4 IHn0) + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn4 : + cupicef.Term cupicef.prop (Coq__Init__Peano.le m m0) => + le__mn5 : + cupicef.Term cupicef.prop (Coq__Init__Peano.le m n1) => + Coq__Init__Peano.match____le cupicef.prop m + (n4 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n4) => + cupicef.prod + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I Coq__Init__Datatypes.nat + (n5 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n4 (Coq__Init__Datatypes.S n5)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n4 (Coq__Init__Datatypes.S n5)) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (Coq__Init__Peano.le m n5) + (le__mn6 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le m n5) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (Coq__Init__Peano.le m n5) + (le__mn7 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le m n5) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n5)) + le__mn6 le__mn7))))) + (IHn1 : + (n5 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n4 (Coq__Init__Datatypes.S n5)) -> + le__mn6 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le m n5) -> + le__mn7 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le m n5) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n5)) + le__mn6 le__mn7)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n4 (Coq__Init__Datatypes.S m0)) + (def__n3 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n4 (Coq__Init__Datatypes.S m0)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Peano.le m + (Coq__Init__Datatypes.S m0))) + (Coq__Init__Peano.le__S m m0 le__mn4) + (Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n4 (Coq__Init__Peano.le m) l + (Coq__Init__Datatypes.S m0) def__n3)))) + (IHn1 : + (n4 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m (Coq__Init__Datatypes.S n4)) -> + le__mn6 : + cupicef.Term cupicef.prop (Coq__Init__Peano.le m n4) -> + le__mn7 : + cupicef.Term cupicef.prop (Coq__Init__Peano.le m n4) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n4)) + le__mn6 le__mn7)) => + def__n0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m (Coq__Init__Datatypes.S m0)) => + Coq__Logic__EqdepFacts.internal__eq__rew__r__dep + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m (Coq__Init__Datatypes.S m0) + (cupicef.cast + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I Coq__Init__Datatypes.nat + (m1 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m1 (Coq__Init__Datatypes.S m0)) + (def__n1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m1 (Coq__Init__Datatypes.S m0)) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I Coq__Init__Datatypes.nat + (y : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + y (Coq__Init__Datatypes.S m0)) + (H3 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + y (Coq__Init__Datatypes.S m0)) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I))) + cupicef.I + (m1 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + def__n1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m1 (Coq__Init__Datatypes.S m0)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (Coq__Init__Peano.le m1 m0) + (le__mn6 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le m1 m0) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Peano.le m1 + (Coq__Init__Datatypes.S m0))) + (Coq__Init__Peano.le__S m1 m0 le__mn6) + (Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m1 (Coq__Init__Peano.le m1) + (Coq__Init__Peano.le__n m1) + (Coq__Init__Datatypes.S m0) def__n1)))) + (le__mn0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S m0) + m0) => + Coq__Init__Logic.match____False cupicef.prop + (__ : + cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Peano.le + (Coq__Init__Datatypes.S m0) + (Coq__Init__Datatypes.S m0))) + (Coq__Init__Peano.le__S + (Coq__Init__Datatypes.S m0) m0 le__mn0) + (Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) + (Coq__Init__Peano.le + (Coq__Init__Datatypes.S m0)) + (Coq__Init__Peano.le__n + (Coq__Init__Datatypes.S m0)) + (Coq__Init__Datatypes.S m0) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0)))) + (let__f0 m n n1 IHn0 n2 H H1 H2 le__mn1 le__mn2 n3 + le__mn3 m0 le__mn4 le__mn5 IHn1 def__n0 le__mn0)) + def__n0 le__mn4) + (m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__mn6 : + cupicef.Term cupicef.prop (Coq__Init__Peano.le m m1) => + IHn1 : + (n4 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) + (Coq__Init__Datatypes.S n4)) -> + le__mn7 : + cupicef.Term cupicef.prop (Coq__Init__Peano.le m n4) -> + le__mn8 : + cupicef.Term cupicef.prop (Coq__Init__Peano.le m n4) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n4)) + le__mn7 le__mn8)) => + def__n0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) + (Coq__Init__Datatypes.S m0)) => + (H3 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m1 m0) => + Coq__Init__Logic.eq__ind__r + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m0 + (m2 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I Coq__Init__Datatypes.nat + (n4 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m2) + (Coq__Init__Datatypes.S n4)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m2) + (Coq__Init__Datatypes.S n4)) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (Coq__Init__Peano.le m n4) + (le__mn7 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le m n4) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (Coq__Init__Peano.le m n4) + (le__mn8 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le m n4) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Peano.le m n4)) + le__mn7 le__mn8))))) + (IHn2 : + (n4 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m2) + (Coq__Init__Datatypes.S n4)) -> + le__mn7 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le m n4) -> + le__mn8 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le m n4) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n4)) + le__mn7 le__mn8)) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (Coq__Init__Peano.le m m2) + (le__mn7 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le m m2) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m2) + (Coq__Init__Datatypes.S m0)) + (def__n1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m2) + (Coq__Init__Datatypes.S m0)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Peano.le m + (Coq__Init__Datatypes.S m0))) + (Coq__Init__Peano.le__S m m0 le__mn4) + (Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m2) + (Coq__Init__Peano.le m) + (Coq__Init__Peano.le__S m m2 le__mn7) + (Coq__Init__Datatypes.S m0) def__n1))))) + (IHn2 : + (n4 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) + (Coq__Init__Datatypes.S n4)) -> + le__mn7 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le m n4) -> + le__mn8 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le m n4) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n4)) + le__mn7 le__mn8)) => + le__mn7 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le m m0) => + def__n1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) + (Coq__Init__Datatypes.S m0)) => + Coq__Init__Logic.eq__ind__r + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) + (Coq__Init__Datatypes.S m0))) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0)) + (def__n3 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) + (Coq__Init__Datatypes.S m0)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Peano.le m + (Coq__Init__Datatypes.S m0))) + (Coq__Init__Peano.le__S m m0 le__mn4) + (Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) + (Coq__Init__Peano.le m) + (Coq__Init__Peano.le__S m m0 le__mn7) + (Coq__Init__Datatypes.S m0) def__n3)) + (cupicef.cast cupicef.prop cupicef.prop + ((le__mn8 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le m m0) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Peano.le m + (Coq__Init__Datatypes.S m0))) + (Coq__Init__Peano.le__S m m0 le__mn8) + (Coq__Init__Peano.le__S m m0 le__mn7)) + le__mn4) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Peano.le m + (Coq__Init__Datatypes.S m0))) + (Coq__Init__Peano.le__S m m0 le__mn4) + (Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0) + (Coq__Init__Peano.le m) + (Coq__Init__Peano.le__S m m0 le__mn7) + (Coq__Init__Datatypes.S m0) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0)))) + cupicef.I + (Coq__Init__Logic.eq__ind__r + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m m0)) + le__mn7 + (le__mn8 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le m m0) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Peano.le m + (Coq__Init__Datatypes.S m0))) + (Coq__Init__Peano.le__S m m0 le__mn8) + (Coq__Init__Peano.le__S m m0 le__mn7)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Peano.le m + (Coq__Init__Datatypes.S m0))) + (Coq__Init__Peano.le__S m m0 le__mn7)) + le__mn4 + (let__H2 m n n1 IHn0 n2 H H1 H2 le__mn1 + le__mn2 n3 le__mn3 m0 le__mn4 le__mn5 m1 + le__mn6 IHn1 def__n0 H3 IHn2 le__mn7 + def__n1))) + def__n1 + (UIP__nat (Coq__Init__Datatypes.S m0) + (Coq__Init__Datatypes.S m0) def__n1 + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m0)))) + m1 H3 IHn1 le__mn6 def__n0) + (let__H m n n1 IHn0 n2 H H1 H2 le__mn1 le__mn2 n3 + le__mn3 m0 le__mn4 le__mn5 m1 le__mn6 IHn1 def__n0)) + n1 le__mn5 IHn0) + n3 le__mn3) + n1 le__mn1 le__mn2 + (let__def__n2 m n n1 IHn0 n2 H H1 H2 le__mn1 le__mn2)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Peano.le m n1)) + le__mn2)) + n2 H1 H) + (let__H1 m n n1 IHn0 n2 H)) + (let__n0 m n) n + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n)). + +(; End of translation. ;) diff --git a/out/Coq__Arith__Plus.dk b/out/Coq__Arith__Plus.dk new file mode 100644 index 000000000..f54e9a827 --- /dev/null +++ b/out/Coq__Arith__Plus.dk @@ -0,0 +1,2831 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def plus__Snm__nSm : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n) m) + (Coq__Init__Nat.add n (Coq__Init__Datatypes.S m))) + +:= Coq__Init__Peano.plus__n__Sm. + +thm plus__assoc__reverse : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Nat.add n m) p) + (Coq__Init__Nat.add n (Coq__Init__Nat.add m p))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add n (Coq__Init__Nat.add m p)) + (Coq__Init__Nat.add (Coq__Init__Nat.add n m) p) + (Coq__Arith__PeanoNat.Nat__add__assoc n m p). + +def let__H : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0)) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0))) => + __) + (Coq__Arith__PeanoNat.Nat__add__cancel__l n0 m0 p0). + +thm plus__reg__l : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add p n) (Coq__Init__Nat.add p m)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H n m p n m p. + +def let__H0 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) => + Coq__Init__Peano.le n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le + (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) => + Coq__Init__Peano.le n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) => + Coq__Init__Peano.le n0 m0)) + (__ : + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => + __0) + (Coq__Arith__PeanoNat.Nat__add__le__mono__l n0 m0 p0). + +thm plus__le__reg__l : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Nat.add p n) (Coq__Init__Nat.add p m)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H0 n m p n m p. + +def let__H1 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) => + Coq__Init__Peano.lt n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt + (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) => + Coq__Init__Peano.lt n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) => + Coq__Init__Peano.lt n0 m0)) + (__ : + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => + __0) + (Coq__Arith__PeanoNat.Nat__add__lt__mono__l n0 m0 p0). + +thm plus__lt__reg__l : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Nat.add p n) (Coq__Init__Nat.add p m)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H1 n m p n m p. + +def let__H2 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) => + Coq__Init__Peano.le n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le + (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) => + Coq__Init__Peano.le n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0))) + (__ : + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => + __) + (Coq__Arith__PeanoNat.Nat__add__le__mono__l n0 m0 p0). + +thm plus__le__compat__l : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Nat.add p n) (Coq__Init__Nat.add p m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H2 n m p n m p. + +def let__H3 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add n0 p0) + (Coq__Arith__PeanoNat.Nat__add m0 p0)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add n0 p0) + (Coq__Arith__PeanoNat.Nat__add m0 p0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add n0 p0) + (Coq__Arith__PeanoNat.Nat__add m0 p0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add n0 p0) + (Coq__Arith__PeanoNat.Nat__add m0 p0)) => + Coq__Init__Peano.le n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add n0 p0) + (Coq__Arith__PeanoNat.Nat__add m0 p0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add n0 p0) + (Coq__Arith__PeanoNat.Nat__add m0 p0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le + (Coq__Arith__PeanoNat.Nat__add n0 p0) + (Coq__Arith__PeanoNat.Nat__add m0 p0)) => + Coq__Init__Peano.le n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.le n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => + Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add n0 p0) + (Coq__Arith__PeanoNat.Nat__add m0 p0))) + (__ : + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add n0 p0) + (Coq__Arith__PeanoNat.Nat__add m0 p0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__add n0 p0) + (Coq__Arith__PeanoNat.Nat__add m0 p0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => + __) + (Coq__Arith__PeanoNat.Nat__add__le__mono__r n0 m0 p0). + +thm plus__le__compat__r : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Nat.add n p) (Coq__Init__Nat.add m p)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H3 n m p n m p. + +def let__H4 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) => + Coq__Init__Peano.lt n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt + (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) => + Coq__Init__Peano.lt n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0))) + (__ : + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) + (Coq__Arith__PeanoNat.Nat__add p0 m0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => + __) + (Coq__Arith__PeanoNat.Nat__add__lt__mono__l n0 m0 p0). + +thm plus__lt__compat__l : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Nat.add p n) (Coq__Init__Nat.add p m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H4 n m p n m p. + +def let__H5 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p0) + (Coq__Arith__PeanoNat.Nat__add m0 p0)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p0) + (Coq__Arith__PeanoNat.Nat__add m0 p0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p0) + (Coq__Arith__PeanoNat.Nat__add m0 p0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p0) + (Coq__Arith__PeanoNat.Nat__add m0 p0)) => + Coq__Init__Peano.lt n0 m0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p0) + (Coq__Arith__PeanoNat.Nat__add m0 p0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p0) + (Coq__Arith__PeanoNat.Nat__add m0 p0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt + (Coq__Arith__PeanoNat.Nat__add n0 p0) + (Coq__Arith__PeanoNat.Nat__add m0 p0)) => + Coq__Init__Peano.lt n0 m0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n0 m0) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => + Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p0) + (Coq__Arith__PeanoNat.Nat__add m0 p0))) + (__ : + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p0) + (Coq__Arith__PeanoNat.Nat__add m0 p0))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p0) + (Coq__Arith__PeanoNat.Nat__add m0 p0)) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => + __) + (Coq__Arith__PeanoNat.Nat__add__lt__mono__r n0 m0 p0). + +thm plus__lt__compat__r : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Nat.add n p) (Coq__Init__Nat.add m p)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + let__H5 n m p n m p. + +thm plus__le__compat : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> + __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le p q) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Nat.add n p) (Coq__Init__Nat.add m q)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Arith__PeanoNat.Nat__add__le__mono n m p q. + +thm plus__le__lt__compat : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> + __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt p q) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Nat.add n p) (Coq__Init__Nat.add m q)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Arith__PeanoNat.Nat__add__le__lt__mono n m p q. + +thm plus__lt__le__compat : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) -> + __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le p q) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Nat.add n p) (Coq__Init__Nat.add m q)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Arith__PeanoNat.Nat__add__lt__le__mono n m p q. + +thm plus__lt__compat : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) -> + __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt p q) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (Coq__Init__Nat.add n p) (Coq__Init__Nat.add m q)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Arith__PeanoNat.Nat__add__lt__mono n m p q. + +thm le__plus__l : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n (Coq__Init__Nat.add n m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Arith__PeanoNat.Nat__le__add__r n m. + +thm le__plus__r : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le m (Coq__Init__Nat.add n m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq__ind__r + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Arith__PeanoNat.Nat__add m n) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Peano.le m n0) + (Coq__Arith__PeanoNat.Nat__le__add__r m n) + (Coq__Arith__PeanoNat.Nat__add n m) + (Coq__Arith__PeanoNat.Nat__add__comm n m). + +thm le__plus__trans : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n (Coq__Init__Nat.add m p)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) => + (lemma : + cupicef.Term cupicef.prop + (Coq__Init__Peano.le m (Coq__Arith__PeanoNat.Nat__add m p)) => + Coq__Classes__Morphisms.trans__contra__inv__impl__morphism + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Peano.le + (Coq__Classes__RelationClasses.PreOrder__Transitive + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Peano.le Coq__Arith__PeanoNat.Nat__le__preorder) + n (Coq__Init__Nat.add m p) m lemma) + (Coq__Arith__PeanoNat.Nat__le__add__r m p) H. + +thm lt__plus__trans : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n (Coq__Init__Nat.add m p)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) => + Coq__Arith__PeanoNat.Nat__lt__le__trans n m (Coq__Init__Nat.add m p) H + (Coq__Arith__PeanoNat.Nat__le__add__r m p). + +def let__H7 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) + Coq__Init__Datatypes.O) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) + Coq__Init__Datatypes.O) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) + Coq__Init__Datatypes.O) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) + Coq__Init__Datatypes.O) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) + Coq__Init__Datatypes.O) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) + Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat (cupicef.type (cupicef.s cupicef.z)) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.True) + e) + Coq__Init__Logic.I Coq__Init__Datatypes.O H1. + +def let__H6 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) + Coq__Init__Datatypes.O) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) + Coq__Init__Datatypes.O) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) + (y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) y) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + y Coq__Init__Datatypes.O) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + y Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) + Coq__Init__Datatypes.O) => + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) + Coq__Init__Datatypes.O) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) + (let__H7 n m n0 H H0 H1)) + H0) + Coq__Init__Datatypes.O H. + +def let__H9 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) + Coq__Init__Datatypes.O) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) + Coq__Init__Datatypes.O) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) + Coq__Init__Datatypes.O) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) + Coq__Init__Datatypes.O) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) + Coq__Init__Datatypes.O) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) + Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat (cupicef.type (cupicef.s cupicef.z)) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.True) + e) + Coq__Init__Logic.I Coq__Init__Datatypes.O H1. + +def let__H8 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) + Coq__Init__Datatypes.O) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m Coq__Init__Datatypes.O) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) + Coq__Init__Datatypes.O) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) + (y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) y) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + y Coq__Init__Datatypes.O) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + y Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m Coq__Init__Datatypes.O)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) + Coq__Init__Datatypes.O) => + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) + Coq__Init__Datatypes.O) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m Coq__Init__Datatypes.O) + (let__H9 n m n0 H H0 H1)) + H0) + Coq__Init__Datatypes.O H. + +thm plus__is__O : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add n m) Coq__Init__Datatypes.O) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m Coq__Init__Datatypes.O)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.prop + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add n0 m) Coq__Init__Datatypes.O) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add n0 m) Coq__Init__Datatypes.O) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m Coq__Init__Datatypes.O))) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add Coq__Init__Datatypes.O m) + Coq__Init__Datatypes.O) => + Coq__Init__Logic.conj + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O) + H) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) + Coq__Init__Datatypes.O) => + Coq__Init__Logic.conj + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m Coq__Init__Datatypes.O) + (let__H6 n m n0 H + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O)) + (let__H8 n m n0 H + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O))) + n. + +def let__H10 : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add + (Coq__Init__Datatypes.S (Coq__Init__Datatypes.S m1)) n) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add + (Coq__Init__Datatypes.S (Coq__Init__Datatypes.S m1)) n) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S (Coq__Init__Datatypes.S m1)) + n) + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat (cupicef.type (cupicef.s cupicef.z)) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat + (cupicef.type (cupicef.s cupicef.z)) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.True) + n0) + e) + Coq__Init__Logic.I (Coq__Init__Datatypes.S Coq__Init__Datatypes.O) H. + +def plus__is__one : + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add m n) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n Coq__Init__Datatypes.O))) + +:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.set + (cupicef.rule cupicef.prop cupicef.set) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add n0 n) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add n0 n) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) => + Coq__Init__Specif.sumbool + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n Coq__Init__Datatypes.O)))) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add Coq__Init__Datatypes.O n) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) => + Coq__Init__Specif.left + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n Coq__Init__Datatypes.O)) + (Coq__Init__Logic.conj + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O) + H)) + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.set + (cupicef.rule cupicef.prop cupicef.set) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) n) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) n) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) => + Coq__Init__Specif.sumbool + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n Coq__Init__Datatypes.O)))) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O) n) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) => + Coq__Init__Specif.right + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O) + Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n Coq__Init__Datatypes.O)) + (Coq__Init__Logic.conj + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Peano.eq__add__S n Coq__Init__Datatypes.O H))) + (m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add + (Coq__Init__Datatypes.S (Coq__Init__Datatypes.S m1)) n) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) => + Coq__Init__Logic.False__rec + (Coq__Init__Specif.sumbool + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S (Coq__Init__Datatypes.S m1)) + Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S (Coq__Init__Datatypes.S m1)) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n Coq__Init__Datatypes.O))) + (let__H10 m n m0 m1 H)) + m0) + m. + +def tail__plus : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat))))) + (c => + c + (tail__plus0 => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + m + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + tail__plus0 n0 (Coq__Init__Datatypes.S m)) + n)) + cupicef.0. + +thm plus__tail__plus : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add n m) (tail__plus n m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__ind + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I + Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add n0 m) (tail__plus n0 m))) + (cupicef.cast (cupicef.rule cupicef.set cupicef.prop) + (cupicef.rule cupicef.set cupicef.prop) + (cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I + Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m m)) + (cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I + Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add Coq__Init__Datatypes.O m) + (tail__plus Coq__Init__Datatypes.O m))) + cupicef.I + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + m)) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add n0 m) (tail__plus n0 m))) => + cupicef.cast (cupicef.rule cupicef.set cupicef.prop) + (cupicef.rule cupicef.set cupicef.prop) + (cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I + Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S (Coq__Init__Nat.add n0 m)) n1) + (tail__plus n0 (Coq__Init__Datatypes.S m)))) + (cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I + Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m) + (tail__plus (Coq__Init__Datatypes.S n0) m))) + cupicef.I + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add n0 (Coq__Init__Datatypes.S m)) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S (Coq__Init__Nat.add n0 m)) n1) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S (Coq__Init__Nat.add n0 m)) + (Coq__Init__Nat.add n0 (Coq__Init__Datatypes.S m))) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S (Coq__Init__Nat.add n0 m)) + (Coq__Init__Nat.add n0 (Coq__Init__Datatypes.S m))) + cupicef.I (Coq__Init__Peano.plus__n__Sm n0 m)) + (tail__plus n0 (Coq__Init__Datatypes.S m)) + (IHn (Coq__Init__Datatypes.S m)))) + n. + +thm succ__plus__discr : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n (Coq__Init__Datatypes.S (Coq__Init__Nat.add m n)))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Arith__PeanoNat.Nat__succ__add__discr m n. + +thm n__SSn : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n (Coq__Init__Datatypes.S (Coq__Init__Datatypes.S n)))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + succ__plus__discr n (Coq__Init__Datatypes.S Coq__Init__Datatypes.O). + +thm n__SSSn : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S (Coq__Init__Datatypes.S n))))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + succ__plus__discr n + (Coq__Init__Datatypes.S (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)). + +thm n__SSSSn : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S (Coq__Init__Datatypes.S n)))))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + succ__plus__discr n + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))). + +(; End of translation. ;) diff --git a/out/Coq__Arith__Wf__nat.dk b/out/Coq__Arith__Wf__nat.dk new file mode 100644 index 000000000..87e087299 --- /dev/null +++ b/out/Coq__Arith__Wf__nat.dk @@ -0,0 +1,3746 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def ltof : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Peano.lt (f a) (f b). + +def gtof : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Peano.gt (f b) (f a). + +def let__H : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat) -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt (f a) n) -> + cupicef.Term cupicef.prop (Coq__Init__Wf.Acc A (ltof A f) a) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat) => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__ind + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (f a) n0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt (f a) n0) => + Coq__Init__Wf.Acc A (ltof A f) a))) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (f a) Coq__Init__Datatypes.O) => + Coq__Init__Logic.False__ind (Coq__Init__Wf.Acc A (ltof A f) a) + (Coq__Arith__PeanoNat.Nat__nlt__0__r (f a) H)) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt (f a) n0) -> + cupicef.Term cupicef.prop (Coq__Init__Wf.Acc A (ltof A f) a)) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Ha : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (f a) (Coq__Init__Datatypes.S n0)) => + Coq__Init__Wf.Acc__intro A (ltof A f) a + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (f b) (f a)) + (Hb : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (f b) (f a)) => + Coq__Init__Wf.Acc A (ltof A f) b))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (ltof A f y a) + (__ : cupicef.Term cupicef.prop (ltof A f y a) => + Coq__Init__Wf.Acc A (ltof A f) y))) + cupicef.I + (b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hb : cupicef.Term cupicef.prop (Coq__Init__Peano.lt (f b) (f a)) => + IHn b + (Coq__Arith__PeanoNat.Nat__lt__le__trans (f b) (f a) n0 Hb + (Coq__Arith__Lt.lt__n__Sm__le (f a) n0 Ha))))) + n. + +def well__founded__ltof : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat) -> + cupicef.Term cupicef.prop (Coq__Init__Wf.well__founded A (ltof A f)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Wf.Acc A (ltof A f) a)) + (Coq__Init__Wf.well__founded A (ltof A f)) cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + let__H A f (Coq__Init__Datatypes.S (f a)) a + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S (f a)) + (Coq__Init__Datatypes.S (f a))) + (Coq__Init__Peano.lt (f a) (Coq__Init__Datatypes.S (f a))) + cupicef.I (Coq__Init__Peano.le__n (Coq__Init__Datatypes.S (f a))))). + +def well__founded__gtof : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat) -> + cupicef.Term cupicef.prop (Coq__Init__Wf.well__founded A (gtof A f)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat) => + well__founded__ltof A f. + +def let__H0 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.set) -> + F : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (ltof A f y x) -> + cupicef.Term cupicef.set (P y)) -> + cupicef.Term cupicef.set (P x)) -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt (f a) n) -> + cupicef.Term cupicef.set (P a) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.set) => + F : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (ltof A f y x) -> + cupicef.Term cupicef.set (P y)) -> + cupicef.Term cupicef.set (P x)) => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__rect + (cupicef.cast + (cupicef.rule cupicef.set + (cupicef.axiom + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.set + (cupicef.axiom + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule cupicef.set + (cupicef.axiom + (cupicef.sup cupicef.set + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I Coq__Init__Datatypes.nat + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.univ + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup cupicef.set + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I)) + (cupicef.prod cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I Coq__Init__Datatypes.nat + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.set) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.set)) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.set + (cupicef.rule cupicef.prop cupicef.set) cupicef.I + (Coq__Init__Peano.lt (f a) n0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt (f a) n0) => + P a)))) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (f a) Coq__Init__Datatypes.O) => + Coq__Init__Logic.False__rec (P a) + (Coq__Arith__PeanoNat.Nat__nlt__0__r (f a) H)) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt (f a) n0) -> + cupicef.Term cupicef.set (P a)) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Ha : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (f a) (Coq__Init__Datatypes.S n0)) => + F a + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.set)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.set)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.set) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.set)) + cupicef.I A + (b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.set + (cupicef.rule cupicef.prop cupicef.set) cupicef.I + (Coq__Init__Peano.lt (f b) (f a)) + (Hb : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (f b) (f a)) => + P b))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.set) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.set)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.set + (cupicef.rule cupicef.prop cupicef.set) cupicef.I + (ltof A f y a) + (__ : cupicef.Term cupicef.prop (ltof A f y a) => P y))) + cupicef.I + (b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hb : cupicef.Term cupicef.prop (Coq__Init__Peano.lt (f b) (f a)) => + IHn b + (Coq__Arith__PeanoNat.Nat__lt__le__trans (f b) (f a) n0 Hb + (Coq__Arith__Lt.lt__n__Sm__le (f a) n0 Ha))))) + n. + +def induction__ltof1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.set) -> + __ : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (ltof A f y x) -> + cupicef.Term cupicef.set (P y)) -> + cupicef.Term cupicef.set (P x)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set (P a) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.set) => + F : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (ltof A f y x) -> + cupicef.Term cupicef.set (P y)) -> + cupicef.Term cupicef.set (P x)) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + let__H0 A f P F (Coq__Init__Datatypes.S (f a)) a + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S (f a)) + (Coq__Init__Datatypes.S (f a))) + (Coq__Init__Peano.lt (f a) (Coq__Init__Datatypes.S (f a))) cupicef.I + (Coq__Init__Peano.le__n (Coq__Init__Datatypes.S (f a)))). + +def induction__gtof1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.set) -> + __ : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (gtof A f y x) -> + cupicef.Term cupicef.set (P y)) -> + cupicef.Term cupicef.set (P x)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set (P a) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat) => + induction__ltof1 A f. + +def induction__ltof2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.set) -> + __ : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (ltof A f y x) -> + cupicef.Term cupicef.set (P y)) -> + cupicef.Term cupicef.set (P x)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set (P a) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat) => + Coq__Init__Wf.well__founded__induction A (ltof A f) + (well__founded__ltof A f). + +def induction__gtof2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.set) -> + __ : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (gtof A f y x) -> + cupicef.Term cupicef.set (P y)) -> + cupicef.Term cupicef.set (P x)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set (P a) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat) => + induction__ltof2 A f. + +def let__H1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + H__compat : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt (f x) (f y))) -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt (f a) n) -> + cupicef.Term cupicef.prop (Coq__Init__Wf.Acc A R a) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + H__compat : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt (f x) (f y))) => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__ind + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt (f a) n0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt (f a) n0) => + Coq__Init__Wf.Acc A R a))) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (f a) Coq__Init__Datatypes.O) => + Coq__Init__Logic.False__ind (Coq__Init__Wf.Acc A R a) + (Coq__Arith__PeanoNat.Nat__nlt__0__r (f a) H)) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt (f a) n0) -> + cupicef.Term cupicef.prop (Coq__Init__Wf.Acc A R a)) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Ha : + cupicef.Term cupicef.prop + (Coq__Init__Peano.lt (f a) (Coq__Init__Datatypes.S n0)) => + Coq__Init__Wf.Acc__intro A R a + (b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hb : cupicef.Term cupicef.prop (R b a) => + IHn b + (Coq__Arith__PeanoNat.Nat__lt__le__trans (f b) (f a) n0 + (H__compat b a Hb) (Coq__Arith__Lt.lt__n__Sm__le (f a) n0 Ha)))) + n. + +def well__founded__lt__compat : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + H__compat : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt (f x) (f y))) -> + cupicef.Term cupicef.prop (Coq__Init__Wf.well__founded A R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + H__compat : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt (f x) (f y))) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Wf.Acc A R a)) + (Coq__Init__Wf.well__founded A R) cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + let__H1 A f R H__compat (Coq__Init__Datatypes.S (f a)) a + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S (f a)) + (Coq__Init__Datatypes.S (f a))) + (Coq__Init__Peano.lt (f a) (Coq__Init__Datatypes.S (f a))) + cupicef.I (Coq__Init__Peano.le__n (Coq__Init__Datatypes.S (f a))))). + +def lt__wf : + cupicef.Term cupicef.prop + (Coq__Init__Wf.well__founded + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Peano.lt) + +:= well__founded__ltof + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => m). + +def lt__wf__rec1 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.set) -> + __ : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n0) -> + cupicef.Term cupicef.set (P m)) -> + cupicef.Term cupicef.set (P n0)) -> + cupicef.Term cupicef.set (P n) + +:= p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.set) => + F : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) -> + cupicef.Term cupicef.set (P m)) -> + cupicef.Term cupicef.set (P n)) => + induction__ltof1 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => m) P F p. + +def lt__wf__rec : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.set) -> + __ : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n0) -> + cupicef.Term cupicef.set (P m)) -> + cupicef.Term cupicef.set (P n0)) -> + cupicef.Term cupicef.set (P n) + +:= p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.set) => + F : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) -> + cupicef.Term cupicef.set (P m)) -> + cupicef.Term cupicef.set (P n)) => + induction__ltof2 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => m) P F p. + +thm lt__wf__ind : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + __ : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n0) -> + cupicef.Term cupicef.prop (P m)) -> + cupicef.Term cupicef.prop (P n0)) -> + cupicef.Term cupicef.prop (P n) + +:= p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + H : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) -> + cupicef.Term cupicef.prop (P m)) -> + cupicef.Term cupicef.prop (P n)) => + Coq__Init__Wf.Acc__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Peano.lt + (p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => P p0) + (x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : + (y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt y x) -> + cupicef.Term cupicef.prop + (Coq__Init__Wf.Acc + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Peano.lt y)) => + H1 : + (y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt y x) -> + cupicef.Term cupicef.prop (P y)) => + H x H1) + p (lt__wf p). + +def gt__wf__rec : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.set) -> + __ : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.gt n0 m) -> + cupicef.Term cupicef.set (P m)) -> + cupicef.Term cupicef.set (P n0)) -> + cupicef.Term cupicef.set (P n) + +:= lt__wf__rec. + +thm gt__wf__ind : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + __ : + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.gt n0 m) -> + cupicef.Term cupicef.prop (P m)) -> + cupicef.Term cupicef.prop (P n0)) -> + cupicef.Term cupicef.prop (P n) + +:= lt__wf__ind. + +def lt__wf__double__rec : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.set) -> + __ : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt p n) -> + cupicef.Term cupicef.set (P p q)) -> + __0 : + (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt p m) -> + cupicef.Term cupicef.set (P n p)) -> + cupicef.Term cupicef.set (P n m)) -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set (P n m) + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.set) => + Hrec : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt p n) -> + cupicef.Term cupicef.set (P p q)) -> + __0 : + (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt p m) -> + cupicef.Term cupicef.set (P n p)) -> + cupicef.Term cupicef.set (P n m)) => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.cast cupicef.set (cupicef.rule cupicef.set cupicef.set) + ((n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => P n m)) + p) + (cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => P p m)) + cupicef.I + (lt__wf__rec p + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => P n m)) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set (P m m0)) => + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.cast cupicef.set cupicef.set + ((n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => P n n0) + q) + (P n q) cupicef.I + (lt__wf__rec q + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + P n n0) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n0) -> + cupicef.Term cupicef.set (P n m)) => + Hrec n n0 + (p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + q0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H1 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt p0 n) => + H p0 H1 q0) + H0)))). + +thm lt__wf__double__ind : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + __ : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt p n) -> + cupicef.Term cupicef.prop (P p q)) -> + __0 : + (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt p m) -> + cupicef.Term cupicef.prop (P n p)) -> + cupicef.Term cupicef.prop (P n m)) -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (P n m) + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + Hrec : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt p n) -> + cupicef.Term cupicef.prop (P p q)) -> + __0 : + (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt p m) -> + cupicef.Term cupicef.prop (P n p)) -> + cupicef.Term cupicef.prop (P n m)) => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.cast cupicef.prop (cupicef.rule cupicef.set cupicef.prop) + ((n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I + Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => P n m)) + p) + (cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I + Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => P p m)) + cupicef.I + (lt__wf__ind p + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I + Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => P n m)) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (P m m0)) => + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.cast cupicef.prop cupicef.prop + ((n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => P n n0) + q) + (P n q) cupicef.I + (lt__wf__ind q + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + P n n0) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n0) -> + cupicef.Term cupicef.prop (P n m)) => + Hrec n n0 + (p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + q0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H1 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt p0 n) => + H p0 H1 q0) + H0)))). + +def inv__lt__rel : + A : cupicef.Univ cupicef.set -> + F : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + x : cupicef.Term cupicef.set A -> + y : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ cupicef.set => + F : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + x : cupicef.Term cupicef.set A => + y : cupicef.Term cupicef.set A => + Coq__Init__Logic.ex2 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => F x n) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (F y m) + (__ : cupicef.Term cupicef.prop (F y m) => Coq__Init__Peano.lt n m))). + +def let__i : + A : cupicef.Univ cupicef.set -> + R : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) -> + F : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + F__compat : + (x : cupicef.Term cupicef.set A -> + y : cupicef.Term cupicef.set A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (inv__lt__rel A F x y)) -> + x : cupicef.Term cupicef.set A -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => F x n)) -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + fxn : cupicef.Term cupicef.prop (F x n) -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H0 : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n0) -> + x0 : cupicef.Term cupicef.set A -> + fxn0 : cupicef.Term cupicef.prop (F x0 m) -> + cupicef.Term cupicef.prop + (Coq__Init__Wf.Acc + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + R x0)) -> + x0 : cupicef.Term cupicef.set A -> + fxn0 : cupicef.Term cupicef.prop (F x0 n0) -> + y : cupicef.Term cupicef.set A -> + H1 : cupicef.Term cupicef.prop (R y x0) -> + cupicef.Term cupicef.prop (inv__lt__rel A F y x0) + +:= A : cupicef.Univ cupicef.set => + R : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) => + F : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + F__compat : + (x : cupicef.Term cupicef.set A -> + y : cupicef.Term cupicef.set A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (inv__lt__rel A F x y)) => + x : cupicef.Term cupicef.set A => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => F x n)) => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + fxn : cupicef.Term cupicef.prop (F x n) => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n0) -> + x0 : cupicef.Term cupicef.set A -> + fxn0 : cupicef.Term cupicef.prop (F x0 m) -> + cupicef.Term cupicef.prop + (Coq__Init__Wf.Acc + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + R x0)) => + x0 : cupicef.Term cupicef.set A => + fxn0 : cupicef.Term cupicef.prop (F x0 n0) => + y : cupicef.Term cupicef.set A => + H1 : cupicef.Term cupicef.prop (R y x0) => + F__compat y x0 H1. + +thm acc__lt__rel : + A : cupicef.Univ cupicef.set -> + R : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) -> + F : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + F__compat : + (x : cupicef.Term cupicef.set A -> + y : cupicef.Term cupicef.set A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (inv__lt__rel A F x y)) -> + x : cupicef.Term cupicef.set A -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => F x n)) -> + cupicef.Term cupicef.prop + (Coq__Init__Wf.Acc + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I A) + R x) + +:= A : cupicef.Univ cupicef.set => + R : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) => + F : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + F__compat : + (x : cupicef.Term cupicef.set A -> + y : cupicef.Term cupicef.set A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (inv__lt__rel A F x y)) => + x : cupicef.Term cupicef.set A => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => F x n)) => + Coq__Init__Logic.match____ex cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => F x n) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => F x n)) => + Coq__Init__Wf.Acc + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I A) + R x) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + fxn : cupicef.Term cupicef.prop (F x n) => + cupicef.cast cupicef.prop + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + ((n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x0 : cupicef.Term cupicef.set A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (F x0 n0) + (fxn0 : cupicef.Term cupicef.prop (F x0 n0) => + Coq__Init__Wf.Acc + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + R x0))) + n) + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x0 : cupicef.Term cupicef.set A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (F x0 n) + (fxn0 : cupicef.Term cupicef.prop (F x0 n) => + Coq__Init__Wf.Acc + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + R x0))) + cupicef.I + (lt__wf__ind n + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x0 : cupicef.Term cupicef.set A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + F x0 n0) + (fxn0 : cupicef.Term cupicef.prop (F x0 n0) => + Coq__Init__Wf.Acc + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + R x0))) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n0) -> + x0 : cupicef.Term cupicef.set A -> + fxn0 : cupicef.Term cupicef.prop (F x0 m) -> + cupicef.Term cupicef.prop + (Coq__Init__Wf.Acc + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + R x0)) => + x0 : cupicef.Term cupicef.set A => + fxn0 : cupicef.Term cupicef.prop (F x0 n0) => + Coq__Init__Wf.Acc__intro + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + R x0 + (y : cupicef.Term cupicef.set A => + H1 : cupicef.Term cupicef.prop (R y x0) => + Coq__Init__Logic.match____ex2 cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + F y n1) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (F x0 m) + (__ : cupicef.Term cupicef.prop (F x0 m) => + Coq__Init__Peano.lt n1 m))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex2 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n1 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + F y n1) + (n1 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (m : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (F x0 m) + (__ : cupicef.Term cupicef.prop (F x0 m) => + Coq__Init__Peano.lt n1 m)))) => + Coq__Init__Wf.Acc + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + R y) + (x1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H2 : cupicef.Term cupicef.prop (F y x1) => + H3 : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (F x0 m) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt x1 m)) => + H0 x1 (H3 n0 fxn0) y H2) + (let__i A R F F__compat x H n fxn n0 H0 x0 fxn0 y H1)))) + x fxn) + H. + +thm well__founded__inv__lt__rel__compat : + A : cupicef.Univ cupicef.set -> + R : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) -> + F : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + F__compat : + (x : cupicef.Term cupicef.set A -> + y : cupicef.Term cupicef.set A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (inv__lt__rel A F x y)) -> + cupicef.Term cupicef.prop + (Coq__Init__Wf.well__founded + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I A) + R) + +:= A : cupicef.Univ cupicef.set => + R : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) => + F : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + F__compat : + (x : cupicef.Term cupicef.set A -> + y : cupicef.Term cupicef.set A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (inv__lt__rel A F x y)) => + a : cupicef.Term cupicef.set A => + Coq__Init__Wf.Acc__intro + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I A) + R a + (y : cupicef.Term cupicef.set A => + H : cupicef.Term cupicef.prop (R y a) => + Coq__Init__Logic.match____ex2 cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => F y n) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (F a m) + (__ : cupicef.Term cupicef.prop (F a m) => + Coq__Init__Peano.lt n m))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex2 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + F y n) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (F a m) + (__ : cupicef.Term cupicef.prop (F a m) => + Coq__Init__Peano.lt n m)))) => + Coq__Init__Wf.Acc + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + R y) + (x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (F y x) => + H1 : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (F a m) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt x m)) => + acc__lt__rel A R F F__compat y + (Coq__Init__Logic.ex__intro + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => F y n) + x H0)) + (F__compat y a H)). + +thm well__founded__inv__rel__inv__lt__rel : + A : cupicef.Univ cupicef.set -> + F : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + cupicef.Term cupicef.prop + (Coq__Init__Wf.well__founded + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I A) + (inv__lt__rel A F)) + +:= A : cupicef.Univ cupicef.set => + F : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + well__founded__inv__lt__rel__compat A (inv__lt__rel A F) F + (x : cupicef.Term cupicef.set A => + y : cupicef.Term cupicef.set A => + H : cupicef.Term cupicef.prop (inv__lt__rel A F x y) => + H). + +def has__unique__least__element : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Coq__Init__Logic.ex A + (Coq__Init__Logic.unique A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.and (P x) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + P x_27_) + (__ : cupicef.Term cupicef.prop (P x_27_) => R x x_27_))))). + +def let__o : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + Pdec : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (P n) (Coq__Init__Logic.not (P n)))) -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => P n)) -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + HPn0 : cupicef.Term cupicef.prop (P n0) -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and (Coq__Init__Peano.lt n_27_ n1) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_)))))) + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (P n_27_) + (__ : cupicef.Term cupicef.prop (P n_27_) => + Coq__Init__Peano.le n1 n_27_)))) -> + IH : + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (P n_27_) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n1 n_27_)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (P n1) (Coq__Init__Logic.not (P n1))) + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + Pdec : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (P n) (Coq__Init__Logic.not (P n)))) => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => P n)) => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + HPn0 : cupicef.Term cupicef.prop (P n0) => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and (Coq__Init__Peano.lt n_27_ n1) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_)))))) + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (P n_27_) + (__ : cupicef.Term cupicef.prop (P n_27_) => + Coq__Init__Peano.le n1 n_27_)))) => + IH : + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (P n_27_) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n1 n_27_)) => + Pdec n1. + +def let__H3 : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + Pdec : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (P n) (Coq__Init__Logic.not (P n)))) -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => P n)) -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + HPn0 : cupicef.Term cupicef.prop (P n0) -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and (Coq__Init__Peano.lt n_27_ n1) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_)))))) + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (P n_27_) + (__ : cupicef.Term cupicef.prop (P n_27_) => + Coq__Init__Peano.le n1 n_27_)))) -> + IH : + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (P n_27_) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n1 n_27_)) -> + HP : cupicef.Term cupicef.prop (Coq__Init__Logic.not (P n1)) -> + n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + Hn_27_ : cupicef.Term cupicef.prop (P n_27_) -> + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Coq__Init__Peano.le n2 m) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n2 m))) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n2 m) + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + Pdec : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (P n) (Coq__Init__Logic.not (P n)))) => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => P n)) => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + HPn0 : cupicef.Term cupicef.prop (P n0) => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and (Coq__Init__Peano.lt n_27_ n1) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_)))))) + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (P n_27_) + (__ : cupicef.Term cupicef.prop (P n_27_) => + Coq__Init__Peano.le n1 n_27_)))) => + IH : + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (P n_27_) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n1 n_27_)) => + HP : cupicef.Term cupicef.prop (Coq__Init__Logic.not (P n1)) => + n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Hn_27_ : cupicef.Term cupicef.prop (P n_27_) => + n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n2 m) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n2 m) => + Coq__Init__Logic.and (Coq__Init__Peano.le n2 m) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n2 m)))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (Coq__Init__Peano.le n2 m) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n2 m))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Coq__Init__Peano.le n2 m) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n2 m))) => + Coq__Init__Peano.lt n2 m)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Peano.lt n2 m) + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n2 m) => + Coq__Init__Logic.and (Coq__Init__Peano.le n2 m) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n2 m)))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (Coq__Init__Peano.le n2 m) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n2 m))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Coq__Init__Peano.le n2 m) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n2 m))) => + Coq__Init__Peano.lt n2 m))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (Coq__Init__Peano.le n2 m) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n2 m))) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Coq__Init__Peano.le n2 m) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n2 m))) => + Coq__Init__Peano.lt n2 m)) + (__ : + (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n2 m) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Coq__Init__Peano.le n2 m) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n2 m)))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Coq__Init__Peano.le n2 m) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n2 m))) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.lt n2 m)) => + __0) + (Coq__Arith__PeanoNat.Nat__le__neq n2 m). + +def let__H2 : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + Pdec : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (P n) (Coq__Init__Logic.not (P n)))) -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => P n)) -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + HPn0 : cupicef.Term cupicef.prop (P n0) -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and (Coq__Init__Peano.lt n_27_ n) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_)))))) + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (P n_27_) + (__ : cupicef.Term cupicef.prop (P n_27_) => + Coq__Init__Peano.le n n_27_)))) + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + Pdec : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (P n) (Coq__Init__Logic.not (P n)))) => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => P n)) => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + HPn0 : cupicef.Term cupicef.prop (P n0) => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__ind + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.or + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and (Coq__Init__Peano.lt n_27_ n1) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_)))))) + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (P n_27_) + (__ : cupicef.Term cupicef.prop (P n_27_) => + Coq__Init__Peano.le n1 n_27_)))) + (Coq__Init__Logic.or__intror + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and + (Coq__Init__Peano.lt n_27_ Coq__Init__Datatypes.O) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_)))))) + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (P n_27_) + (__ : cupicef.Term cupicef.prop (P n_27_) => + Coq__Init__Peano.le Coq__Init__Datatypes.O n_27_))) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (P n_27_) => + Coq__Arith__PeanoNat.Nat__le__0__l n_27_)) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and (Coq__Init__Peano.lt n_27_ n1) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_)))))) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P n_27_) + (__ : cupicef.Term cupicef.prop (P n_27_) => + Coq__Init__Peano.le n1 n_27_)))) => + Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and (Coq__Init__Peano.lt n_27_ n1) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_)))))) + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (P n_27_) + (__ : cupicef.Term cupicef.prop (P n_27_) => + Coq__Init__Peano.le n1 n_27_))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and (Coq__Init__Peano.lt n_27_ n1) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_)))))) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P n_27_) + (__ : cupicef.Term cupicef.prop (P n_27_) => + Coq__Init__Peano.le n1 n_27_)))) => + Coq__Init__Logic.or + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and + (Coq__Init__Peano.lt n_27_ (Coq__Init__Datatypes.S n1)) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P n_27__27_) + (__0 : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_)))))) + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + P n_27_) + (__0 : cupicef.Term cupicef.prop (P n_27_) => + Coq__Init__Peano.le (Coq__Init__Datatypes.S n1) n_27_)))) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and (Coq__Init__Peano.lt n_27_ n1) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_)))))) => + Coq__Init__Logic.match____ex cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and (Coq__Init__Peano.lt n_27_ n1) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_))))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and (Coq__Init__Peano.lt n_27_ n1) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_)))))) => + Coq__Init__Logic.or + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and + (Coq__Init__Peano.lt n_27_ (Coq__Init__Datatypes.S n1)) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P n_27__27_) + (__0 : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_)))))) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P n_27_) + (__0 : cupicef.Term cupicef.prop (P n_27_) => + Coq__Init__Peano.le (Coq__Init__Datatypes.S n1) n_27_)))) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Coq__Init__Peano.lt n_27_ n1) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_))))) => + Coq__Init__Logic.match____and cupicef.prop + (Coq__Init__Peano.lt n_27_ n1) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_)))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Coq__Init__Peano.lt n_27_ n1) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_))))) => + Coq__Init__Logic.or + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_0 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and + (Coq__Init__Peano.lt n_27_0 + (Coq__Init__Datatypes.S n1)) + (Coq__Init__Logic.and (P n_27_0) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P n_27__27_) + (__0 : + cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_0 n_27__27_)))))) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27_0 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P n_27_0) + (__0 : cupicef.Term cupicef.prop (P n_27_0) => + Coq__Init__Peano.le (Coq__Init__Datatypes.S n1) + n_27_0)))) + (IH1 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n_27_ n1) => + IH2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_)))) => + Coq__Init__Logic.or__introl + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_0 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and + (Coq__Init__Peano.lt n_27_0 + (Coq__Init__Datatypes.S n1)) + (Coq__Init__Logic.and (P n_27_0) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_0 n_27__27_)))))) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27_0 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P n_27_0) + (__ : cupicef.Term cupicef.prop (P n_27_0) => + Coq__Init__Peano.le (Coq__Init__Datatypes.S n1) + n_27_0))) + (Coq__Init__Logic.ex__intro + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_0 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and + (Coq__Init__Peano.lt n_27_0 + (Coq__Init__Datatypes.S n1)) + (Coq__Init__Logic.and (P n_27_0) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_0 n_27__27_))))) + n_27_ + (Coq__Init__Logic.conj + (Coq__Init__Peano.lt n_27_ (Coq__Init__Datatypes.S n1)) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_)))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n_27_) + (Coq__Init__Datatypes.S n1)) + (Coq__Init__Peano.lt n_27_ + (Coq__Init__Datatypes.S n1)) + cupicef.I + (Coq__Init__Peano.le__S + (Coq__Init__Datatypes.S n_27_) n1 IH1)) + IH2))) + H1) + H0) + (IH : + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (P n_27_) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n1 n_27_)) => + Coq__Init__Logic.match____or cupicef.prop (P n1) + (Coq__Init__Logic.not (P n1)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (P n1) (Coq__Init__Logic.not (P n1))) => + Coq__Init__Logic.or + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and + (Coq__Init__Peano.lt n_27_ (Coq__Init__Datatypes.S n1)) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P n_27__27_) + (__0 : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_)))))) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P n_27_) + (__0 : cupicef.Term cupicef.prop (P n_27_) => + Coq__Init__Peano.le (Coq__Init__Datatypes.S n1) n_27_)))) + (HP : cupicef.Term cupicef.prop (P n1) => + Coq__Init__Logic.or__introl + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and + (Coq__Init__Peano.lt n_27_ (Coq__Init__Datatypes.S n1)) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_)))))) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P n_27_) + (__ : cupicef.Term cupicef.prop (P n_27_) => + Coq__Init__Peano.le (Coq__Init__Datatypes.S n1) n_27_))) + (Coq__Init__Logic.ex__intro + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and + (Coq__Init__Peano.lt n_27_ (Coq__Init__Datatypes.S n1)) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_))))) + n1 + (Coq__Init__Logic.conj + (Coq__Init__Peano.lt n1 (Coq__Init__Datatypes.S n1)) + (Coq__Init__Logic.and (P n1) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n1 n_27__27_)))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Peano.le (Coq__Init__Datatypes.S n1) + (Coq__Init__Datatypes.S n1)) + (Coq__Init__Peano.lt n1 (Coq__Init__Datatypes.S n1)) + cupicef.I + (Coq__Init__Peano.le__n (Coq__Init__Datatypes.S n1))) + (Coq__Init__Logic.conj (P n1) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n1 n_27__27_))) + HP IH)))) + (HP : cupicef.Term cupicef.prop (Coq__Init__Logic.not (P n1)) => + Coq__Init__Logic.or__intror + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and + (Coq__Init__Peano.lt n_27_ (Coq__Init__Datatypes.S n1)) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_)))))) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P n_27_) + (__ : cupicef.Term cupicef.prop (P n_27_) => + Coq__Init__Peano.le (Coq__Init__Datatypes.S n1) n_27_))) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Hn_27_ : cupicef.Term cupicef.prop (P n_27_) => + let__H3 P Pdec H n0 HPn0 n n1 IHn IH HP n_27_ Hn_27_ n1 n_27_ + (Coq__Init__Logic.conj (Coq__Init__Peano.le n1 n_27_) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n_27_)) + (IH n_27_ Hn_27_) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n_27_) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n_27_) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n_27_)) + cupicef.I + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 n_27_) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 + (n_27_0 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P n_27_0) + (Hn_27_0 : cupicef.Term cupicef.prop (P n_27_0) => + Coq__Init__Logic.False)) + (Hn_27_0 : cupicef.Term cupicef.prop (P n1) => + HP Hn_27_0) + n_27_ H0 Hn_27_))))) + (let__o P Pdec H n0 HPn0 n n1 IHn IH)) + IHn) + n. + +def let__o0 : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + Pdec : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (P n) (Coq__Init__Logic.not (P n)))) -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => P n)) -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + HPn0 : cupicef.Term cupicef.prop (P n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and (Coq__Init__Peano.lt n_27_ n0) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_)))))) + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (P n_27_) + (__ : cupicef.Term cupicef.prop (P n_27_) => + Coq__Init__Peano.le n0 n_27_)))) + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + Pdec : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (P n) (Coq__Init__Logic.not (P n)))) => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => P n)) => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + HPn0 : cupicef.Term cupicef.prop (P n0) => + let__H2 P Pdec H n0 HPn0 n0. + +thm dec__inh__nat__subset__has__unique__least__element : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + __ : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (P n) (Coq__Init__Logic.not (P n)))) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => P n)) -> + cupicef.Term cupicef.prop + (has__unique__least__element + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Peano.le P) + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + Pdec : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (P n) (Coq__Init__Logic.not (P n)))) => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => P n)) => + Coq__Init__Logic.match____ex cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => P n) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => P n)) => + has__unique__least__element + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Peano.le P) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + HPn0 : cupicef.Term cupicef.prop (P n0) => + Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and (Coq__Init__Peano.lt n_27_ n0) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_)))))) + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (P n_27_) + (__ : cupicef.Term cupicef.prop (P n_27_) => + Coq__Init__Peano.le n0 n_27_))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and (Coq__Init__Peano.lt n_27_ n0) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_)))))) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P n_27_) + (__ : cupicef.Term cupicef.prop (P n_27_) => + Coq__Init__Peano.le n0 n_27_)))) => + has__unique__least__element + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Peano.le P) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and (Coq__Init__Peano.lt n_27_ n0) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_)))))) => + Coq__Init__Logic.match____ex cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and (Coq__Init__Peano.lt n_27_ n0) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_))))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and (Coq__Init__Peano.lt n_27_ n0) + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n_27_ n_27__27_)))))) => + has__unique__least__element + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Peano.le P) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Coq__Init__Peano.lt n n0) + (Coq__Init__Logic.and (P n) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n n_27__27_))))) => + Coq__Init__Logic.match____and cupicef.prop + (Coq__Init__Peano.lt n n0) + (Coq__Init__Logic.and (P n) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n n_27__27_)))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Coq__Init__Peano.lt n n0) + (Coq__Init__Logic.and (P n) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n n_27__27_))))) => + has__unique__least__element + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Peano.le P) + (H2 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n n0) => + H3 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (P n) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n n_27__27_)))) => + Coq__Init__Logic.match____and cupicef.prop (P n) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n n_27__27_))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (P n) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n_27__27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P n_27__27_) + (__ : cupicef.Term cupicef.prop (P n_27__27_) => + Coq__Init__Peano.le n n_27__27_)))) => + has__unique__least__element + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Peano.le P) + (H4 : cupicef.Term cupicef.prop (P n) => + H5 : + (n_27__27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (P n_27__27_) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le n n_27__27_)) => + Coq__Init__Logic.ex__intro + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Logic.unique + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and (P x) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (x_27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P x_27_) + (__ : cupicef.Term cupicef.prop (P x_27_) => + Coq__Init__Peano.le x x_27_))))) + n + (Coq__Init__Logic.conj + (Coq__Init__Logic.and (P n) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (x_27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P x_27_) + (__ : cupicef.Term cupicef.prop (P x_27_) => + Coq__Init__Peano.le n x_27_)))) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (x_27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I + (Coq__Init__Logic.and (P x_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (x_27_0 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P x_27_0) + (__ : + cupicef.Term cupicef.prop (P x_27_0) => + Coq__Init__Peano.le x_27_ x_27_0)))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (P x_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop + cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (x_27_0 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop + cupicef.prop + (cupicef.rule cupicef.prop + cupicef.prop) + cupicef.I (P x_27_0) + (__ : + cupicef.Term cupicef.prop + (P x_27_0) => + Coq__Init__Peano.le x_27_ x_27_0)))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n x_27_))) + (Coq__Init__Logic.conj (P n) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (x_27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P x_27_) + (__ : cupicef.Term cupicef.prop (P x_27_) => + Coq__Init__Peano.le n x_27_))) + H4 H5) + (n_27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H6 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (x_27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P x_27_) + (__ : + cupicef.Term cupicef.prop (P x_27_) => + Coq__Init__Peano.le n_27_ x_27_)))) => + Coq__Init__Logic.match____and cupicef.prop (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (x_27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P x_27_) + (__ : cupicef.Term cupicef.prop (P x_27_) => + Coq__Init__Peano.le n_27_ x_27_))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (x_27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop + cupicef.prop) + cupicef.I (P x_27_) + (__ : + cupicef.Term cupicef.prop + (P x_27_) => + Coq__Init__Peano.le n_27_ x_27_)))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n n_27_) + (HPn_27_ : cupicef.Term cupicef.prop (P n_27_) => + Hn_27_ : + (x_27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (P x_27_) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le n_27_ x_27_)) => + Coq__Arith__PeanoNat.Nat__le__antisymm n n_27_ + (H5 n_27_ HPn_27_) (Hn_27_ n H4)) + H6))) + H3) + H1) + H0) + (H0 : + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (P n_27_) -> + cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 n_27_)) => + Coq__Init__Logic.ex__intro + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Logic.unique + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.and (P x) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (x_27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P x_27_) + (__ : cupicef.Term cupicef.prop (P x_27_) => + Coq__Init__Peano.le x x_27_))))) + n0 + (Coq__Init__Logic.conj + (Coq__Init__Logic.and (P n0) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (x_27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P x_27_) + (__ : cupicef.Term cupicef.prop (P x_27_) => + Coq__Init__Peano.le n0 x_27_)))) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (x_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (P x_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (x_27_0 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P x_27_0) + (__ : cupicef.Term cupicef.prop (P x_27_0) => + Coq__Init__Peano.le x_27_ x_27_0)))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (P x_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (x_27_0 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P x_27_0) + (__ : cupicef.Term cupicef.prop (P x_27_0) => + Coq__Init__Peano.le x_27_ x_27_0)))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 x_27_))) + (Coq__Init__Logic.conj (P n0) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (x_27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P x_27_) + (__ : cupicef.Term cupicef.prop (P x_27_) => + Coq__Init__Peano.le n0 x_27_))) + HPn0 H0) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (x_27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P x_27_) + (__ : cupicef.Term cupicef.prop (P x_27_) => + Coq__Init__Peano.le n_27_ x_27_)))) => + Coq__Init__Logic.match____and cupicef.prop (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (x_27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P x_27_) + (__ : cupicef.Term cupicef.prop (P x_27_) => + Coq__Init__Peano.le n_27_ x_27_))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (P n_27_) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (x_27_ : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P x_27_) + (__ : cupicef.Term cupicef.prop (P x_27_) => + Coq__Init__Peano.le n_27_ x_27_)))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 n_27_) + (HPn_27_ : cupicef.Term cupicef.prop (P n_27_) => + Hn_27_ : + (x_27_ : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (P x_27_) -> + cupicef.Term cupicef.prop + (Coq__Init__Peano.le n_27_ x_27_)) => + Coq__Arith__PeanoNat.Nat__le__antisymm n0 n_27_ + (H0 n_27_ HPn_27_) (Hn_27_ n0 HPn0)) + H1))) + (let__o0 P Pdec H n0 HPn0)) + H. + +(; End of translation. ;) diff --git a/out/Coq__Bool__Bool.dk b/out/Coq__Bool__Bool.dk new file mode 100644 index 000000000..2f62500c9 --- /dev/null +++ b/out/Coq__Bool__Bool.dk @@ -0,0 +1,48232 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def Is__true : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Univ cupicef.prop + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True Coq__Init__Logic.False b. + +def let__H0 : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + x : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + x : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True Coq__Init__Logic.False e) + Coq__Init__Logic.I Coq__Init__Datatypes.false H. + +def let__H1 : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + x : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + x : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False Coq__Init__Logic.True e) + Coq__Init__Logic.I Coq__Init__Datatypes.true H. + +def bool__dec : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 b2) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 b2))) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.bool__rec + (b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.bool + (x : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 x) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 x)))) + (x : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.set + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true b) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true b))) + (Coq__Init__Specif.left + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (Coq__Init__Specif.right + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + cupicef.I + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H0 b1 b2 x H)))) + x) + (x : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.set + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false b) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false b))) + (Coq__Init__Specif.right + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + cupicef.I + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H1 b1 b2 x H)))) + (Coq__Init__Specif.left + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + x) + b1 b2. + +def let__H2 : + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True Coq__Init__Logic.False e) + Coq__Init__Logic.I Coq__Init__Datatypes.false H. + +thm diff__true__false : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + +:= cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + cupicef.I + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.False__ind Coq__Init__Logic.False (let__H2 H)). + +def let__H3 : + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False Coq__Init__Logic.True e) + Coq__Init__Logic.I Coq__Init__Datatypes.true H. + +thm diff__false__true : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + +:= cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + cupicef.I + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.False__ind Coq__Init__Logic.False (let__H3 H)). + +def let__H4 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True Coq__Init__Logic.False e) + Coq__Init__Logic.I Coq__Init__Datatypes.false H2. + +def let__H5 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False Coq__Init__Logic.True e) + Coq__Init__Logic.I Coq__Init__Datatypes.true H1. + +thm eq__true__false__abs : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.true) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.true) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.false) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.false) => + Coq__Init__Logic.False))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop Coq__Init__Logic.False + Coq__Init__Logic.False cupicef.I + (Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H4 b H H0 H1 H2))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop Coq__Init__Logic.False + Coq__Init__Logic.False cupicef.I + (Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H5 b H H0 H1 H2))) + b H H0. + +def let__H7 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true. + +def let__H6 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H0 (let__H7 b H H0). + +thm not__true__is__false : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.true)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.false)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + cupicef.I + ((H1 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + H1) + (let__H6 b H H0))) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b H. + +def let__H9 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false. + +def let__H8 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) => + H0 (let__H9 b H H0). + +thm not__false__is__true : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.false)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.true)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + cupicef.I + ((H1 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + H1) + (let__H8 b H H0))) + b H. + +def let__H11 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true. + +def let__H10 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H (let__H11 b H). + +thm not__true__iff__false : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false)) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.iff + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.true)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.false)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + (H1 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + H1) + (let__H10 b H)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + cupicef.I + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop Coq__Init__Logic.False + Coq__Init__Logic.False cupicef.I + (diff__false__true + (Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false H)))))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + cupicef.I + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop Coq__Init__Logic.False + Coq__Init__Logic.False cupicef.I (diff__false__true H0))))) + b. + +def let__H13 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false. + +def let__H12 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) => + H (let__H13 b H). + +thm not__false__iff__true : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.iff + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.false)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + cupicef.I + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop Coq__Init__Logic.False + Coq__Init__Logic.False cupicef.I + (diff__false__true + (Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false + H0)))))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false))) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) => + (H1 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + H1) + (let__H12 b H)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + cupicef.I + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop Coq__Init__Logic.False + Coq__Init__Logic.False cupicef.I (diff__false__true H))))) + b. + +def leb : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Univ cupicef.prop + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true) + Coq__Init__Logic.True b1. + +thm leb__implb : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (leb b1 b2) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.implb b1 b2) Coq__Init__Datatypes.true)) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.iff (leb b1 b) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.implb b1 b) Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.iff (leb b Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.implb b Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (Coq__Init__Logic.iff + (leb Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.implb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s 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Coq__Init__Logic.True => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.I))) 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(cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (Coq__Init__Logic.iff + (leb Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.implb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + H) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + H))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.True + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.True))) + (Coq__Init__Logic.iff + (leb Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.implb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.True + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.True)) + (H : cupicef.Term cupicef.prop Coq__Init__Logic.True => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.I))) + b1) + b2. + +def eqb : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.set + (b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.match____bool cupicef.set + (b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false b2) + (Coq__Init__Datatypes.match____bool cupicef.set + (b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true b2) + b1. + +def let__H14 : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Univ cupicef.prop) -> + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b1 b2) Coq__Init__Datatypes.true) -> + H0 : cupicef.Term cupicef.prop (P b1) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b1 Coq__Init__Datatypes.true) Coq__Init__Datatypes.true) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) -> + H3 : cupicef.Term cupicef.prop (P Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Univ cupicef.prop) => + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b1 b2) Coq__Init__Datatypes.true) => + H0 : cupicef.Term cupicef.prop (P b1) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b1 Coq__Init__Datatypes.true) Coq__Init__Datatypes.true) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) => + H3 : cupicef.Term cupicef.prop (P Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False Coq__Init__Logic.True e) + Coq__Init__Logic.I Coq__Init__Datatypes.true H2. + +def let__H15 : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Univ cupicef.prop) -> + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b1 b2) Coq__Init__Datatypes.true) -> + H0 : cupicef.Term cupicef.prop (P b1) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b1 Coq__Init__Datatypes.false) Coq__Init__Datatypes.true) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) -> + H3 : cupicef.Term cupicef.prop (P Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Univ cupicef.prop) => + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b1 b2) Coq__Init__Datatypes.true) => + H0 : cupicef.Term cupicef.prop (P b1) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b1 Coq__Init__Datatypes.false) Coq__Init__Datatypes.true) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) => + H3 : cupicef.Term cupicef.prop (P Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False Coq__Init__Logic.True e) + Coq__Init__Logic.I Coq__Init__Datatypes.true H2. + +thm eqb__subst : + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Univ cupicef.prop) -> + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b1 b2) Coq__Init__Datatypes.true) -> + __0 : cupicef.Term cupicef.prop (P b1) -> + cupicef.Term cupicef.prop (P b2) + +:= P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Univ cupicef.prop) => + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b1 b2) Coq__Init__Datatypes.true) => + H0 : cupicef.Term cupicef.prop (P b1) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b1 b) Coq__Init__Datatypes.true) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b1 b) Coq__Init__Datatypes.true) => + P b)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b1 Coq__Init__Datatypes.true) Coq__Init__Datatypes.true) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b Coq__Init__Datatypes.true) Coq__Init__Datatypes.true) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (P b) + (H3 : cupicef.Term cupicef.prop (P b) => + P Coq__Init__Datatypes.true))) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) => + H3 : cupicef.Term cupicef.prop (P Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop (P Coq__Init__Datatypes.true) + (P Coq__Init__Datatypes.true) cupicef.I H3) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) => + H3 : cupicef.Term cupicef.prop (P Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop (P Coq__Init__Datatypes.true) + (P Coq__Init__Datatypes.true) cupicef.I + (Coq__Init__Logic.False__ind (P Coq__Init__Datatypes.true) + (let__H14 P b1 b2 H H0 H1 H2 H3))) + b1 H1 H0) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b1 Coq__Init__Datatypes.false) Coq__Init__Datatypes.true) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b Coq__Init__Datatypes.false) Coq__Init__Datatypes.true) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (P b) + (H3 : cupicef.Term cupicef.prop (P b) => + P Coq__Init__Datatypes.false))) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) => + H3 : cupicef.Term cupicef.prop (P Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop + (P Coq__Init__Datatypes.false) (P Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.False__ind (P Coq__Init__Datatypes.false) + (let__H15 P b1 b2 H H0 H1 H2 H3))) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) => + H3 : cupicef.Term cupicef.prop (P Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop + (P Coq__Init__Datatypes.false) (P Coq__Init__Datatypes.false) + cupicef.I H3) + b1 H1 H0) + b2 H. + +thm eqb__reflx : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b b) Coq__Init__Datatypes.true) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b0 b0) Coq__Init__Datatypes.true) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + b. + +def let__H16 : + a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb a b) Coq__Init__Datatypes.true) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb a Coq__Init__Datatypes.false) Coq__Init__Datatypes.true) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb a b) Coq__Init__Datatypes.true) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb a Coq__Init__Datatypes.false) Coq__Init__Datatypes.true) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False Coq__Init__Logic.True e) + Coq__Init__Logic.I Coq__Init__Datatypes.true H1. + +thm eqb__prop : + a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb a b) Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + a b) + +:= a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb a b) Coq__Init__Datatypes.true) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb a b0) Coq__Init__Datatypes.true) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb a b0) Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + a b0)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb a Coq__Init__Datatypes.true) Coq__Init__Datatypes.true) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b0 Coq__Init__Datatypes.true) Coq__Init__Datatypes.true) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b0 Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.true)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + cupicef.I H1) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + cupicef.I H1) + a H0) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb a Coq__Init__Datatypes.false) Coq__Init__Datatypes.true) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b0 Coq__Init__Datatypes.false) Coq__Init__Datatypes.true) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b0 Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.false)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (let__H16 a b H H0 H1))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + a H0) + b H. + +thm eqb__true__iff : + a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb a b) Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + a b)) + +:= a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb a b0) Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + a b0)) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b0 Coq__Init__Datatypes.true) Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) 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cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s 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(cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + 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(cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b0 Coq__Init__Datatypes.false) Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.false)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true H)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false H)))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true))) + a) + b. + +def let__H18 : + a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + +:= a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true. + +def let__H17 : + a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H (let__H18 a b H). + +def let__H20 : + a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + +:= a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false. + +def let__H19 : + a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) => + H (let__H20 a b H). + +thm eqb__false__iff : + a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb a b) Coq__Init__Datatypes.false) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + a b))) + +:= a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb a b0) Coq__Init__Datatypes.false) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + a b0))) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b0 Coq__Init__Datatypes.true) Coq__Init__Datatypes.false) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.true))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + cupicef.I + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop + Coq__Init__Logic.False Coq__Init__Logic.False cupicef.I + (diff__false__true + (Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.false H)))) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + (H1 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + H1) + (let__H17 a b H)))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + cupicef.I + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop + Coq__Init__Logic.False Coq__Init__Logic.False cupicef.I + (diff__false__true H0))) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false))) + a) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b0 Coq__Init__Datatypes.false) Coq__Init__Datatypes.false) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.false))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + cupicef.I + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop + Coq__Init__Logic.False Coq__Init__Logic.False cupicef.I + (diff__false__true + (Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.false H0)))) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + cupicef.I + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop + Coq__Init__Logic.False Coq__Init__Logic.False cupicef.I + (diff__false__true + (Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.false H)))) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + (H1 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + H1) + (let__H19 a b H)))) + a) + b. + +def ifb : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.set + (b4 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.bool) + b2 b3 b1. + +thm negb__orb : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb (Coq__Init__Datatypes.orb b1 b2)) + (Coq__Init__Datatypes.andb (Coq__Init__Datatypes.negb b1) + (Coq__Init__Datatypes.negb b2))) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb (Coq__Init__Datatypes.orb b1 b)) + (Coq__Init__Datatypes.andb (Coq__Init__Datatypes.negb b1) + (Coq__Init__Datatypes.negb b))) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.andb (Coq__Init__Datatypes.negb b) + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b1) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.andb (Coq__Init__Datatypes.negb b) + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + b1) + b2. + +thm negb__andb : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb (Coq__Init__Datatypes.andb b1 b2)) + (Coq__Init__Datatypes.orb (Coq__Init__Datatypes.negb b1) + (Coq__Init__Datatypes.negb b2))) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb (Coq__Init__Datatypes.andb b1 b)) + (Coq__Init__Datatypes.orb (Coq__Init__Datatypes.negb b1) + (Coq__Init__Datatypes.negb b))) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.orb (Coq__Init__Datatypes.negb b) + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + b1) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.orb (Coq__Init__Datatypes.negb b) + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + b1) + b2. + +thm negb__involutive : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb (Coq__Init__Datatypes.negb b)) b) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb (Coq__Init__Datatypes.negb b0)) b0) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true)) + Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false)) + Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b. + +thm negb__involutive__reverse : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.negb (Coq__Init__Datatypes.negb b))) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 (Coq__Init__Datatypes.negb (Coq__Init__Datatypes.negb b0))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.negb + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.negb + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b. + +thm negb__sym : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b_27_ (Coq__Init__Datatypes.negb b)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.negb b_27_)) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b_27_ (Coq__Init__Datatypes.negb b)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 (Coq__Init__Datatypes.negb b)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 (Coq__Init__Datatypes.negb b)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.negb b0))) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (Coq__Init__Datatypes.negb b)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (Coq__Init__Datatypes.negb b0)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (Coq__Init__Datatypes.negb b0)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true)) + cupicef.I H1) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b H0) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false (Coq__Init__Datatypes.negb b)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false (Coq__Init__Datatypes.negb b0)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false (Coq__Init__Datatypes.negb b0)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false)) + cupicef.I H1) + b H0) + b_27_ H. + +def let__H21 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False Coq__Init__Logic.True e) + Coq__Init__Logic.I Coq__Init__Datatypes.true H. + +def let__H22 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True Coq__Init__Logic.False e) + Coq__Init__Logic.I Coq__Init__Datatypes.false H. + +thm no__fixpoint__negb : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb b) b)) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb b0) b0)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true)) + cupicef.I + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + cupicef.I + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.False__ind Coq__Init__Logic.False (let__H21 b H)))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false)) + cupicef.I + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + cupicef.I + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.False__ind Coq__Init__Logic.False (let__H22 b H)))) + b. + +thm eqb__negb1 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb (Coq__Init__Datatypes.negb b) b) Coq__Init__Datatypes.false) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb (Coq__Init__Datatypes.negb b0) b0) Coq__Init__Datatypes.false) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b. + +thm eqb__negb2 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b (Coq__Init__Datatypes.negb b)) Coq__Init__Datatypes.false) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb b0 (Coq__Init__Datatypes.negb b0)) Coq__Init__Datatypes.false) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true)) + Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (eqb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false)) + Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b. + +thm if__negb : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => A) x y + (Coq__Init__Datatypes.negb b)) + (Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => A) y x + b)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => A) x y + (Coq__Init__Datatypes.negb b0)) + (Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => A) y x + b0)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y y) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => A) + x y (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true)) + y) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => A) + x y (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false)) + x) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + b. + +thm negb__true__iff : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb b) Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false)) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb b0) Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.false)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true H)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false H)))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true))) + b. + +thm negb__false__iff : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb b) Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb b0) Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false H)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true H)))) + b. + +thm orb__true__iff : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 b2) Coq__Init__Datatypes.true) + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true))) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 b) Coq__Init__Datatypes.true) + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true))) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.or__introl + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + Coq__Init__Logic.or__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true) + H))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.or__intror + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + Coq__Init__Logic.or__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true) + H))) + b1) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.or__introl + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) => + Coq__Init__Logic.or__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true) + H))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.or__introl + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + H) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) => + Coq__Init__Logic.or__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + H0) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + H0) + H))) + b1) + b2. + +thm orb__false__iff : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 b2) Coq__Init__Datatypes.false) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.false))) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 b) Coq__Init__Datatypes.false) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false))) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.conj + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + H H) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) => + Coq__Init__Logic.and__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + H1) + H))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.conj + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false) + H) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) => + Coq__Init__Logic.and__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + H1) + H))) + b1) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.conj + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + H + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.and__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + H0) + H))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + Coq__Init__Logic.conj + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.and__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false) + H))) + b1) + b2. + +def orb__true__elim : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 b2) Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true)) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.set + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.set) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.set)) + cupicef.I Coq__Init__Datatypes.bool + (b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.set + (cupicef.rule cupicef.prop cupicef.set) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b b2) Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b b2) Coq__Init__Datatypes.true) => + Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true)))) + (cupicef.cast + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.set)) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.set)) + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.set) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.set)) + cupicef.I Coq__Init__Datatypes.bool + (b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.set + (cupicef.rule cupicef.prop cupicef.set) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true)))) + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.set) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.set)) + cupicef.I Coq__Init__Datatypes.bool + (b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.set + (cupicef.rule cupicef.prop cupicef.set) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true b2) + Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true b2) + Coq__Init__Datatypes.true) => + Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true)))) + cupicef.I + (b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Specif.left + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true) + H)) + (cupicef.cast + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.set)) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.set)) + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.set) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.set)) + cupicef.I Coq__Init__Datatypes.bool + (b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.set + (cupicef.rule cupicef.prop cupicef.set) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true) => + Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true)))) + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.set) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.set)) + cupicef.I Coq__Init__Datatypes.bool + (b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.set + (cupicef.rule cupicef.prop cupicef.set) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false b2) + Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false b2) + Coq__Init__Datatypes.true) => + Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true)))) + cupicef.I + (b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true) => + Coq__Init__Specif.right + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true) + H)) + b1. + +def let__H23 : + a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb a b) Coq__Init__Datatypes.true) -> + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 b2) Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true)) + +:= a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb a b) Coq__Init__Datatypes.true) => + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 b2) Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 b2) Coq__Init__Datatypes.true) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 b2) Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 b2) Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 b2) + Coq__Init__Datatypes.true) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 b2) Coq__Init__Datatypes.true))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 b2) Coq__Init__Datatypes.true) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 b2) Coq__Init__Datatypes.true) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true))) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 b2) Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 b2) Coq__Init__Datatypes.true)) => + __) + (orb__true__iff b1 b2). + +thm orb__prop : + a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb a b) Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + a Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) + +:= a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb a b) Coq__Init__Datatypes.true) => + let__H23 a b H a b H. + +def let__H24 : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true)) -> + b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b4 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b3 b4) Coq__Init__Datatypes.true) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true)) => + b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b4 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b3 b4) Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b3 b4) Coq__Init__Datatypes.true) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b3 b4) Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b3 b4) Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b3 b4) + Coq__Init__Datatypes.true) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b3 b4) Coq__Init__Datatypes.true))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.true)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b3 b4) Coq__Init__Datatypes.true)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b3 b4) Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.true))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b3 b4) Coq__Init__Datatypes.true)) => + __0) + (orb__true__iff b3 b4). + +thm orb__true__intro : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 b2) Coq__Init__Datatypes.true) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true)) => + let__H24 b1 b2 H b1 b2 H. + +thm orb__false__intro : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.false) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 b2) Coq__Init__Datatypes.false) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.false) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq__ind__r + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b3 b2) Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq__ind__r + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false b3) + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false) + b2 H0) + b1 H. + +def let__H25 : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 b2) Coq__Init__Datatypes.false) -> + b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b4 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b3 b4) Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false)) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 b2) Coq__Init__Datatypes.false) => + b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b4 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b3 b4) Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b3 b4) Coq__Init__Datatypes.false) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b3 b4) Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b3 b4) + Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b3 b4) + Coq__Init__Datatypes.false) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b3 b4) + Coq__Init__Datatypes.false))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b3 b4) Coq__Init__Datatypes.false) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b3 b4) Coq__Init__Datatypes.false) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false))) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b3 b4) Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b3 b4) Coq__Init__Datatypes.false)) => + __) + (orb__false__iff b3 b4). + +thm orb__false__elim : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 b2) Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.false)) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 b2) Coq__Init__Datatypes.false) => + let__H25 b1 b2 H b1 b2 H. + +thm orb__diag : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b b) b) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b0 b0) b0) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b. + +thm orb__true__r : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b0 Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + b. + +thm orb__true__l : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true b) + Coq__Init__Datatypes.true) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true. + +thm orb__false__r : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.false) b) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b0 Coq__Init__Datatypes.false) b0) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b. + +thm orb__false__l : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false b) b) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false b0) b0) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b. + +thm orb__negb__r : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b (Coq__Init__Datatypes.negb b)) + Coq__Init__Datatypes.true) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b0 (Coq__Init__Datatypes.negb b0)) + Coq__Init__Datatypes.true) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true)) + Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false)) + Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + b. + +thm orb__comm : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 b2) (Coq__Init__Datatypes.orb b2 b1)) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 b) (Coq__Init__Datatypes.orb b b1)) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true b)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + b1) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false b)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b1) + b2. + +thm orb__assoc : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 (Coq__Init__Datatypes.orb b2 b3)) + (Coq__Init__Datatypes.orb (Coq__Init__Datatypes.orb b1 b2) b3)) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 (Coq__Init__Datatypes.orb b2 b)) + (Coq__Init__Datatypes.orb (Coq__Init__Datatypes.orb b1 b2) b)) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.orb (Coq__Init__Datatypes.orb b1 b) + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + b1) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + b1) + b2) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.orb (Coq__Init__Datatypes.orb b1 b) + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + b1) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b1) + b2) + b3. + +thm andb__true__iff : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b1 b2) Coq__Init__Datatypes.true) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true))) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b1 b) Coq__Init__Datatypes.true) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true))) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.conj + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + Coq__Init__Logic.and__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true) + H))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.conj + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + H + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + Coq__Init__Logic.and__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + H0) + H))) + b1) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.conj + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true) + H) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) => + Coq__Init__Logic.and__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + H1) + H))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.conj + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + H H) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) => + Coq__Init__Logic.and__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + H1) + H))) + b1) + b2. + +thm andb__false__iff : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b1 b2) Coq__Init__Datatypes.false) + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.false))) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b1 b) Coq__Init__Datatypes.false) + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false))) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.or__introl + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + H) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) => + Coq__Init__Logic.or__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + H0) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + H0) + H))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + Coq__Init__Logic.or__introl + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) => + Coq__Init__Logic.or__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false) + H))) + b1) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + Coq__Init__Logic.or__intror + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.or__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false) + H))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + Coq__Init__Logic.or__introl + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.or__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false) + H))) + b1) + b2. + +def let__H26 : + a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (Coq__Init__Datatypes.andb a b)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb a Coq__Init__Datatypes.true)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (Coq__Init__Datatypes.andb a b)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb a Coq__Init__Datatypes.true)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True Coq__Init__Logic.False e) + Coq__Init__Logic.I Coq__Init__Datatypes.false H1. + +def let__H27 : + a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (Coq__Init__Datatypes.andb a b)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb a Coq__Init__Datatypes.false)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (Coq__Init__Datatypes.andb a b)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb a Coq__Init__Datatypes.false)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True Coq__Init__Logic.False e) + Coq__Init__Logic.I Coq__Init__Datatypes.false H1. + +def let__H28 : + a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (Coq__Init__Datatypes.andb a b)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb a Coq__Init__Datatypes.false)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (Coq__Init__Datatypes.andb a b)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb a Coq__Init__Datatypes.false)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True Coq__Init__Logic.False e) + Coq__Init__Logic.I Coq__Init__Datatypes.false H1. + +def andb__true__eq : + a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (Coq__Init__Datatypes.andb a b)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true a) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true b)) + +:= a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (Coq__Init__Datatypes.andb a b)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (Coq__Init__Datatypes.andb a b0)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (Coq__Init__Datatypes.andb a b0)) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true a) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true b0))) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb a Coq__Init__Datatypes.true)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb b0 Coq__Init__Datatypes.true)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb b0 Coq__Init__Datatypes.true)) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true b0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.conj + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + H1 H1)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.False__ind + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (let__H26 a b H H0 H1))) + a H0) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb a Coq__Init__Datatypes.false)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb b0 Coq__Init__Datatypes.false)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb b0 Coq__Init__Datatypes.false)) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true b0) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.False__ind + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (let__H27 a b H H0 H1))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.False__ind + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (let__H28 a b H H0 H1))) + a H0) + b H. + +def let__H29 : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.false) -> + b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b4 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b3 b4) Coq__Init__Datatypes.false) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.false) => + b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b4 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b3 b4) Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b3 b4) Coq__Init__Datatypes.false) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b3 b4) Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b3 b4) + Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b3 b4) + Coq__Init__Datatypes.false) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b3 b4) + Coq__Init__Datatypes.false))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b3 b4) Coq__Init__Datatypes.false)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b3 b4) Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b3 b4) Coq__Init__Datatypes.false)) => + __0) + (andb__false__iff b3 b4). + +thm andb__false__intro1 : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b1 b2) Coq__Init__Datatypes.false) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.false) => + let__H29 b1 b2 H b1 b2 + (Coq__Init__Logic.or__introl + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.false) + H). + +def let__H30 : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.false) -> + b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b4 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b3 b4) Coq__Init__Datatypes.false) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.false) => + b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b4 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b3 b4) Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b3 b4) Coq__Init__Datatypes.false) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b3 b4) Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b3 b4) + Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b3 b4) + Coq__Init__Datatypes.false) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b3 b4) + Coq__Init__Datatypes.false))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b3 b4) Coq__Init__Datatypes.false)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b3 b4) Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b3 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b4 Coq__Init__Datatypes.false)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b3 b4) Coq__Init__Datatypes.false)) => + __0) + (andb__false__iff b3 b4). + +thm andb__false__intro2 : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b1 b2) Coq__Init__Datatypes.false) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.false) => + let__H30 b1 b2 H b1 b2 + (Coq__Init__Logic.or__intror + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.false) + H). + +thm andb__false__r : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b0 Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b. + +thm andb__false__l : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false b) + Coq__Init__Datatypes.false) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false. + +thm andb__diag : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b b) b) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b0 b0) b0) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b. + +thm andb__true__r : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.true) b) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b0 Coq__Init__Datatypes.true) b0) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b. + +thm andb__true__l : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true b) b) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b. + +def andb__false__elim : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b1 b2) Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.false)) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.set + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.set) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.set)) + cupicef.I Coq__Init__Datatypes.bool + (b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.set + (cupicef.rule cupicef.prop cupicef.set) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b b2) Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b b2) + Coq__Init__Datatypes.false) => + Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.false)))) + (cupicef.cast + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.set)) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.set)) + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.set) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.set)) + cupicef.I Coq__Init__Datatypes.bool + (b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.set + (cupicef.rule cupicef.prop cupicef.set) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.false) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.false) => + Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.false)))) + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.set) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.set)) + cupicef.I Coq__Init__Datatypes.bool + (b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.set + (cupicef.rule cupicef.prop cupicef.set) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true b2) + Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true b2) + Coq__Init__Datatypes.false) => + Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.false)))) + cupicef.I + (b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.false) => + Coq__Init__Specif.right + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.false) + H)) + (cupicef.cast + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.set)) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.set)) + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.set) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.set)) + cupicef.I Coq__Init__Datatypes.bool + (b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.set + (cupicef.rule cupicef.prop cupicef.set) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.false)))) + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.set) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.set)) + cupicef.I Coq__Init__Datatypes.bool + (b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.set + (cupicef.rule cupicef.prop cupicef.set) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false b2) + Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + b2) + Coq__Init__Datatypes.false) => + Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.false)))) + cupicef.I + (b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + Coq__Init__Specif.left + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.false) + H)) + b1. + +thm andb__negb__r : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b (Coq__Init__Datatypes.negb b)) + Coq__Init__Datatypes.false) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b0 (Coq__Init__Datatypes.negb b0)) + Coq__Init__Datatypes.false) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true)) + Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false)) + Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b. + +thm andb__comm : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b1 b2) (Coq__Init__Datatypes.andb b2 b1)) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b1 b) (Coq__Init__Datatypes.andb b b1)) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true b)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b1) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false b)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b1) + b2. + +thm andb__assoc : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b1 (Coq__Init__Datatypes.andb b2 b3)) + (Coq__Init__Datatypes.andb (Coq__Init__Datatypes.andb b1 b2) b3)) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b1 (Coq__Init__Datatypes.andb b2 b)) + (Coq__Init__Datatypes.andb (Coq__Init__Datatypes.andb b1 b2) b)) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b1 + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.andb (Coq__Init__Datatypes.andb b1 b) + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b1) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b1) + b2) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b1 + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.andb (Coq__Init__Datatypes.andb b1 b) + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b1) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b1) + b2) + b3. + +thm andb__orb__distrib__r : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b1 (Coq__Init__Datatypes.orb b2 b3)) + (Coq__Init__Datatypes.orb (Coq__Init__Datatypes.andb b1 b2) + (Coq__Init__Datatypes.andb b1 b3))) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b1 (Coq__Init__Datatypes.orb b2 b)) + (Coq__Init__Datatypes.orb (Coq__Init__Datatypes.andb b1 b2) + (Coq__Init__Datatypes.andb b1 b))) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b1 + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.orb (Coq__Init__Datatypes.andb b1 b) + (Coq__Init__Datatypes.andb b1 Coq__Init__Datatypes.true))) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.true))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b1) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.true))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b1) + b2) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b1 + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.orb (Coq__Init__Datatypes.andb b1 b) + (Coq__Init__Datatypes.andb b1 Coq__Init__Datatypes.false))) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.false))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b1) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.false))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b1) + b2) + b3. + +thm andb__orb__distrib__l : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb (Coq__Init__Datatypes.orb b1 b2) b3) + (Coq__Init__Datatypes.orb (Coq__Init__Datatypes.andb b1 b3) + (Coq__Init__Datatypes.andb b2 b3))) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb (Coq__Init__Datatypes.orb b1 b2) b) + (Coq__Init__Datatypes.orb (Coq__Init__Datatypes.andb b1 b) + (Coq__Init__Datatypes.andb b2 b))) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s 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(Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b1) + b2) + b3. + +thm orb__andb__distrib__r : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 (Coq__Init__Datatypes.andb b2 b3)) + (Coq__Init__Datatypes.andb (Coq__Init__Datatypes.orb b1 b2) + (Coq__Init__Datatypes.orb b1 b3))) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 (Coq__Init__Datatypes.andb b2 b)) + (Coq__Init__Datatypes.andb (Coq__Init__Datatypes.orb b1 b2) + (Coq__Init__Datatypes.orb b1 b))) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.andb (Coq__Init__Datatypes.orb b1 b) + (Coq__Init__Datatypes.orb b1 Coq__Init__Datatypes.true))) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.true))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + b1) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.true))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b1) + b2) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.andb (Coq__Init__Datatypes.orb b1 b) + (Coq__Init__Datatypes.orb b1 Coq__Init__Datatypes.false))) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.false))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b1) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.false))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b1) + b2) + b3. + +thm orb__andb__distrib__l : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb (Coq__Init__Datatypes.andb b1 b2) b3) + (Coq__Init__Datatypes.andb (Coq__Init__Datatypes.orb b1 b3) + (Coq__Init__Datatypes.orb b2 b3))) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b3 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb (Coq__Init__Datatypes.andb b1 b2) b) + (Coq__Init__Datatypes.andb (Coq__Init__Datatypes.orb b1 b) + (Coq__Init__Datatypes.orb b2 b))) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb (Coq__Init__Datatypes.andb b1 b) + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.orb b1 Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.true))) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + b1) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.andb + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type 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(cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b1 (Coq__Init__Datatypes.orb b1 b)) b1) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.true)) + b) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b1) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb b + (Coq__Init__Datatypes.orb b Coq__Init__Datatypes.false)) + b) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b1) + b2. + +thm absorption__orb : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 (Coq__Init__Datatypes.andb b1 b2)) b1) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b1 (Coq__Init__Datatypes.andb b1 b)) b1) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.true)) + b) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b1) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb b + (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.false)) + b) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b1) + b2. + +thm xorb__false__r : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b Coq__Init__Datatypes.false) b) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.false) b0) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b. + +thm xorb__false__l : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false b) b) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false b0) b0) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b. + +thm xorb__true__r : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.negb b)) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb b). + +thm xorb__true__l : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true b) + (Coq__Init__Datatypes.negb b)) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb b). + +thm xorb__nilpotent : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b) Coq__Init__Datatypes.false) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 b0) Coq__Init__Datatypes.false) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b. + +thm xorb__comm : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) + (Coq__Init__Datatypes.xorb b_27_ b)) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b0) (Coq__Init__Datatypes.xorb b0 b)) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true b0)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + b) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false b0)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b) + b_27_. + +thm xorb__assoc__reverse : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27__27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb (Coq__Init__Datatypes.xorb b b_27_) + b_27__27_) + (Coq__Init__Datatypes.xorb b + (Coq__Init__Datatypes.xorb b_27_ b_27__27_))) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27__27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb (Coq__Init__Datatypes.xorb b b_27_) b0) + (Coq__Init__Datatypes.xorb b (Coq__Init__Datatypes.xorb b_27_ b0))) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb (Coq__Init__Datatypes.xorb b b0) + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.xorb b + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.true))) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.xorb b0 + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.xorb b0 + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + b) + b_27_) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb (Coq__Init__Datatypes.xorb b b0) + Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.xorb b + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.false))) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.xorb b0 + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + b) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.xorb b0 + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b) + b_27_) + b_27__27_. + +def let__H31 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) Coq__Init__Datatypes.false) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) Coq__Init__Datatypes.false) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True Coq__Init__Logic.False e) + Coq__Init__Logic.I Coq__Init__Datatypes.false H1. + +thm xorb__eq : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b b_27_) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) Coq__Init__Datatypes.false) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b0) Coq__Init__Datatypes.false) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b0) Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b b0)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.true)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (let__H31 b b_27_ H H0 H1))) + b H0) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.false)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + cupicef.I H1) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + cupicef.I H1) + b H0) + b_27_ H. + +def let__H32 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27__27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) b_27__27_) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) Coq__Init__Datatypes.true) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27__27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) b_27__27_) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) Coq__Init__Datatypes.true) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False Coq__Init__Logic.True e) + Coq__Init__Logic.I Coq__Init__Datatypes.true H2. + +def let__H33 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27__27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) b_27__27_) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) Coq__Init__Datatypes.false) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27__27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) b_27__27_) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) Coq__Init__Datatypes.false) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True Coq__Init__Logic.False e) + Coq__Init__Logic.I Coq__Init__Datatypes.false H2. + +thm xorb__move__l__r__1 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27__27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) b_27__27_) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b_27_ (Coq__Init__Datatypes.xorb b b_27__27_)) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27__27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) b_27__27_) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) b0) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) b0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b_27_ (Coq__Init__Datatypes.xorb b b0))) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) Coq__Init__Datatypes.true) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b0) Coq__Init__Datatypes.true) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b0) Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 (Coq__Init__Datatypes.xorb b Coq__Init__Datatypes.true))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.true))) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (let__H32 b b_27_ b_27__27_ H H0 H1 H2))) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + cupicef.I H2) + b H1) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.true))) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + cupicef.I H2) + b H1) + b_27_ H0) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) Coq__Init__Datatypes.false) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b0) Coq__Init__Datatypes.false) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b0) + Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 (Coq__Init__Datatypes.xorb b Coq__Init__Datatypes.false))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.false))) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + cupicef.I H2) + b H1) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.false))) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (let__H33 b b_27_ b_27__27_ H H0 H1 H2))) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + cupicef.I H2) + b H1) + b_27_ H0) + b_27__27_ H. + +def let__H34 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27__27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) b_27__27_) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) Coq__Init__Datatypes.true) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27__27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) b_27__27_) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) Coq__Init__Datatypes.true) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False Coq__Init__Logic.True e) + Coq__Init__Logic.I Coq__Init__Datatypes.true H2. + +def let__H35 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27__27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) b_27__27_) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) Coq__Init__Datatypes.false) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27__27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) b_27__27_) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) Coq__Init__Datatypes.false) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True Coq__Init__Logic.False e) + Coq__Init__Logic.I Coq__Init__Datatypes.false H2. + +thm xorb__move__l__r__2 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27__27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) b_27__27_) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27__27_ b_27_)) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27__27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) b_27__27_) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) b0) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) b0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b0 b_27_))) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) Coq__Init__Datatypes.true) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b0) Coq__Init__Datatypes.true) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b0) Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true b0))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true))) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (let__H34 b b_27_ b_27__27_ H H0 H1 H2))) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b H1) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false))) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + cupicef.I H2) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + cupicef.I H2) + b H1) + b_27_ H0) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27_) Coq__Init__Datatypes.false) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b0) Coq__Init__Datatypes.false) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b0) + Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false b0))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true))) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (let__H35 b b_27_ b_27__27_ H H0 H1 H2))) + b H1) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false))) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + cupicef.I H2) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + cupicef.I H2) + b H1) + b_27_ H0) + b_27__27_ H. + +def let__H36 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27__27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ b_27__27_)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ Coq__Init__Datatypes.true)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27__27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ b_27__27_)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ Coq__Init__Datatypes.true)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True Coq__Init__Logic.False e) + Coq__Init__Logic.I Coq__Init__Datatypes.false H2. + +def let__H37 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27__27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ b_27__27_)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ Coq__Init__Datatypes.false)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27__27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ b_27__27_)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ Coq__Init__Datatypes.false)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False Coq__Init__Logic.True e) + Coq__Init__Logic.I Coq__Init__Datatypes.true H2. + +thm xorb__move__r__l__1 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27__27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ b_27__27_)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b_27_ b) b_27__27_) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27__27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ b_27__27_)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ b0)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ b0)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b_27_ b) b0)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ Coq__Init__Datatypes.true)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.true)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 b) Coq__Init__Datatypes.true)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true b0) + Coq__Init__Datatypes.true)) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (let__H36 b b_27_ b_27__27_ H H0 H1 H2))) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + b H1) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false b0) + Coq__Init__Datatypes.true)) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + cupicef.I H2) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) + cupicef.I H2) + b H1) + b_27_ H0) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ Coq__Init__Datatypes.false)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.false)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 b) Coq__Init__Datatypes.false)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true b0) + Coq__Init__Datatypes.false)) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (let__H37 b b_27_ b_27__27_ H H0 H1 H2))) + b H1) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false b0) + Coq__Init__Datatypes.false)) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) + cupicef.I H2) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + cupicef.I H2) + b H1) + b_27_ H0) + b_27__27_ H. + +def let__H38 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27__27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ b_27__27_)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ Coq__Init__Datatypes.true)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27__27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ b_27__27_)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ Coq__Init__Datatypes.true)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True Coq__Init__Logic.False e) + Coq__Init__Logic.I Coq__Init__Datatypes.false H2. + +def let__H39 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27__27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ b_27__27_)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ Coq__Init__Datatypes.true)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27__27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ b_27__27_)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ Coq__Init__Datatypes.true)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False Coq__Init__Logic.True e) + Coq__Init__Logic.I Coq__Init__Datatypes.true H2. + +thm xorb__move__r__l__2 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27__27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ b_27__27_)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b_27__27_) b_27_) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27__27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ b_27__27_)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ b0)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ b0)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b b0) b_27_)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ Coq__Init__Datatypes.true)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.true)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b Coq__Init__Datatypes.true) b0)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true)) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (let__H38 b b_27_ b_27__27_ H H0 H1 H2))) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + b H1) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false)) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (let__H39 b b_27_ b_27__27_ H H0 H1 H2))) + b H1) + b_27_ H0) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b_27_ Coq__Init__Datatypes.false)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.false)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b Coq__Init__Datatypes.false) b0)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true)) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) + cupicef.I H2) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) + cupicef.I H2) + b H1) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb b0 Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false)) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + cupicef.I H2) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false) + cupicef.I H2) + b H1) + b_27_ H0) + b_27__27_ H. + +thm negb__xorb__l : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb (Coq__Init__Datatypes.xorb b b_27_)) + (Coq__Init__Datatypes.xorb (Coq__Init__Datatypes.negb b) b_27_)) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I + Coq__Init__Datatypes.bool + (b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb (Coq__Init__Datatypes.xorb b0 b_27_)) + (Coq__Init__Datatypes.xorb (Coq__Init__Datatypes.negb b0) b_27_))) + (b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true b0)) + (Coq__Init__Datatypes.xorb + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true) b0)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true) + Coq__Init__Datatypes.false)) + b_27_) + (b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false b0)) + (Coq__Init__Datatypes.xorb + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false) b0)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false) + Coq__Init__Datatypes.false)) + b_27_) + b. + +thm negb__xorb__r : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb (Coq__Init__Datatypes.xorb b b_27_)) + (Coq__Init__Datatypes.xorb b (Coq__Init__Datatypes.negb b_27_))) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I + Coq__Init__Datatypes.bool + (b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb (Coq__Init__Datatypes.xorb b0 b_27_)) + (Coq__Init__Datatypes.xorb b0 (Coq__Init__Datatypes.negb b_27_)))) + (b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true b0)) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.negb b0))) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true))) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false))) + b_27_) + (b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false b0)) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.negb b0))) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true))) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false))) + b_27_) + b. + +thm xorb__negb__negb : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb (Coq__Init__Datatypes.negb b) + (Coq__Init__Datatypes.negb b_27_)) + (Coq__Init__Datatypes.xorb b b_27_)) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I + Coq__Init__Datatypes.bool + (b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb (Coq__Init__Datatypes.negb b0) + (Coq__Init__Datatypes.negb b_27_)) + (Coq__Init__Datatypes.xorb b0 b_27_))) + (b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true) + (Coq__Init__Datatypes.negb b0)) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true b0)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + b_27_) + (b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false) + (Coq__Init__Datatypes.negb b0)) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false b0)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.xorb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + b_27_) + b. + +def let__H40 : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + H0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + H1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true. + +def let__H : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + H0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + H1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H0 (let__H40 b1 b2 H H0 H1). + +def let__H41 : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + H0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + H1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + H1 H2. + +def let__H43 : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + H0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + H1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true. + +def let__H42 : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + H0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + H1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) => + H1 (let__H43 b1 b2 H H0 H1). + +def let__H44 : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + H0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + H1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + H0 H2. + +def let__H46 : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) -> + H0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) -> + H1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) => + H0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) => + H1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true. + +def let__H45 : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) -> + H0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) -> + H1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) => + H0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) => + H1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H0 (let__H46 b1 b2 H H0 H1). + +def let__H47 : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) -> + H0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) -> + H1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) => + H0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) => + H1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + H1 H2. + +thm eq__iff__eq__true : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 b2) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true))) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 b) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true))) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (H0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true) + (let__H b1 b2 H H0 H1)) + H))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + H)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (H0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) => + H2) (let__H42 b1 b2 H H0 H1)) + H))) + b1) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast 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(cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false H)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (H0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + H1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true + H2)) + (let__H45 b1 b2 H H0 H1)) + H))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false))) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.false) => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + H0) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + H0)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (H0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + H1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false) + H))) + b1) + b2. + +def let__H48 : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 b2) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 b2) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 b2) => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 b2)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 b2) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 b2) => + Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 b2))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 b2)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 b2) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true))) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 b2)) => + __0) + (eq__iff__eq__true b1 b2). + +thm eq__true__iff__eq : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b2 Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 b2) + +:= let__H48. + +def let__H50 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb b) Coq__Init__Datatypes.true)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb b) Coq__Init__Datatypes.true)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true. + +def let__H49 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb b) Coq__Init__Datatypes.true)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb b) Coq__Init__Datatypes.true)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true)) => + H0 (let__H50 b H H0). + +thm eq__true__negb__classical : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb b) Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb b) Coq__Init__Datatypes.true)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb b0) Coq__Init__Datatypes.true)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb b0) Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.true)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + cupicef.I + ((H1 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + H1) + (let__H49 b H H0))) + b H. + +def let__H52 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true. + +def let__H51 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H0 (let__H52 b H H0). + +thm eq__true__not__negb : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb b) Coq__Init__Datatypes.true) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.true)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb b0) Coq__Init__Datatypes.true)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true) + Coq__Init__Datatypes.true) + cupicef.I + ((H1 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + H1) + (let__H51 b H H0))) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false) + Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + b H. + +thm absurd__eq__bool : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : cupicef.Term cupicef.prop Coq__Init__Logic.False -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b b_27_) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b b_27_) + H. + +thm absurd__eq__true : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : cupicef.Term cupicef.prop Coq__Init__Logic.False -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true) + H. + +thm trans__eq__bool : + x : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + y : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + z : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + x y) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + y z) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + x z) + +:= Coq__Init__Logic.eq__trans + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool). + +thm Is__true__eq__true : + x : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : cupicef.Term cupicef.prop (Is__true x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + x Coq__Init__Datatypes.true) + +:= x : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : cupicef.Term cupicef.prop (Is__true x) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (Is__true b) + (H0 : cupicef.Term cupicef.prop (Is__true b) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) + (H0 : cupicef.Term cupicef.prop (Is__true Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (H0 : cupicef.Term cupicef.prop (Is__true Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + H0)) + x H. + +thm Is__true__eq__left : + x : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + x Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop (Is__true x) + +:= x : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + x Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__ind__r + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (x0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => Is__true x0) + (cupicef.cast cupicef.prop cupicef.prop Coq__Init__Logic.True + (Is__true Coq__Init__Datatypes.true) cupicef.I Coq__Init__Logic.I) + x H. + +thm Is__true__eq__right : + x : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true x) -> + cupicef.Term cupicef.prop (Is__true x) + +:= x : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true x) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (x0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => Is__true x0) + (cupicef.cast cupicef.prop cupicef.prop Coq__Init__Logic.True + (Is__true Coq__Init__Datatypes.true) cupicef.I Coq__Init__Logic.I) + x H. + +thm eqb__refl : + x : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop (Is__true (eqb x x)) + +:= x : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Is__true (eqb b b)) + (cupicef.cast cupicef.prop cupicef.prop Coq__Init__Logic.True + (Is__true (eqb Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + cupicef.I Coq__Init__Logic.I) + (cupicef.cast cupicef.prop cupicef.prop Coq__Init__Logic.True + (Is__true (eqb Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + cupicef.I Coq__Init__Logic.I) + x. + +thm eqb__eq : + x : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + y : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : cupicef.Term cupicef.prop (Is__true (eqb x y)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + x y) + +:= x : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + y : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : cupicef.Term cupicef.prop (Is__true (eqb x y)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Is__true (eqb x b)) + (H0 : cupicef.Term cupicef.prop (Is__true (eqb x b)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + x b)) + (H0 : + cupicef.Term cupicef.prop + (Is__true (eqb x Coq__Init__Datatypes.true)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Is__true (eqb b Coq__Init__Datatypes.true)) + (H1 : + cupicef.Term cupicef.prop + (Is__true (eqb b Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) + (H1 : + cupicef.Term cupicef.prop + (Is__true + (eqb Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (H1 : + cupicef.Term cupicef.prop + (Is__true + (eqb Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + H1)) + x H0) + (H0 : + cupicef.Term cupicef.prop + (Is__true (eqb x Coq__Init__Datatypes.false)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Is__true (eqb b Coq__Init__Datatypes.false)) + (H1 : + cupicef.Term cupicef.prop + (Is__true (eqb b Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false)) + (H1 : + cupicef.Term cupicef.prop + (Is__true + (eqb Coq__Init__Datatypes.true Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + H1)) + (H1 : + cupicef.Term cupicef.prop + (Is__true + (eqb Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + x H0) + y H. + +thm orb__prop__elim : + a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : cupicef.Term cupicef.prop (Is__true (Coq__Init__Datatypes.orb a b)) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.or (Is__true a) (Is__true b)) + +:= a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : cupicef.Term cupicef.prop (Is__true (Coq__Init__Datatypes.orb a b)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Is__true (Coq__Init__Datatypes.orb a b0)) + (H0 : + cupicef.Term cupicef.prop + (Is__true (Coq__Init__Datatypes.orb a b0)) => + Coq__Init__Logic.or (Is__true a) (Is__true b0))) + (H0 : + cupicef.Term cupicef.prop + (Is__true (Coq__Init__Datatypes.orb a Coq__Init__Datatypes.true)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Is__true (Coq__Init__Datatypes.orb b0 Coq__Init__Datatypes.true)) + (H1 : + cupicef.Term cupicef.prop + (Is__true + (Coq__Init__Datatypes.orb b0 Coq__Init__Datatypes.true)) => + Coq__Init__Logic.or (Is__true b0) + (Is__true Coq__Init__Datatypes.true))) + (H1 : + cupicef.Term cupicef.prop + (Is__true + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.or Coq__Init__Logic.True Coq__Init__Logic.True) + (Coq__Init__Logic.or (Is__true Coq__Init__Datatypes.true) + (Is__true Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.or__introl Coq__Init__Logic.True + Coq__Init__Logic.True Coq__Init__Logic.I)) + (H1 : + cupicef.Term cupicef.prop + (Is__true + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.or Coq__Init__Logic.False Coq__Init__Logic.True) + (Coq__Init__Logic.or (Is__true Coq__Init__Datatypes.false) + (Is__true Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.or__intror Coq__Init__Logic.False + Coq__Init__Logic.True Coq__Init__Logic.I)) + a H0) + (H0 : + cupicef.Term cupicef.prop + (Is__true (Coq__Init__Datatypes.orb a Coq__Init__Datatypes.false)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Is__true (Coq__Init__Datatypes.orb b0 Coq__Init__Datatypes.false)) + (H1 : + cupicef.Term cupicef.prop + (Is__true + (Coq__Init__Datatypes.orb b0 Coq__Init__Datatypes.false)) => + Coq__Init__Logic.or (Is__true b0) + (Is__true Coq__Init__Datatypes.false))) + (H1 : + cupicef.Term cupicef.prop + (Is__true + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.or Coq__Init__Logic.True Coq__Init__Logic.False) + (Coq__Init__Logic.or (Is__true Coq__Init__Datatypes.true) + (Is__true Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.or__introl Coq__Init__Logic.True + Coq__Init__Logic.False Coq__Init__Logic.I)) + (H1 : + cupicef.Term cupicef.prop + (Is__true + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.or Coq__Init__Logic.False Coq__Init__Logic.False) + (Coq__Init__Logic.or (Is__true Coq__Init__Datatypes.false) + (Is__true Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.False__ind + (Coq__Init__Logic.or Coq__Init__Logic.False + Coq__Init__Logic.False) + H1)) + a H0) + b H. + +thm orb__prop__intro : + a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop (Coq__Init__Logic.or (Is__true a) (Is__true b)) -> + cupicef.Term cupicef.prop (Is__true (Coq__Init__Datatypes.orb a b)) + +:= a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Is__true a) (Is__true b)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or (Is__true a) (Is__true b0)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Is__true a) (Is__true b0)) => + Is__true (Coq__Init__Datatypes.orb a b0))) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Is__true a) + (Is__true Coq__Init__Datatypes.true)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or (Is__true b0) + (Is__true Coq__Init__Datatypes.true)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Is__true b0) + (Is__true Coq__Init__Datatypes.true)) => + Is__true (Coq__Init__Datatypes.orb b0 Coq__Init__Datatypes.true))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Is__true Coq__Init__Datatypes.true) + (Is__true Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop Coq__Init__Logic.True + (Is__true + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + cupicef.I Coq__Init__Logic.I) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Is__true Coq__Init__Datatypes.false) + (Is__true Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop Coq__Init__Logic.True + (Is__true + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + cupicef.I Coq__Init__Logic.I) + a H0) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Is__true a) + (Is__true Coq__Init__Datatypes.false)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or (Is__true b0) + (Is__true Coq__Init__Datatypes.false)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Is__true b0) + (Is__true Coq__Init__Datatypes.false)) => + Is__true (Coq__Init__Datatypes.orb b0 Coq__Init__Datatypes.false))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Is__true Coq__Init__Datatypes.true) + (Is__true Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop Coq__Init__Logic.True + (Is__true + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + cupicef.I Coq__Init__Logic.I) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Is__true Coq__Init__Datatypes.false) + (Is__true Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop Coq__Init__Logic.False + (Is__true + (Coq__Init__Datatypes.orb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.or__ind Coq__Init__Logic.False + Coq__Init__Logic.False Coq__Init__Logic.False + (H2 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind Coq__Init__Logic.False H2) + (H2 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind Coq__Init__Logic.False H2) + H1)) + a H0) + b H. + +thm andb__prop__intro : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Is__true b1) (Is__true b2)) -> + cupicef.Term cupicef.prop (Is__true (Coq__Init__Datatypes.andb b1 b2)) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Is__true b1) (Is__true b2)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (Is__true b1) (Is__true b)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Is__true b1) (Is__true b)) => + Is__true (Coq__Init__Datatypes.andb b1 b))) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Is__true b1) + (Is__true Coq__Init__Datatypes.true)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (Is__true b) + (Is__true Coq__Init__Datatypes.true)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Is__true b) + (Is__true Coq__Init__Datatypes.true)) => + Is__true (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.true))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Is__true Coq__Init__Datatypes.true) + (Is__true Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop Coq__Init__Logic.True + (Is__true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + cupicef.I Coq__Init__Logic.I) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Is__true Coq__Init__Datatypes.false) + (Is__true Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop Coq__Init__Logic.False + (Is__true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.and__ind Coq__Init__Logic.False + Coq__Init__Logic.True Coq__Init__Logic.False + (H2 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + H3 : cupicef.Term cupicef.prop Coq__Init__Logic.True => + Coq__Init__Logic.False__ind Coq__Init__Logic.False H2) + H1)) + b1 H0) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Is__true b1) + (Is__true Coq__Init__Datatypes.false)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (Is__true b) + (Is__true Coq__Init__Datatypes.false)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Is__true b) + (Is__true Coq__Init__Datatypes.false)) => + Is__true (Coq__Init__Datatypes.andb b Coq__Init__Datatypes.false))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Is__true Coq__Init__Datatypes.true) + (Is__true Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop Coq__Init__Logic.False + (Is__true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.and__ind Coq__Init__Logic.True + Coq__Init__Logic.False Coq__Init__Logic.False + (H2 : cupicef.Term cupicef.prop Coq__Init__Logic.True => + H3 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind Coq__Init__Logic.False H3) + H1)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Is__true Coq__Init__Datatypes.false) + (Is__true Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop Coq__Init__Logic.False + (Is__true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.and__ind Coq__Init__Logic.False + Coq__Init__Logic.False Coq__Init__Logic.False + (H2 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + H3 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind Coq__Init__Logic.False H3) + H1)) + b1 H0) + b2 H. + +thm andb__prop__elim : + a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : cupicef.Term cupicef.prop (Is__true (Coq__Init__Datatypes.andb a b)) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.and (Is__true a) (Is__true b)) + +:= a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : cupicef.Term cupicef.prop (Is__true (Coq__Init__Datatypes.andb a b)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Is__true (Coq__Init__Datatypes.andb a b0)) + (H0 : + cupicef.Term cupicef.prop + (Is__true (Coq__Init__Datatypes.andb a b0)) => + Coq__Init__Logic.and (Is__true a) (Is__true b0))) + (H0 : + cupicef.Term cupicef.prop + (Is__true (Coq__Init__Datatypes.andb a Coq__Init__Datatypes.true)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Is__true (Coq__Init__Datatypes.andb b0 Coq__Init__Datatypes.true)) + (H1 : + cupicef.Term cupicef.prop + (Is__true + (Coq__Init__Datatypes.andb b0 Coq__Init__Datatypes.true)) => + Coq__Init__Logic.and (Is__true b0) + (Is__true Coq__Init__Datatypes.true))) + (H1 : + cupicef.Term cupicef.prop + (Is__true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and Coq__Init__Logic.True Coq__Init__Logic.True) + (Coq__Init__Logic.and (Is__true Coq__Init__Datatypes.true) + (Is__true Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.conj Coq__Init__Logic.True Coq__Init__Logic.True + H1 H1)) + (H1 : + cupicef.Term cupicef.prop + (Is__true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and Coq__Init__Logic.False Coq__Init__Logic.True) + (Coq__Init__Logic.and (Is__true Coq__Init__Datatypes.false) + (Is__true Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.conj Coq__Init__Logic.False + Coq__Init__Logic.True H1 Coq__Init__Logic.I)) + a H0) + (H0 : + cupicef.Term cupicef.prop + (Is__true (Coq__Init__Datatypes.andb a Coq__Init__Datatypes.false)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Is__true + (Coq__Init__Datatypes.andb b0 Coq__Init__Datatypes.false)) + (H1 : + cupicef.Term cupicef.prop + (Is__true + (Coq__Init__Datatypes.andb b0 Coq__Init__Datatypes.false)) => + Coq__Init__Logic.and (Is__true b0) + (Is__true Coq__Init__Datatypes.false))) + (H1 : + cupicef.Term cupicef.prop + (Is__true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and Coq__Init__Logic.True Coq__Init__Logic.False) + (Coq__Init__Logic.and (Is__true Coq__Init__Datatypes.true) + (Is__true Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.conj Coq__Init__Logic.True + Coq__Init__Logic.False Coq__Init__Logic.I H1)) + (H1 : + cupicef.Term cupicef.prop + (Is__true + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and Coq__Init__Logic.False + Coq__Init__Logic.False) + (Coq__Init__Logic.and (Is__true Coq__Init__Datatypes.false) + (Is__true Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.conj Coq__Init__Logic.False + Coq__Init__Logic.False H1 H1)) + a H0) + b H. + +def let__H54 : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) (Is__true b2)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) + (Is__true Coq__Init__Datatypes.true)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true Coq__Init__Datatypes.false) + (Is__true Coq__Init__Datatypes.true)) -> + H2 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False -> + cupicef.Term cupicef.prop Coq__Init__Logic.True) -> + H3 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.True + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) (Is__true b2)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) + (Is__true Coq__Init__Datatypes.true)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true Coq__Init__Datatypes.false) + (Is__true Coq__Init__Datatypes.true)) => + H2 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False -> + cupicef.Term cupicef.prop Coq__Init__Logic.True) => + H3 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.I. + +def let__H53 : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) (Is__true b2)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) + (Is__true Coq__Init__Datatypes.true)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true Coq__Init__Datatypes.false) + (Is__true Coq__Init__Datatypes.true)) -> + H2 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False -> + cupicef.Term cupicef.prop Coq__Init__Logic.True) -> + H3 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) (Is__true b2)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) + (Is__true Coq__Init__Datatypes.true)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true Coq__Init__Datatypes.false) + (Is__true Coq__Init__Datatypes.true)) => + H2 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False -> + cupicef.Term cupicef.prop Coq__Init__Logic.True) => + H3 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 (let__H54 b1 b2 H H0 H1 H2 H3). + +def let__H55 : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) (Is__true b2)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) + (Is__true Coq__Init__Datatypes.true)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true Coq__Init__Datatypes.false) + (Is__true Coq__Init__Datatypes.true)) -> + H2 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False -> + cupicef.Term cupicef.prop Coq__Init__Logic.True) -> + H3 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H4 : cupicef.Term cupicef.prop Coq__Init__Logic.False -> + cupicef.Term cupicef.prop Coq__Init__Logic.True + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) (Is__true b2)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) + (Is__true Coq__Init__Datatypes.true)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true Coq__Init__Datatypes.false) + (Is__true Coq__Init__Datatypes.true)) => + H2 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False -> + cupicef.Term cupicef.prop Coq__Init__Logic.True) => + H3 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + H2 H4. + +def let__H57 : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) (Is__true b2)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) + (Is__true Coq__Init__Datatypes.false)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true Coq__Init__Datatypes.true) + (Is__true Coq__Init__Datatypes.false)) -> + H2 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H3 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False -> + cupicef.Term cupicef.prop Coq__Init__Logic.True) -> + cupicef.Term cupicef.prop Coq__Init__Logic.True + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) (Is__true b2)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) + (Is__true Coq__Init__Datatypes.false)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true Coq__Init__Datatypes.true) + (Is__true Coq__Init__Datatypes.false)) => + H2 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False -> + cupicef.Term cupicef.prop Coq__Init__Logic.True) => + Coq__Init__Logic.I. + +def let__H56 : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) (Is__true b2)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) + (Is__true Coq__Init__Datatypes.false)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true Coq__Init__Datatypes.true) + (Is__true Coq__Init__Datatypes.false)) -> + H2 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H3 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False -> + cupicef.Term cupicef.prop Coq__Init__Logic.True) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) (Is__true b2)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) + (Is__true Coq__Init__Datatypes.false)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true Coq__Init__Datatypes.true) + (Is__true Coq__Init__Datatypes.false)) => + H2 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False -> + cupicef.Term cupicef.prop Coq__Init__Logic.True) => + H2 (let__H57 b1 b2 H H0 H1 H2 H3). + +def let__H58 : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) (Is__true b2)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) + (Is__true Coq__Init__Datatypes.false)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true Coq__Init__Datatypes.true) + (Is__true Coq__Init__Datatypes.false)) -> + H2 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H3 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False -> + cupicef.Term cupicef.prop Coq__Init__Logic.True) -> + H4 : cupicef.Term cupicef.prop Coq__Init__Logic.False -> + cupicef.Term cupicef.prop Coq__Init__Logic.True + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) (Is__true b2)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) + (Is__true Coq__Init__Datatypes.false)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true Coq__Init__Datatypes.true) + (Is__true Coq__Init__Datatypes.false)) => + H2 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False -> + cupicef.Term cupicef.prop Coq__Init__Logic.True) => + H4 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + H3 H4. + +thm eq__bool__prop__intro : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) (Is__true b2)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 b2) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) (Is__true b2)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.iff (Is__true b1) (Is__true b)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) (Is__true b)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 b)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) + (Is__true Coq__Init__Datatypes.true)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.iff (Is__true b) + (Is__true Coq__Init__Datatypes.true)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b) + (Is__true Coq__Init__Datatypes.true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true Coq__Init__Datatypes.true) + (Is__true Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true Coq__Init__Datatypes.false) + (Is__true Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + cupicef.I + (Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.False + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.True)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.True + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True => + Coq__Init__Logic.False)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (H2 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False -> + cupicef.Term cupicef.prop Coq__Init__Logic.True) => + H3 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + (H4 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + H4) + (let__H53 b1 b2 H H0 H1 H2 H3)) + H1)) + b1 H0) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) + (Is__true Coq__Init__Datatypes.false)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.iff (Is__true b) + (Is__true Coq__Init__Datatypes.false)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b) + (Is__true Coq__Init__Datatypes.false)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true Coq__Init__Datatypes.true) + (Is__true Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.True + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True => + Coq__Init__Logic.False)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.False + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.True)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (H2 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False -> + cupicef.Term cupicef.prop Coq__Init__Logic.True) => + (H4 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + H4) + (let__H56 b1 b2 H H0 H1 H2 H3)) + H1)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true Coq__Init__Datatypes.false) + (Is__true Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + b1 H0) + b2 H. + +def let__H59 : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 b2) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 b2) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False Coq__Init__Logic.True e) + Coq__Init__Logic.I Coq__Init__Datatypes.true H1. + +def let__H60 : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 b2) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.false) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 b2) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.false) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True Coq__Init__Logic.False e) + Coq__Init__Logic.I Coq__Init__Datatypes.false H1. + +thm eq__bool__prop__elim : + b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 b2) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (Is__true b1) (Is__true b2)) + +:= b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b2 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 b2) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 b) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 b) => + Coq__Init__Logic.iff (Is__true b1) (Is__true b))) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.true) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true) => + Coq__Init__Logic.iff (Is__true b) + (Is__true Coq__Init__Datatypes.true))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.True + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True => + Coq__Init__Logic.True)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.True + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True => + Coq__Init__Logic.True))) + (Coq__Init__Logic.iff (Is__true Coq__Init__Datatypes.true) + (Is__true Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.True + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True => + Coq__Init__Logic.True)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.True + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True => + Coq__Init__Logic.True)) + (H2 : cupicef.Term cupicef.prop Coq__Init__Logic.True => + Coq__Init__Logic.I) + (H2 : cupicef.Term cupicef.prop Coq__Init__Logic.True => + Coq__Init__Logic.I))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.iff Coq__Init__Logic.False Coq__Init__Logic.True) + (Coq__Init__Logic.iff (Is__true Coq__Init__Datatypes.false) + (Is__true Coq__Init__Datatypes.true)) + cupicef.I + (Coq__Init__Logic.False__ind + (Coq__Init__Logic.iff Coq__Init__Logic.False + Coq__Init__Logic.True) + (let__H59 b1 b2 H H0 H1))) + b1 H0) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b1 Coq__Init__Datatypes.false) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false) => + Coq__Init__Logic.iff (Is__true b) + (Is__true Coq__Init__Datatypes.false))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.iff Coq__Init__Logic.True Coq__Init__Logic.False) + (Coq__Init__Logic.iff (Is__true Coq__Init__Datatypes.true) + (Is__true Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.False__ind + (Coq__Init__Logic.iff Coq__Init__Logic.True + Coq__Init__Logic.False) + (let__H60 b1 b2 H H0 H1))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.False + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.False + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False))) + (Coq__Init__Logic.iff (Is__true Coq__Init__Datatypes.false) + (Is__true Coq__Init__Datatypes.false)) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.False + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.False + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False)) + (H2 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind Coq__Init__Logic.False H2) + (H2 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind Coq__Init__Logic.False H2))) + b1 H0) + b2 H. + +thm negb__prop__elim : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : cupicef.Term cupicef.prop (Is__true (Coq__Init__Datatypes.negb b)) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Is__true b)) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : cupicef.Term cupicef.prop (Is__true (Coq__Init__Datatypes.negb b)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Is__true (Coq__Init__Datatypes.negb b0)) + (H0 : + cupicef.Term cupicef.prop + (Is__true (Coq__Init__Datatypes.negb b0)) => + Coq__Init__Logic.not (Is__true b0))) + (H0 : + cupicef.Term cupicef.prop + (Is__true (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.not Coq__Init__Logic.True) + (Coq__Init__Logic.not (Is__true Coq__Init__Datatypes.true)) cupicef.I + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.True + (H1 : cupicef.Term cupicef.prop Coq__Init__Logic.True => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not Coq__Init__Logic.True) cupicef.I + (H1 : cupicef.Term cupicef.prop Coq__Init__Logic.True => + Coq__Init__Logic.False__ind Coq__Init__Logic.False H0))) + (H0 : + cupicef.Term cupicef.prop + (Is__true (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.not Coq__Init__Logic.False) + (Coq__Init__Logic.not (Is__true Coq__Init__Datatypes.false)) + cupicef.I + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.False + (H1 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not Coq__Init__Logic.False) cupicef.I + (H1 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind Coq__Init__Logic.False H1))) + b H. + +def let__H62 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.not (Is__true b)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Is__true Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop Coq__Init__Logic.True + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.not (Is__true b)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Is__true Coq__Init__Datatypes.true)) => + Coq__Init__Logic.I. + +def let__H61 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.not (Is__true b)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Is__true Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.not (Is__true b)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Is__true Coq__Init__Datatypes.true)) => + H0 (let__H62 b H H0). + +thm negb__prop__intro : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : cupicef.Term cupicef.prop (Coq__Init__Logic.not (Is__true b)) -> + cupicef.Term cupicef.prop (Is__true (Coq__Init__Datatypes.negb b)) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.not (Is__true b)) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Is__true b0)) + (H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not (Is__true b0)) => + Is__true (Coq__Init__Datatypes.negb b0))) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Is__true Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.prop cupicef.prop Coq__Init__Logic.False + (Is__true (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true)) + cupicef.I + ((H1 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind Coq__Init__Logic.False H1) + (let__H61 b H H0))) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Is__true Coq__Init__Datatypes.false)) => + cupicef.cast cupicef.prop cupicef.prop Coq__Init__Logic.True + (Is__true (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false)) + cupicef.I Coq__Init__Logic.I) + b H. + +def let__H64 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Is__true (Coq__Init__Datatypes.negb b))) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Is__true (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false))) -> + cupicef.Term cupicef.prop Coq__Init__Logic.True + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Is__true (Coq__Init__Datatypes.negb b))) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Is__true (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false))) => + Coq__Init__Logic.I. + +def let__H63 : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Is__true (Coq__Init__Datatypes.negb b))) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Is__true (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false))) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Is__true (Coq__Init__Datatypes.negb b))) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Is__true (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false))) => + H0 (let__H64 b H H0). + +thm negb__prop__classical : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Is__true (Coq__Init__Datatypes.negb b))) -> + cupicef.Term cupicef.prop (Is__true b) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Is__true (Coq__Init__Datatypes.negb b))) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Is__true (Coq__Init__Datatypes.negb b0))) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Is__true (Coq__Init__Datatypes.negb b0))) => + Is__true b0)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Is__true (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true))) => + cupicef.cast cupicef.prop cupicef.prop Coq__Init__Logic.True + (Is__true Coq__Init__Datatypes.true) cupicef.I Coq__Init__Logic.I) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Is__true (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false))) => + cupicef.cast cupicef.prop cupicef.prop Coq__Init__Logic.False + (Is__true Coq__Init__Datatypes.false) cupicef.I + ((H1 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind Coq__Init__Logic.False H1) + (let__H63 b H H0))) + b H. + +thm negb__prop__involutive : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : cupicef.Term cupicef.prop (Is__true b) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Is__true (Coq__Init__Datatypes.negb b))) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : cupicef.Term cupicef.prop (Is__true b) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (Is__true b0) + (H0 : cupicef.Term cupicef.prop (Is__true b0) => + Coq__Init__Logic.not (Is__true (Coq__Init__Datatypes.negb b0)))) + (H0 : cupicef.Term cupicef.prop (Is__true Coq__Init__Datatypes.true) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.not Coq__Init__Logic.False) + (Coq__Init__Logic.not + (Is__true (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true))) + cupicef.I + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.False + (H1 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not Coq__Init__Logic.False) cupicef.I + (H1 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind Coq__Init__Logic.False H1))) + (H0 : cupicef.Term cupicef.prop (Is__true Coq__Init__Datatypes.false) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.not Coq__Init__Logic.True) + (Coq__Init__Logic.not + (Is__true (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false))) + cupicef.I + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.True + (H1 : cupicef.Term cupicef.prop Coq__Init__Logic.True => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not Coq__Init__Logic.True) cupicef.I + (H1 : cupicef.Term cupicef.prop Coq__Init__Logic.True => + Coq__Init__Logic.False__ind Coq__Init__Logic.False H0))) + b H. + +thm andb__if : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + a_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => A) a + a_27_ (Coq__Init__Datatypes.andb b b_27_)) + (Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => A) + (Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b_27_0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + A) + a a_27_ b_27_) + a_27_ b)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + a_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => A) a + a_27_ (Coq__Init__Datatypes.andb b b0)) + (Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => A) + (Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b_27_0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + A) + a a_27_ b0) + a_27_ b)) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => A) + a a_27_ + (Coq__Init__Datatypes.andb b0 Coq__Init__Datatypes.true)) + (Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => A) + a a_27_ b0)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a a) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + A) + a a_27_ + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.true)) + a) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a_27_ a_27_) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + A) + a a_27_ + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.true)) + a_27_) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a_27_)) + b) + (Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => A) + a a_27_ + (Coq__Init__Datatypes.andb b0 Coq__Init__Datatypes.false)) + (Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => A) + a_27_ a_27_ b0)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a_27_ a_27_) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + A) + a a_27_ + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.true + Coq__Init__Datatypes.false)) + a_27_) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a_27_)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a_27_ a_27_) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + A) + a a_27_ + (Coq__Init__Datatypes.andb Coq__Init__Datatypes.false + Coq__Init__Datatypes.false)) + a_27_) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a_27_)) + b) + b_27_. + +thm negb__if : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + a_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => A) a + a_27_ (Coq__Init__Datatypes.negb b)) + (Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => A) + a_27_ a b)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + a_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => A) a + a_27_ (Coq__Init__Datatypes.negb b0)) + (Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b1 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => A) + a_27_ a b0)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a_27_ a_27_) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => A) + a a_27_ (Coq__Init__Datatypes.negb Coq__Init__Datatypes.true)) + a_27_) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a_27_)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a a) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => A) + a a_27_ (Coq__Init__Datatypes.negb Coq__Init__Datatypes.false)) + a) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a)) + b. + +thm andb__lazy__alt : + a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.andb a b) + (Coq__Init__Datatypes.match____bool cupicef.set + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false a)) + +:= a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.match____bool cupicef.set + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false a). + +thm orb__lazy__alt : + a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.orb a b) + (Coq__Init__Datatypes.match____bool cupicef.set + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true b a)) + +:= a : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.match____bool cupicef.set + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true b a). + +def reflect : + P : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Univ cupicef.set. + +def ReflectT : + P : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop P -> + cupicef.Term cupicef.set (reflect P Coq__Init__Datatypes.true). + +def ReflectF : + P : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) -> + cupicef.Term cupicef.set (reflect P Coq__Init__Datatypes.false). + +[ ] cupicef.code_guarded? _ (ReflectT _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (ReflectF _ _) --> cupicef.guarded. + +def match____reflect : + s : cupicef.Sort -> + P : cupicef.Univ cupicef.prop -> + P0 : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.set (reflect P __) -> + cupicef.Univ s) -> + case__ReflectT : + (__ : cupicef.Term cupicef.prop P -> + cupicef.Term s (P0 Coq__Init__Datatypes.true (ReflectT P __))) -> + case__ReflectF : + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) -> + cupicef.Term s (P0 Coq__Init__Datatypes.false (ReflectF P __))) -> + __ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + x : cupicef.Term cupicef.set (reflect P __) -> + cupicef.Term s (P0 __ x). + +[ s, P, P0, case__ReflectT, case__ReflectF, __] + match____reflect s P P0 case__ReflectT case__ReflectF + {Coq__Init__Datatypes.true} (ReflectT P __) + --> + case__ReflectT __. + +[ s, P, P0, case__ReflectT, case__ReflectF, __] + match____reflect s P P0 case__ReflectT case__ReflectF + {Coq__Init__Datatypes.false} (ReflectF P __) + --> + case__ReflectF __. + +[ s, P, P0, s'] + match____reflect s P + (__ => x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P0 __ x))) + --> + match____reflect s' P (__ => x => P0 __ x). + +def reflect__rect : + P : cupicef.Univ cupicef.prop -> + P0 : + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + r : cupicef.Term cupicef.set (reflect P b) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : + (p : cupicef.Term cupicef.prop P -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P0 Coq__Init__Datatypes.true (ReflectT P p))) -> + f0 : + (n : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P0 Coq__Init__Datatypes.false (ReflectF P n))) -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + r : cupicef.Term cupicef.set (reflect P b) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P0 b r) + +:= P : cupicef.Univ cupicef.prop => + P0 : + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + r : cupicef.Term cupicef.set (reflect P b) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : + (p : cupicef.Term cupicef.prop P -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P0 Coq__Init__Datatypes.true (ReflectT P p))) => + f0 : + (n : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P0 Coq__Init__Datatypes.false (ReflectF P n))) => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + r : cupicef.Term cupicef.set (reflect P b) => + match____reflect (cupicef.type (cupicef.s cupicef.z)) P + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + r0 : cupicef.Term cupicef.set (reflect P b0) => + P0 b0 r0) + f f0 b r. + +def reflect__ind : + P : cupicef.Univ cupicef.prop -> + P0 : + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + r : cupicef.Term cupicef.set (reflect P b) -> + cupicef.Univ cupicef.prop) -> + f : + (p : cupicef.Term cupicef.prop P -> + cupicef.Term cupicef.prop (P0 Coq__Init__Datatypes.true (ReflectT P p))) -> + f0 : + (n : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) -> + cupicef.Term cupicef.prop (P0 Coq__Init__Datatypes.false (ReflectF P n))) -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + r : cupicef.Term cupicef.set (reflect P b) -> + cupicef.Term cupicef.prop (P0 b r) + +:= P : cupicef.Univ cupicef.prop => + P0 : + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + r : cupicef.Term cupicef.set (reflect P b) -> + cupicef.Univ cupicef.prop) => + f : + (p : cupicef.Term cupicef.prop P -> + cupicef.Term cupicef.prop (P0 Coq__Init__Datatypes.true (ReflectT P p))) => + f0 : + (n : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) -> + cupicef.Term cupicef.prop + (P0 Coq__Init__Datatypes.false (ReflectF P n))) => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + r : cupicef.Term cupicef.set (reflect P b) => + match____reflect cupicef.prop P + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + r0 : cupicef.Term cupicef.set (reflect P b0) => + P0 b0 r0) + f f0 b r. + +def reflect__rec : + P : cupicef.Univ cupicef.prop -> + P0 : + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + r : cupicef.Term cupicef.set (reflect P b) -> + cupicef.Univ cupicef.set) -> + f : + (p : cupicef.Term cupicef.prop P -> + cupicef.Term cupicef.set (P0 Coq__Init__Datatypes.true (ReflectT P p))) -> + f0 : + (n : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) -> + cupicef.Term cupicef.set (P0 Coq__Init__Datatypes.false (ReflectF P n))) -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + r : cupicef.Term cupicef.set (reflect P b) -> + cupicef.Term cupicef.set (P0 b r) + +:= P : cupicef.Univ cupicef.prop => + P0 : + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + r : cupicef.Term cupicef.set (reflect P b) -> + cupicef.Univ cupicef.set) => + reflect__rect P + (cupicef.cast + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I Coq__Init__Datatypes.bool + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.set (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (reflect P b) + (r : cupicef.Term cupicef.set (reflect P b) => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I Coq__Init__Datatypes.bool + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (reflect P b) + (r : cupicef.Term cupicef.set (reflect P b) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I))) + cupicef.I P0). + +def let__H65 : + P : cupicef.Univ cupicef.prop -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : cupicef.Term cupicef.set (reflect P b) -> + n : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) -> + H0 : cupicef.Term cupicef.prop P -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= P : cupicef.Univ cupicef.prop => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : cupicef.Term cupicef.set (reflect P b) => + n : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) => + H0 : cupicef.Term cupicef.prop P => + n H0. + +def let__H66 : + P : cupicef.Univ cupicef.prop -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : cupicef.Term cupicef.set (reflect P b) -> + n : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= P : cupicef.Univ cupicef.prop => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : cupicef.Term cupicef.set (reflect P b) => + n : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False Coq__Init__Logic.True e) + Coq__Init__Logic.I Coq__Init__Datatypes.true H0. + +thm reflect__iff : + P : cupicef.Univ cupicef.prop -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : cupicef.Term cupicef.set (reflect P b) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff P + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) + +:= P : cupicef.Univ cupicef.prop => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : cupicef.Term cupicef.set (reflect P b) => + match____reflect cupicef.prop P + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + r : cupicef.Term cupicef.set (reflect P b0) => + Coq__Init__Logic.iff P + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.true)) + (p : cupicef.Term cupicef.prop P => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I P + (__ : cupicef.Term cupicef.prop P => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + P)) + (H0 : cupicef.Term cupicef.prop P => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + p)) + (n : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I P + (__ : cupicef.Term cupicef.prop P => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + P)) + (H0 : cupicef.Term cupicef.prop P => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (let__H65 P b H n H0)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.False__ind P (let__H66 P b H n H0))) + b H. + +def let__H68 : + P : cupicef.Univ cupicef.prop -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff P + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff P + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + H1 : + (__ : cupicef.Term cupicef.prop P -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop P) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + +:= P : cupicef.Univ cupicef.prop => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff P + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff P + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H1 : + (__ : cupicef.Term cupicef.prop P -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop P) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true. + +def let__H67 : + P : cupicef.Univ cupicef.prop -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff P + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff P + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + H1 : + (__ : cupicef.Term cupicef.prop P -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop P) -> + cupicef.Term cupicef.prop P + +:= P : cupicef.Univ cupicef.prop => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff P + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff P + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H1 : + (__ : cupicef.Term cupicef.prop P -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop P) => + H2 (let__H68 P b H H0 H1 H2). + +def let__H69 : + P : cupicef.Univ cupicef.prop -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff P + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff P + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + H1 : + (__ : cupicef.Term cupicef.prop P -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) -> + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop P) -> + H3 : cupicef.Term cupicef.prop P -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + +:= P : cupicef.Univ cupicef.prop => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff P + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff P + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H1 : + (__ : cupicef.Term cupicef.prop P -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop P) => + H3 : cupicef.Term cupicef.prop P => + H1 H3. + +def iff__reflect : + P : cupicef.Univ cupicef.prop -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff P + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) -> + cupicef.Term cupicef.set (reflect P b) + +:= P : cupicef.Univ cupicef.prop => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff P + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true)) => + Coq__Init__Datatypes.match____bool cupicef.set + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.set + (cupicef.rule cupicef.prop cupicef.set) cupicef.I + (Coq__Init__Logic.iff P + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.true)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff P + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.true)) => + reflect P b0)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff P + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.set cupicef.set + (reflect P Coq__Init__Datatypes.true) + (reflect P Coq__Init__Datatypes.true) cupicef.I + (Coq__Init__Logic.and__rec + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I P + (__ : cupicef.Term cupicef.prop P => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + P)) + (reflect P Coq__Init__Datatypes.true) + (H1 : + (__ : cupicef.Term cupicef.prop P -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) => + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop P) => + (H3 : cupicef.Term cupicef.prop P => + cupicef.cast cupicef.set cupicef.set + (reflect P Coq__Init__Datatypes.true) + (reflect P Coq__Init__Datatypes.true) cupicef.I + (ReflectT P H3)) + (let__H67 P b H H0 H1 H2)) + H0)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff P + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) => + cupicef.cast cupicef.set cupicef.set + (reflect P Coq__Init__Datatypes.false) + (reflect P Coq__Init__Datatypes.false) cupicef.I + (Coq__Init__Logic.and__rec + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I P + (__ : cupicef.Term cupicef.prop P => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + P)) + (reflect P Coq__Init__Datatypes.false) + (H1 : + (__ : cupicef.Term cupicef.prop P -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) => + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop P) => + cupicef.cast cupicef.set cupicef.set + (reflect P Coq__Init__Datatypes.false) + (reflect P Coq__Init__Datatypes.false) cupicef.I + (ReflectF P + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I P + (H3 : cupicef.Term cupicef.prop P => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not P) cupicef.I + (H3 : cupicef.Term cupicef.prop P => + diff__false__true (H1 H3))))) + H0)) + b H. + +def reflect__dec : + P : cupicef.Univ cupicef.prop -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : cupicef.Term cupicef.set (reflect P b) -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool P (Coq__Init__Logic.not P)) + +:= P : cupicef.Univ cupicef.prop => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : cupicef.Term cupicef.set (reflect P b) => + match____reflect cupicef.set P + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + r : cupicef.Term cupicef.set (reflect P b0) => + Coq__Init__Specif.sumbool P (Coq__Init__Logic.not P)) + (p : cupicef.Term cupicef.prop P => + Coq__Init__Specif.left P (Coq__Init__Logic.not P) p) + (n : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) => + Coq__Init__Specif.right P (Coq__Init__Logic.not P) n) + b H. + +(; End of translation. ;) diff --git a/out/Coq__Bool__BoolEq.dk b/out/Coq__Bool__BoolEq.dk new file mode 100644 index 000000000..cfac036fc --- /dev/null +++ b/out/Coq__Bool__BoolEq.dk @@ -0,0 +1,1120 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def beq__eq__true : + A : cupicef.Univ cupicef.set -> + beq : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool) -> + beq__refl : + (x : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (beq x x))) -> + x : cupicef.Term cupicef.set A -> + y : cupicef.Term cupicef.set A -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (beq x y)) + +:= A : cupicef.Univ cupicef.set => + beq : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool) => + beq__refl : + (x : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (beq x x))) => + x : cupicef.Term cupicef.set A => + y : cupicef.Term cupicef.set A => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (y0 : cupicef.Term cupicef.set A => + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (beq x y0)) + (beq__refl x) y H. + +def let__H0 : + A : cupicef.Univ cupicef.set -> + beq : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool) -> + beq__refl : + (x : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (beq x x))) -> + x : cupicef.Term cupicef.set A -> + y : cupicef.Term cupicef.set A -> + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.set => + beq : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool) => + beq__refl : + (x : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (beq x x))) => + x : cupicef.Term cupicef.set A => + y : cupicef.Term cupicef.set A => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (e0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False Coq__Init__Logic.True e0) + Coq__Init__Logic.I Coq__Init__Datatypes.true H. + +def beq__eq__not__false : + A : cupicef.Univ cupicef.set -> + beq : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool) -> + beq__refl : + (x : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (beq x x))) -> + x : cupicef.Term cupicef.set A -> + y : cupicef.Term cupicef.set A -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false (beq x y))) + +:= A : cupicef.Univ cupicef.set => + beq : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool) => + beq__refl : + (x : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (beq x x))) => + x : cupicef.Term cupicef.set A => + y : cupicef.Term cupicef.set A => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false b)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true)) + cupicef.I + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H0 A beq beq__refl x y e H))) + (beq x y) (beq__eq__true A beq beq__refl x y e). + +def beq__false__not__eq : + A : cupicef.Univ cupicef.set -> + beq : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool) -> + beq__refl : + (x : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (beq x x))) -> + x : cupicef.Term cupicef.set A -> + y : cupicef.Term cupicef.set A -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false (beq x y)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + +:= A : cupicef.Univ cupicef.set => + beq : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool) => + beq__refl : + (x : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (beq x x))) => + x : cupicef.Term cupicef.set A => + y : cupicef.Term cupicef.set A => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false (beq x y)) => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + beq__eq__not__false A beq beq__refl x y e H. + +def exists__beq__eq : + A : cupicef.Univ cupicef.set -> + beq : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool) -> + x : cupicef.Term cupicef.set A -> + y : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sig cupicef.set Coq__Init__Datatypes.bool + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (beq x y))) + +:= A : cupicef.Univ cupicef.set => + beq : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool) => + x : cupicef.Term cupicef.set A => + y : cupicef.Term cupicef.set A => + Coq__Init__Specif.exist + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (beq x y)) + (beq x y) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (beq x y)). + +def not__eq__false__beq : + A : cupicef.Univ cupicef.set -> + beq : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool) -> + beq__eq : + (x : cupicef.Term cupicef.set A -> + y : cupicef.Term cupicef.set A -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (beq x y)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) -> + x : cupicef.Term cupicef.set A -> + y : cupicef.Term cupicef.set A -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false (beq x y)) + +:= A : cupicef.Univ cupicef.set => + beq : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool) => + beq__eq : + (x : cupicef.Term cupicef.set A -> + y : cupicef.Term cupicef.set A -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (beq x y)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) => + x : cupicef.Term cupicef.set A => + y : cupicef.Term cupicef.set A => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) => + Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (beq x y) Coq__Init__Datatypes.false + (Coq__Bool__Bool.not__true__is__false (beq x y) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (beq x y) Coq__Init__Datatypes.true) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (beq x y) Coq__Init__Datatypes.true) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (beq x y) Coq__Init__Datatypes.true)) + cupicef.I + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (beq x y) Coq__Init__Datatypes.true) => + H + (beq__eq x y + (Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (beq x y) Coq__Init__Datatypes.true H0))))). + +def eq__dec : + A : cupicef.Univ cupicef.set -> + beq : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool) -> + beq__refl : + (x : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (beq x x))) -> + beq__eq : + (x : cupicef.Term cupicef.set A -> + y : cupicef.Term cupicef.set A -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (beq x y)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) -> + x : cupicef.Term cupicef.set A -> + y : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y))) + +:= A : cupicef.Univ cupicef.set => + beq : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool) => + beq__refl : + (x : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (beq x x))) => + beq__eq : + (x : cupicef.Term cupicef.set A -> + y : cupicef.Term cupicef.set A -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (beq x y)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) => + x : cupicef.Term cupicef.set A => + y : cupicef.Term cupicef.set A => + Coq__Init__Specif.match____sig cupicef.set + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (beq x y)) + (s : + cupicef.Term cupicef.set + (Coq__Init__Specif.sig cupicef.set Coq__Init__Datatypes.bool + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b (beq x y))) => + Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y))) + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.set + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.set + (cupicef.rule cupicef.prop cupicef.set) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 (beq x y)) + (e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 (beq x y)) => + Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true (beq x y)) => + Coq__Init__Specif.left + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (beq__eq x y H)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false (beq x y)) => + Coq__Init__Specif.right + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (beq__false__not__eq A beq beq__refl x y H)) + b) + (exists__beq__eq A beq x y). + +(; End of translation. ;) diff --git a/out/Coq__Bool__DecBool.dk b/out/Coq__Bool__DecBool.dk new file mode 100644 index 000000000..d6cab6272 --- /dev/null +++ b/out/Coq__Bool__DecBool.dk @@ -0,0 +1,216 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def ifdec : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + H : cupicef.Term cupicef.set (Coq__Init__Specif.sumbool A B) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + H : cupicef.Term cupicef.set (Coq__Init__Specif.sumbool A B) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C => + Coq__Init__Specif.match____sumbool (cupicef.type (cupicef.s cupicef.z)) A + B (H0 : cupicef.Term cupicef.set (Coq__Init__Specif.sumbool A B) => C) + (__ : cupicef.Term cupicef.prop A => x) + (__ : cupicef.Term cupicef.prop B => y) H. + +thm ifdec__left : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.set -> + H : cupicef.Term cupicef.set (Coq__Init__Specif.sumbool A B) -> + __ : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + x : cupicef.Term cupicef.set C -> + y : cupicef.Term cupicef.set C -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + (ifdec A B + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I C) + H x y) + x) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.set => + H : cupicef.Term cupicef.set (Coq__Init__Specif.sumbool A B) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + x : cupicef.Term cupicef.set C => + y : cupicef.Term cupicef.set C => + Coq__Init__Specif.match____sumbool cupicef.prop A B + (s : cupicef.Term cupicef.set (Coq__Init__Specif.sumbool A B) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + (ifdec A B + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I C) + s x y) + x) + (a : cupicef.Term cupicef.prop A => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + x) + (b : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + (ifdec A B + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I C) + (Coq__Init__Specif.right A B b) x y) + x) + (H0 b)) + H. + +thm ifdec__right : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.set -> + H : cupicef.Term cupicef.set (Coq__Init__Specif.sumbool A B) -> + __ : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + x : cupicef.Term cupicef.set C -> + y : cupicef.Term cupicef.set C -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + (ifdec A B + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I C) + H x y) + y) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.set => + H : cupicef.Term cupicef.set (Coq__Init__Specif.sumbool A B) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + x : cupicef.Term cupicef.set C => + y : cupicef.Term cupicef.set C => + Coq__Init__Specif.match____sumbool cupicef.prop A B + (s : cupicef.Term cupicef.set (Coq__Init__Specif.sumbool A B) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + (ifdec A B + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I C) + s x y) + y) + (a : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + (ifdec A B + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I C) + (Coq__Init__Specif.left A B a) x y) + y) + (H0 a)) + (b : cupicef.Term cupicef.prop B => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + y) + H. + +(; End of translation. ;) diff --git a/out/Coq__Bool__IfProp.dk b/out/Coq__Bool__IfProp.dk new file mode 100644 index 000000000..6a704a9d7 --- /dev/null +++ b/out/Coq__Bool__IfProp.dk @@ -0,0 +1,1245 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def IfProp : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Univ cupicef.prop. + +def Iftrue : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop (IfProp A B Coq__Init__Datatypes.true). + +def Iffalse : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop (IfProp A B Coq__Init__Datatypes.false). + +[ ] cupicef.code_guarded? _ (Iftrue _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (Iffalse _ _ _) --> cupicef.guarded. + +def match____IfProp : + s : cupicef.Sort -> + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.prop (IfProp A B __) -> + cupicef.Univ s) -> + case__Iftrue : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term s (P Coq__Init__Datatypes.true (Iftrue A B __))) -> + case__Iffalse : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term s (P Coq__Init__Datatypes.false (Iffalse A B __))) -> + __ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + x : cupicef.Term cupicef.prop (IfProp A B __) -> + cupicef.Term s (P __ x). + +[ s, A, B, P, case__Iftrue, case__Iffalse, __] + match____IfProp s A B P case__Iftrue case__Iffalse + {Coq__Init__Datatypes.true} (Iftrue A B __) + --> + case__Iftrue __. + +[ s, A, B, P, case__Iftrue, case__Iffalse, __] + match____IfProp s A B P case__Iftrue case__Iffalse + {Coq__Init__Datatypes.false} (Iffalse A B __) + --> + case__Iffalse __. + +[ s, A, B, P, s'] + match____IfProp s A B + (__ => x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P __ x))) + --> + match____IfProp s' A B (__ => x => P __ x). + +def IfProp__ind : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Univ cupicef.prop) -> + f : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop (P Coq__Init__Datatypes.true)) -> + f0 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop (P Coq__Init__Datatypes.false)) -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + i : cupicef.Term cupicef.prop (IfProp A B b) -> + cupicef.Term cupicef.prop (P b) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Univ cupicef.prop) => + f : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop (P Coq__Init__Datatypes.true)) => + f0 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop (P Coq__Init__Datatypes.false)) => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + i : cupicef.Term cupicef.prop (IfProp A B b) => + match____IfProp cupicef.prop A B + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + __ : cupicef.Term cupicef.prop (IfProp A B b0) => + P b0) + f f0 b i. + +thm Iftrue__inv : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : cupicef.Term cupicef.prop (IfProp A B b) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop A + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : cupicef.Term cupicef.prop (IfProp A B b) => + match____IfProp cupicef.prop A B + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + __ : cupicef.Term cupicef.prop (IfProp A B b0) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.true) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.true) => + A)) + (H0 : cupicef.Term cupicef.prop A => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + H0) + (H0 : cupicef.Term cupicef.prop B => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.match____False cupicef.prop + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False => A) + (Coq__Bool__Bool.diff__true__false + (Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true H1))) + b H. + +thm Iffalse__inv : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : cupicef.Term cupicef.prop (IfProp A B b) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : cupicef.Term cupicef.prop (IfProp A B b) => + match____IfProp cupicef.prop A B + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + __ : cupicef.Term cupicef.prop (IfProp A B b0) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.false) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.false) => + B)) + (H0 : cupicef.Term cupicef.prop A => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.match____False cupicef.prop + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False => B) + (Coq__Bool__Bool.diff__true__false H1)) + (H0 : cupicef.Term cupicef.prop B => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + H0) + b H. + +def let__H1 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (IfProp A B Coq__Init__Datatypes.true) -> + H0 : cupicef.Term cupicef.prop B -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (IfProp A B Coq__Init__Datatypes.true) => + H0 : cupicef.Term cupicef.prop B => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False Coq__Init__Logic.True e) + Coq__Init__Logic.I Coq__Init__Datatypes.true H2. + +def let__H0 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (IfProp A B Coq__Init__Datatypes.true) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop A + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (IfProp A B Coq__Init__Datatypes.true) => + match____IfProp cupicef.prop A B + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + __ : cupicef.Term cupicef.prop (IfProp A B b) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.true) => + A)) + (H0 : cupicef.Term cupicef.prop A => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + H3 : cupicef.Term cupicef.prop A => + H3) H1 H0) + (H0 : cupicef.Term cupicef.prop B => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.False__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (H3 : cupicef.Term cupicef.prop B => A)) + (let__H1 A B H H0 H1 H2)) + H1 H0) + Coq__Init__Datatypes.true H. + +thm IfProp__true : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (IfProp A B Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop A + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (IfProp A B Coq__Init__Datatypes.true) => + let__H0 A B H + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true). + +def let__H3 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (IfProp A B Coq__Init__Datatypes.false) -> + H0 : cupicef.Term cupicef.prop A -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (IfProp A B Coq__Init__Datatypes.false) => + H0 : cupicef.Term cupicef.prop A => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True Coq__Init__Logic.False e) + Coq__Init__Logic.I Coq__Init__Datatypes.false H2. + +def let__H2 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (IfProp A B Coq__Init__Datatypes.false) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (IfProp A B Coq__Init__Datatypes.false) => + match____IfProp cupicef.prop A B + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + __ : cupicef.Term cupicef.prop (IfProp A B b) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b Coq__Init__Datatypes.false) => + B)) + (H0 : cupicef.Term cupicef.prop A => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.False__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (H3 : cupicef.Term cupicef.prop A => B)) + (let__H3 A B H H0 H1 H2)) + H1 H0) + (H0 : cupicef.Term cupicef.prop B => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + H3 : cupicef.Term cupicef.prop B => + H3) H1 H0) + Coq__Init__Datatypes.false H. + +thm IfProp__false : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (IfProp A B Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (IfProp A B Coq__Init__Datatypes.false) => + let__H2 A B H + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false). + +thm IfProp__or : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : cupicef.Term cupicef.prop (IfProp A B b) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : cupicef.Term cupicef.prop (IfProp A B b) => + match____IfProp cupicef.prop A B + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + __ : cupicef.Term cupicef.prop (IfProp A B b0) => + Coq__Init__Logic.or A B) + (H0 : cupicef.Term cupicef.prop A => Coq__Init__Logic.or__introl A B H0) + (H0 : cupicef.Term cupicef.prop B => Coq__Init__Logic.or__intror A B H0) + b H. + +def let__H5 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : cupicef.Term cupicef.prop (IfProp A B Coq__Init__Datatypes.true) -> + H0 : cupicef.Term cupicef.prop B -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : cupicef.Term cupicef.prop (IfProp A B Coq__Init__Datatypes.true) => + H0 : cupicef.Term cupicef.prop B => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False Coq__Init__Logic.True e) + Coq__Init__Logic.I Coq__Init__Datatypes.true H2. + +def let__H4 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : cupicef.Term cupicef.prop (IfProp A B Coq__Init__Datatypes.true) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) -> + cupicef.Term cupicef.prop A + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : cupicef.Term cupicef.prop (IfProp A B Coq__Init__Datatypes.true) => + match____IfProp cupicef.prop A B + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + __ : cupicef.Term cupicef.prop (IfProp A B b0) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.true) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.true) => + A)) + (H0 : cupicef.Term cupicef.prop A => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + H3 : cupicef.Term cupicef.prop A => + H3) H1 H0) + (H0 : cupicef.Term cupicef.prop B => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.False__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (H3 : cupicef.Term cupicef.prop B => A)) + (let__H5 A B b H H0 H1 H2)) + H1 H0) + Coq__Init__Datatypes.true H. + +def let__H7 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : cupicef.Term cupicef.prop (IfProp A B Coq__Init__Datatypes.false) -> + H0 : cupicef.Term cupicef.prop A -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : cupicef.Term cupicef.prop (IfProp A B Coq__Init__Datatypes.false) => + H0 : cupicef.Term cupicef.prop A => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True Coq__Init__Logic.False e) + Coq__Init__Logic.I Coq__Init__Datatypes.false H2. + +def let__H6 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + H : cupicef.Term cupicef.prop (IfProp A B Coq__Init__Datatypes.false) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + H : cupicef.Term cupicef.prop (IfProp A B Coq__Init__Datatypes.false) => + match____IfProp cupicef.prop A B + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + __ : cupicef.Term cupicef.prop (IfProp A B b0) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.false) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 Coq__Init__Datatypes.false) => + B)) + (H0 : cupicef.Term cupicef.prop A => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.False__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (H3 : cupicef.Term cupicef.prop A => B)) + (let__H7 A B b H H0 H1 H2)) + H1 H0) + (H0 : cupicef.Term cupicef.prop B => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + H3 : cupicef.Term cupicef.prop B => + H3) H1 H0) + Coq__Init__Datatypes.false H. + +thm IfProp__sum : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __ : cupicef.Term cupicef.prop (IfProp A B b) -> + cupicef.Term cupicef.set (Coq__Init__Specif.sumbool A B) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.match____bool cupicef.set + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.set + (cupicef.rule cupicef.prop cupicef.set) cupicef.I (IfProp A B b0) + (__ : cupicef.Term cupicef.prop (IfProp A B b0) => + Coq__Init__Specif.sumbool A B)) + (H : cupicef.Term cupicef.prop (IfProp A B Coq__Init__Datatypes.true) => + Coq__Init__Specif.left A B + (let__H4 A B b H + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true))) + (H : cupicef.Term cupicef.prop (IfProp A B Coq__Init__Datatypes.false) => + Coq__Init__Specif.right A B + (let__H6 A B b H + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false))) + b. + +(; End of translation. ;) diff --git a/out/Coq__Classes__CMorphisms.dk b/out/Coq__Classes__CMorphisms.dk new file mode 100644 index 000000000..e3e063e15 --- /dev/null +++ b/out/Coq__Classes__CMorphisms.dk @@ -0,0 +1,29593 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def Proper : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) -> + m : cupicef.Term (cupicef.type s0) A -> + cupicef.Univ (cupicef.type s1) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) => + m : cupicef.Term (cupicef.type s0) A => + R m m. + +#TYPE_CLASS Proper. + +def proper__prf : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) -> + m : cupicef.Term (cupicef.type s0) A -> + Proper0 : cupicef.Term (cupicef.type s1) (Proper s0 s1 A R m) -> + cupicef.Term (cupicef.type s1) (R m m) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) => + m : cupicef.Term (cupicef.type s0) A => + Proper0 : cupicef.Term (cupicef.type s1) (Proper s0 s1 A R m) => + Proper0. + +def ProperProxy : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) -> + m : cupicef.Term (cupicef.type s0) A -> + cupicef.Univ (cupicef.type s1) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) => + m : cupicef.Term (cupicef.type s0) A => + R m m. + +#TYPE_CLASS ProperProxy. + +def proper__proxy : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) -> + m : cupicef.Term (cupicef.type s0) A -> + ProperProxy0 : cupicef.Term (cupicef.type s1) (ProperProxy s0 s1 A R m) -> + cupicef.Term (cupicef.type s1) (R m m) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) => + m : cupicef.Term (cupicef.type s0) A => + ProperProxy0 : cupicef.Term (cupicef.type s1) (ProperProxy s0 s1 A R m) => + ProperProxy0. + +thm eq__proper__proxy : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) -> + A : cupicef.Univ (cupicef.type s0) -> + x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) + (ProperProxy s0 s1 A + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_0 A))) + x) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) => + A : cupicef.Univ (cupicef.type s0) => + x : cupicef.Term (cupicef.type s0) A => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_0 A) + x. + +thm reflexive__proper__proxy : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Coq__Classes__CRelationClasses.Reflexive s0 s1 A R) -> + x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) (ProperProxy s0 s1 A R x) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Coq__Classes__CRelationClasses.Reflexive s0 s1 A R) => + x : cupicef.Term (cupicef.type s0) A => + (X : cupicef.Term (cupicef.type s1) (R x x) => X) (H x). + +thm proper__proper__proxy : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + x : cupicef.Term (cupicef.type s0) A -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) -> + H : cupicef.Term (cupicef.type s1) (Proper s0 s1 A R x) -> + cupicef.Term (cupicef.type s1) (ProperProxy s0 s1 A R x) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + x : cupicef.Term (cupicef.type s0) A => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) => + H : cupicef.Term (cupicef.type s1) (Proper s0 s1 A R x) => + H. + +def respectful__hetero : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + s7 : cupicef.Nat -> + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s7)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s7)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s7)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s7)) -> + A : cupicef.Univ (cupicef.type s1) -> + B : cupicef.Univ (cupicef.type s2) -> + C : + (__ : cupicef.Term (cupicef.type s1) A -> cupicef.Univ (cupicef.type s3)) -> + D : + (__ : cupicef.Term (cupicef.type s2) B -> cupicef.Univ (cupicef.type s4)) -> + R : + (__ : cupicef.Term (cupicef.type s1) A -> + __0 : cupicef.Term (cupicef.type s2) B -> + cupicef.Univ (cupicef.type s5)) -> + R_27_ : + (x : cupicef.Term (cupicef.type s1) A -> + y : cupicef.Term (cupicef.type s2) B -> + __ : cupicef.Term (cupicef.type s3) (C x) -> + __0 : cupicef.Term (cupicef.type s4) (D y) -> + cupicef.Univ (cupicef.type s6)) -> + __ : + (x : cupicef.Term (cupicef.type s1) A -> + cupicef.Term (cupicef.type s3) (C x)) -> + __0 : + (x : cupicef.Term (cupicef.type s2) B -> + cupicef.Term (cupicef.type s4) (D x)) -> + cupicef.Univ (cupicef.type s7) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + s7 : cupicef.Nat => + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s7)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s7)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s7)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s7)) => + cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s2)) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s5) + (cupicef.type s6))))))))))))) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s2)) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom (cupicef.type s7)))))))))) + (cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s2)) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s5) + (cupicef.type s6)))))))))))) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s2)) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s5) + (cupicef.type s6))))))))))))) + cupicef.I + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (A : cupicef.Univ (cupicef.type s1) => + cupicef.prod (cupicef.axiom (cupicef.type s2)) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s5) + (cupicef.type s6))))))))))) + (cupicef.rule (cupicef.axiom (cupicef.type s2)) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s5) + (cupicef.type s6)))))))))))) + cupicef.I + (cupicef.univ (cupicef.type s2) (cupicef.axiom (cupicef.type s2)) + cupicef.I) + (B : cupicef.Univ (cupicef.type s2) => + cupicef.prod + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s5) + (cupicef.type s6)))))))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s5) + (cupicef.type s6))))))))))) + cupicef.I + (cupicef.prod (cupicef.type s1) + (cupicef.axiom (cupicef.type s3)) + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s3))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s1) A => + cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + (C : + (__ : cupicef.Term (cupicef.type s1) A -> + cupicef.Univ (cupicef.type s3)) => + cupicef.prod + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s5) + (cupicef.type s6))))))))) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s5) + (cupicef.type s6)))))))))) + cupicef.I + (cupicef.prod (cupicef.type s2) + (cupicef.axiom (cupicef.type s4)) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s4))) + cupicef.I B + (__ : cupicef.Term (cupicef.type s2) B => + cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + (D : + (__ : cupicef.Term (cupicef.type s2) B -> + cupicef.Univ (cupicef.type s4)) => + cupicef.prod + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s5) + (cupicef.type s6)))))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s5) + (cupicef.type s6))))))))) + cupicef.I + (cupicef.prod (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5))) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s1) A => + cupicef.prod (cupicef.type s2) + (cupicef.axiom (cupicef.type s5)) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5))) + cupicef.I B + (__0 : cupicef.Term (cupicef.type s2) B => + cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I))) + (R : + (__ : cupicef.Term (cupicef.type s1) A -> + __0 : cupicef.Term (cupicef.type s2) B -> + cupicef.Univ (cupicef.type s5)) => + cupicef.prod + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s5) + (cupicef.type s6))))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s5) + (cupicef.type s6)))))))) + cupicef.I + (cupicef.prod (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6))))) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + cupicef.I A + (x : cupicef.Term (cupicef.type s1) A => + cupicef.prod (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6))))) + cupicef.I B + (y : cupicef.Term (cupicef.type s2) B => + cupicef.prod (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6))) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))) + cupicef.I (C x) + (__ : cupicef.Term (cupicef.type s3) (C x) => + cupicef.prod (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6))) + cupicef.I (D y) + (__0 : + cupicef.Term (cupicef.type s4) (D y) => + cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) + cupicef.I))))) + (R_27_ : + (x : cupicef.Term (cupicef.type s1) A -> + y : cupicef.Term (cupicef.type s2) B -> + __ : cupicef.Term (cupicef.type s3) (C x) -> + __0 : cupicef.Term (cupicef.type s4) (D y) -> + cupicef.Univ (cupicef.type s6)) => + cupicef.prod + (cupicef.rule (cupicef.type s1) (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s5) + (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s5) + (cupicef.type s6))))))) + cupicef.I + (cupicef.prod (cupicef.type s1) (cupicef.type s3) + (cupicef.rule (cupicef.type s1) + (cupicef.type s3)) + cupicef.I A + (x : cupicef.Term (cupicef.type s1) A => C x)) + (f : + (x : cupicef.Term (cupicef.type s1) A -> + cupicef.Term (cupicef.type s3) (C x)) => + cupicef.prod + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s5) + (cupicef.type s6))))) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s5) + (cupicef.type s6)))))) + cupicef.I + (cupicef.prod (cupicef.type s2) + (cupicef.type s4) + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + cupicef.I B + (x : cupicef.Term (cupicef.type s2) B => D x)) + (g : + (x : cupicef.Term (cupicef.type s2) B -> + cupicef.Term (cupicef.type s4) (D x)) => + cupicef.univ + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s5) + (cupicef.type s6)))) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s5) + (cupicef.type s6))))) + cupicef.I))))))))) + (cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s2)) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom (cupicef.type s7))))))))) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s2)) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom (cupicef.type s7)))))))))) + cupicef.I + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (A : cupicef.Univ (cupicef.type s1) => + cupicef.prod (cupicef.axiom (cupicef.type s2)) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom (cupicef.type s7)))))))) + (cupicef.rule (cupicef.axiom (cupicef.type s2)) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom (cupicef.type s7))))))))) + cupicef.I + (cupicef.univ (cupicef.type s2) (cupicef.axiom (cupicef.type s2)) + cupicef.I) + (B : cupicef.Univ (cupicef.type s2) => + cupicef.prod + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom (cupicef.type s7))))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom (cupicef.type s7)))))))) + cupicef.I + (cupicef.prod (cupicef.type s1) + (cupicef.axiom (cupicef.type s3)) + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s3))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s1) A => + cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + (C : + (__ : cupicef.Term (cupicef.type s1) A -> + cupicef.Univ (cupicef.type s3)) => + cupicef.prod + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom (cupicef.type s7)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom (cupicef.type s7))))))) + cupicef.I + (cupicef.prod (cupicef.type s2) + (cupicef.axiom (cupicef.type s4)) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s4))) + cupicef.I B + (__ : cupicef.Term (cupicef.type s2) B => + cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + (D : + (__ : cupicef.Term (cupicef.type s2) B -> + cupicef.Univ (cupicef.type s4)) => + cupicef.prod + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom (cupicef.type s7))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom (cupicef.type s7)))))) + cupicef.I + (cupicef.prod (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5))) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s1) A => + cupicef.prod (cupicef.type s2) + (cupicef.axiom (cupicef.type s5)) + (cupicef.rule (cupicef.type s2) + (cupicef.axiom (cupicef.type s5))) + cupicef.I B + (__0 : cupicef.Term (cupicef.type s2) B => + cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I))) + (R : + (__ : cupicef.Term (cupicef.type s1) A -> + __0 : cupicef.Term (cupicef.type s2) B -> + cupicef.Univ (cupicef.type s5)) => + cupicef.prod + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom (cupicef.type s7))))) + cupicef.I + (cupicef.prod (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6))))) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))))) + cupicef.I A + (x : cupicef.Term (cupicef.type s1) A => + cupicef.prod (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6))))) + cupicef.I B + (y : cupicef.Term (cupicef.type s2) B => + cupicef.prod (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6))) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)))) + cupicef.I (C x) + (__ : cupicef.Term (cupicef.type s3) (C x) => + cupicef.prod (cupicef.type s4) + (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s6))) + cupicef.I (D y) + (__0 : + cupicef.Term (cupicef.type s4) (D y) => + cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) + cupicef.I))))) + (R_27_ : + (x : cupicef.Term (cupicef.type s1) A -> + y : cupicef.Term (cupicef.type s2) B -> + __ : cupicef.Term (cupicef.type s3) (C x) -> + __0 : cupicef.Term (cupicef.type s4) (D y) -> + cupicef.Univ (cupicef.type s6)) => + cupicef.prod + (cupicef.rule (cupicef.type s1) (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.rule (cupicef.type s1) + (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I + (cupicef.prod (cupicef.type s1) (cupicef.type s3) + (cupicef.rule (cupicef.type s1) + (cupicef.type s3)) + cupicef.I A + (x : cupicef.Term (cupicef.type s1) A => C x)) + (__ : + (x : cupicef.Term (cupicef.type s1) A -> + cupicef.Term (cupicef.type s3) (C x)) => + cupicef.prod + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom (cupicef.type s7)) + (cupicef.rule + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + (cupicef.axiom (cupicef.type s7))) + cupicef.I + (cupicef.prod (cupicef.type s2) + (cupicef.type s4) + (cupicef.rule (cupicef.type s2) + (cupicef.type s4)) + cupicef.I B + (x : cupicef.Term (cupicef.type s2) B => D x)) + (__0 : + (x : cupicef.Term (cupicef.type s2) B -> + cupicef.Term (cupicef.type s4) (D x)) => + cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I))))))))) + (cupicef.pair + (cupicef.BoolSome (cupicef.Cumul (cupicef.type s6) (cupicef.type s7))) + cstr_3 + (cupicef.BoolSome (cupicef.Cumul (cupicef.type s5) (cupicef.type s7))) + cstr_2 + (cupicef.BoolSome (cupicef.Cumul (cupicef.type s2) (cupicef.type s7))) + cstr_1 + (cupicef.BoolSome (cupicef.Cumul (cupicef.type s1) (cupicef.type s7))) + cstr_0 cupicef.BoolNone) + (A : cupicef.Univ (cupicef.type s1) => + B : cupicef.Univ (cupicef.type s2) => + C : + (__ : cupicef.Term (cupicef.type s1) A -> + cupicef.Univ (cupicef.type s3)) => + D : + (__ : cupicef.Term (cupicef.type s2) B -> + cupicef.Univ (cupicef.type s4)) => + R : + (__ : cupicef.Term (cupicef.type s1) A -> + __0 : cupicef.Term (cupicef.type s2) B -> + cupicef.Univ (cupicef.type s5)) => + R_27_ : + (x : cupicef.Term (cupicef.type s1) A -> + y : cupicef.Term (cupicef.type s2) B -> + __ : cupicef.Term (cupicef.type s3) (C x) -> + __0 : cupicef.Term (cupicef.type s4) (D y) -> + cupicef.Univ (cupicef.type s6)) => + f : + (x : cupicef.Term (cupicef.type s1) A -> + cupicef.Term (cupicef.type s3) (C x)) => + g : + (x : cupicef.Term (cupicef.type s2) B -> + cupicef.Term (cupicef.type s4) (D x)) => + cupicef.prod (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s5) (cupicef.type s6))) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s5) (cupicef.type s6)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s1) A => + cupicef.prod (cupicef.type s2) + (cupicef.rule (cupicef.type s5) (cupicef.type s6)) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s5) (cupicef.type s6))) + cupicef.I B + (y : cupicef.Term (cupicef.type s2) B => + cupicef.prod (cupicef.type s5) (cupicef.type s6) + (cupicef.rule (cupicef.type s5) (cupicef.type s6)) cupicef.I + (R x y) + (__ : cupicef.Term (cupicef.type s5) (R x y) => + R_27_ x y (f x) (g y))))). + +def respectful : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s4)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s5)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s5)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s4)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s5)) -> + A : cupicef.Univ (cupicef.type s0) -> + B : cupicef.Univ (cupicef.type s2) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s2) (cupicef.axiom (cupicef.type s3))) + (Coq__Classes__CRelationClasses.crelation s2 s3 B) -> + cupicef.Term + (cupicef.sup (cupicef.type s4) (cupicef.axiom (cupicef.type s5))) + (Coq__Classes__CRelationClasses.crelation s4 s5 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s2))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s4) (cupicef.axiom (cupicef.type s4)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s4))) + cstr_3 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s4))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s2) + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => B)))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s4)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s5)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s5)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s4)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s5)) => + cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule (cupicef.axiom (cupicef.type s2)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s3)))))))))) + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule (cupicef.axiom (cupicef.type s2)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s3))) + (cupicef.sup (cupicef.type s4) + (cupicef.axiom (cupicef.type s5))))))) + (cupicef.prod (cupicef.axiom (cupicef.type s0)) + (cupicef.rule (cupicef.axiom (cupicef.type s2)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s3))))))))) + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule (cupicef.axiom (cupicef.type s2)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.type s3)))))))))) + cupicef.I + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (A : cupicef.Univ (cupicef.type s0) => + cupicef.prod (cupicef.axiom (cupicef.type s2)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s3)))))))) + (cupicef.rule (cupicef.axiom (cupicef.type s2)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.type s3))))))))) + cupicef.I + (cupicef.univ (cupicef.type s2) (cupicef.axiom (cupicef.type s2)) + cupicef.I) + (B : cupicef.Univ (cupicef.type s2) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s3))))))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.type s3)))))))) + cupicef.I (Coq__Classes__CRelationClasses.crelation s0 s1 A) + (R : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) => + cupicef.prod + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s3)))))) + (cupicef.rule + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.type s3))))))) + cupicef.I (Coq__Classes__CRelationClasses.crelation s2 s3 B) + (R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s3))) + (Coq__Classes__CRelationClasses.crelation s2 s3 B) => + cupicef.prod + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s3))))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.type s3)))))) + cupicef.I + (cupicef.prod (cupicef.type s0) (cupicef.type s2) + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => B)) + (f : + (x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s2) B) => + cupicef.prod + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s3)))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.type s3))))) + cupicef.I + (cupicef.prod (cupicef.type s0) (cupicef.type s2) + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => B)) + (g : + (x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s2) B) => + cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.type s3)))) + cupicef.I))))))) + (cupicef.prod (cupicef.axiom (cupicef.type s0)) + (cupicef.rule (cupicef.axiom (cupicef.type s2)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s3))) + (cupicef.sup (cupicef.type s4) + (cupicef.axiom (cupicef.type s5)))))) + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule (cupicef.axiom (cupicef.type s2)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s3))) + (cupicef.sup (cupicef.type s4) + (cupicef.axiom (cupicef.type s5))))))) + cupicef.I + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (A : cupicef.Univ (cupicef.type s0) => + cupicef.prod (cupicef.axiom (cupicef.type s2)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s3))) + (cupicef.sup (cupicef.type s4) + (cupicef.axiom (cupicef.type s5))))) + (cupicef.rule (cupicef.axiom (cupicef.type s2)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s3))) + (cupicef.sup (cupicef.type s4) + (cupicef.axiom (cupicef.type s5)))))) + cupicef.I + (cupicef.univ (cupicef.type s2) (cupicef.axiom (cupicef.type s2)) + cupicef.I) + (B : cupicef.Univ (cupicef.type s2) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s3))) + (cupicef.sup (cupicef.type s4) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s3))) + (cupicef.sup (cupicef.type s4) + (cupicef.axiom (cupicef.type s5))))) + cupicef.I (Coq__Classes__CRelationClasses.crelation s0 s1 A) + (R : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) => + cupicef.prod + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s3))) + (cupicef.sup (cupicef.type s4) + (cupicef.axiom (cupicef.type s5))) + (cupicef.rule + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s3))) + (cupicef.sup (cupicef.type s4) + (cupicef.axiom (cupicef.type s5)))) + cupicef.I (Coq__Classes__CRelationClasses.crelation s2 s3 B) + (R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s3))) + (Coq__Classes__CRelationClasses.crelation s2 s3 B) => + Coq__Classes__CRelationClasses.crelation s4 s5 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) + (cupicef.type s4))) + cstr_3 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) + (cupicef.type s4))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s2) + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => B)))))))) + (cupicef.pair + (cupicef.BoolSome (cupicef.Cumul (cupicef.type s3) (cupicef.type s5))) + cstr_4 + (cupicef.BoolSome (cupicef.Cumul (cupicef.type s1) (cupicef.type s5))) + cstr_2 + (cupicef.BoolSome (cupicef.Cumul (cupicef.type s0) (cupicef.type s5))) + cstr_1 cupicef.BoolNone) + (A : cupicef.Univ (cupicef.type s0) => + B : cupicef.Univ (cupicef.type s2) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s2) (cupicef.axiom (cupicef.type s3))) + (Coq__Classes__CRelationClasses.crelation s2 s3 B) => + f : + (x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s2) B) => + g : + (x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s2) B) => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s3))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s3)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s3)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s3))) + cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s1) (cupicef.type s3) + (cupicef.rule (cupicef.type s1) (cupicef.type s3)) cupicef.I + (R x y) + (__ : cupicef.Term (cupicef.type s1) (R x y) => + R_27_ (f x) (g y))))). + +def forall__def : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) -> + A : cupicef.Univ (cupicef.type s0) -> + P : + (__ : cupicef.Term (cupicef.type s0) A -> cupicef.Univ (cupicef.type s1)) -> + cupicef.Univ (cupicef.type s2) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) => + cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s1))))) + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)))) + (cupicef.prod (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s1)))) + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s1))))) + cupicef.I + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (A : cupicef.Univ (cupicef.type s0) => + cupicef.prod + (cupicef.rule (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s1)))) + cupicef.I + (cupicef.prod (cupicef.type s0) (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + (P : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Univ (cupicef.type s1)) => + cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + cupicef.I))) + (cupicef.prod (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)))) + cupicef.I + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (A : cupicef.Univ (cupicef.type s0) => + cupicef.prod + (cupicef.rule (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2))) + cupicef.I + (cupicef.prod (cupicef.type s0) (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + (P : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Univ (cupicef.type s1)) => + cupicef.univ (cupicef.type s2) (cupicef.axiom (cupicef.type s2)) + cupicef.I))) + (cupicef.pair + (cupicef.BoolSome (cupicef.Cumul (cupicef.type s1) (cupicef.type s2))) + cstr_1 + (cupicef.BoolSome (cupicef.Cumul (cupicef.type s0) (cupicef.type s2))) + cstr_0 cupicef.BoolNone) + (A : cupicef.Univ (cupicef.type s0) => + P : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Univ (cupicef.type s1)) => + cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => P x)). + +def forall__relation : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s3)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s4)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s4)) -> + A : cupicef.Univ (cupicef.type s0) -> + P : + (__ : cupicef.Term (cupicef.type s0) A -> cupicef.Univ (cupicef.type s1)) -> + sig : + (a : cupicef.Term (cupicef.type s0) A -> + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s2))) + (Coq__Classes__CRelationClasses.crelation s1 s2 (P a))) -> + cupicef.Term + (cupicef.sup (cupicef.type s3) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s3 s4 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s3) (cupicef.axiom (cupicef.type s3)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s3))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s3))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => P x)))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s3)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s4)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s4)) => + cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2)))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2)))))))) + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2)))) + (cupicef.sup (cupicef.type s3) + (cupicef.axiom (cupicef.type s4)))))) + (cupicef.prod (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2)))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2))))))) + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2)))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2)))))))) + cupicef.I + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (A : cupicef.Univ (cupicef.type s0) => + cupicef.prod + (cupicef.rule (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2)))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2)))))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2)))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2))))))) + cupicef.I + (cupicef.prod (cupicef.type s0) (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + (P : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Univ (cupicef.type s1)) => + cupicef.prod + (cupicef.rule (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2)))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2))))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2)))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2)))))) + cupicef.I + (cupicef.prod (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2)))) + cupicef.I A + (a : cupicef.Term (cupicef.type s0) A => + Coq__Classes__CRelationClasses.crelation s1 s2 (P a))) + (sig : + (a : cupicef.Term (cupicef.type s0) A -> + cupicef.Term + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2))) + (Coq__Classes__CRelationClasses.crelation s1 s2 (P a))) => + cupicef.prod + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2)))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2))))) + cupicef.I + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + cupicef.I A (x : cupicef.Term (cupicef.type s0) A => P x)) + (f : + (x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) (P x)) => + cupicef.prod + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2)))) + cupicef.I + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => P x)) + (g : + (x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) (P x)) => + cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2))) + cupicef.I)))))) + (cupicef.prod (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2)))) + (cupicef.sup (cupicef.type s3) + (cupicef.axiom (cupicef.type s4))))) + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2)))) + (cupicef.sup (cupicef.type s3) + (cupicef.axiom (cupicef.type s4)))))) + cupicef.I + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (A : cupicef.Univ (cupicef.type s0) => + cupicef.prod + (cupicef.rule (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2)))) + (cupicef.sup (cupicef.type s3) + (cupicef.axiom (cupicef.type s4)))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2)))) + (cupicef.sup (cupicef.type s3) + (cupicef.axiom (cupicef.type s4))))) + cupicef.I + (cupicef.prod (cupicef.type s0) (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + (P : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Univ (cupicef.type s1)) => + cupicef.prod + (cupicef.rule (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2)))) + (cupicef.sup (cupicef.type s3) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2)))) + (cupicef.sup (cupicef.type s3) + (cupicef.axiom (cupicef.type s4)))) + cupicef.I + (cupicef.prod (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2)))) + cupicef.I A + (a : cupicef.Term (cupicef.type s0) A => + Coq__Classes__CRelationClasses.crelation s1 s2 (P a))) + (sig : + (a : cupicef.Term (cupicef.type s0) A -> + cupicef.Term + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2))) + (Coq__Classes__CRelationClasses.crelation s1 s2 (P a))) => + Coq__Classes__CRelationClasses.crelation s3 s4 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s3))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s3))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => P x))))))) + (cupicef.pair + (cupicef.BoolSome (cupicef.Cumul (cupicef.type s2) (cupicef.type s4))) + cstr_3 + (cupicef.BoolSome (cupicef.Cumul (cupicef.type s0) (cupicef.type s4))) + cstr_1 cupicef.BoolNone) + (A : cupicef.Univ (cupicef.type s0) => + P : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Univ (cupicef.type s1)) => + sig : + (a : cupicef.Term (cupicef.type s0) A -> + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s2))) + (Coq__Classes__CRelationClasses.crelation s1 s2 (P a))) => + f : + (x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) (P x)) => + g : + (x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) (P x)) => + cupicef.prod (cupicef.type s0) (cupicef.type s2) + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) cupicef.I A + (a : cupicef.Term (cupicef.type s0) A => sig a (f a) (g a))). + +def pointwise__relation : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s3)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s4)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s4)) -> + A : cupicef.Univ (cupicef.type s0) -> + B : cupicef.Univ (cupicef.type s1) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s2))) + (Coq__Classes__CRelationClasses.crelation s1 s2 B) -> + cupicef.Term + (cupicef.sup (cupicef.type s3) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s3 s4 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s3) (cupicef.axiom (cupicef.type s3)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s3))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s3))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => B)))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s3)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s4)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s4)) => + cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2)))))))) + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2))) + (cupicef.sup (cupicef.type s3) + (cupicef.axiom (cupicef.type s4)))))) + (cupicef.prod (cupicef.axiom (cupicef.type s0)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2))))))) + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2)))))))) + cupicef.I + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (A : cupicef.Univ (cupicef.type s0) => + cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.rule + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2)))))) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2))))))) + cupicef.I + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (B : cupicef.Univ (cupicef.type s1) => + cupicef.prod + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2))))) + (cupicef.rule + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2)))))) + cupicef.I (Coq__Classes__CRelationClasses.crelation s1 s2 B) + (R : + cupicef.Term + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2))) + (Coq__Classes__CRelationClasses.crelation s1 s2 B) => + cupicef.prod + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2)))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2))))) + cupicef.I + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + cupicef.I A (__ : cupicef.Term (cupicef.type s0) A => B)) + (f : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) B) => + cupicef.prod + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2)))) + cupicef.I + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => B)) + (g : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) B) => + cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2))) + cupicef.I)))))) + (cupicef.prod (cupicef.axiom (cupicef.type s0)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2))) + (cupicef.sup (cupicef.type s3) + (cupicef.axiom (cupicef.type s4))))) + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2))) + (cupicef.sup (cupicef.type s3) + (cupicef.axiom (cupicef.type s4)))))) + cupicef.I + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (A : cupicef.Univ (cupicef.type s0) => + cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.rule + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2))) + (cupicef.sup (cupicef.type s3) + (cupicef.axiom (cupicef.type s4)))) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2))) + (cupicef.sup (cupicef.type s3) + (cupicef.axiom (cupicef.type s4))))) + cupicef.I + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (B : cupicef.Univ (cupicef.type s1) => + cupicef.prod + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2))) + (cupicef.sup (cupicef.type s3) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2))) + (cupicef.sup (cupicef.type s3) + (cupicef.axiom (cupicef.type s4)))) + cupicef.I (Coq__Classes__CRelationClasses.crelation s1 s2 B) + (R : + cupicef.Term + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2))) + (Coq__Classes__CRelationClasses.crelation s1 s2 B) => + Coq__Classes__CRelationClasses.crelation s3 s4 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s3))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s3))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => B))))))) + (cupicef.pair + (cupicef.BoolSome (cupicef.Cumul (cupicef.type s2) (cupicef.type s4))) + cstr_3 + (cupicef.BoolSome (cupicef.Cumul (cupicef.type s0) (cupicef.type s4))) + cstr_1 cupicef.BoolNone) + (A : cupicef.Univ (cupicef.type s0) => + B : cupicef.Univ (cupicef.type s1) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s2))) + (Coq__Classes__CRelationClasses.crelation s1 s2 B) => + f : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) B) => + g : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) B) => + cupicef.prod (cupicef.type s0) (cupicef.type s2) + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) cupicef.I A + (a : cupicef.Term (cupicef.type s0) A => R (f a) (g a))). + +thm pointwise__pointwise : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + s7 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s2)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s4)) -> + cstr_3 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) -> + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) -> + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) -> + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) -> + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_9 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s4)) -> + cstr_11 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s1)) -> + cstr_12 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s4)) -> + cstr_13 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + B : cupicef.Univ (cupicef.type s6) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s6 s7 B) -> + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.relation__equivalence s1 s2 s3 s4 cstr_0 + cstr_5 cstr_6 cstr_7 cstr_8 cstr_9 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s6))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s6)) + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) (cupicef.type s1))) + cstr_11 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s6) + (cupicef.rule (cupicef.type s0) (cupicef.type s6)) cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => B))) + (pointwise__relation s0 s6 s7 s1 s4 cstr_1 cstr_2 cstr_11 cstr_12 A B + R) + (respectful s0 s5 s6 s7 s1 s4 cstr_1 cstr_2 cstr_10 cstr_11 cstr_12 A + B + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s5))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s5 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_3 A))) + R)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + s7 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s2)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s4)) => + cstr_3 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) => + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) (cupicef.type (cupicef.s cupicef.z))) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) => + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) => + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) => + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_9 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s4)) => + cstr_11 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s1)) => + cstr_12 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s4)) => + cstr_13 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + B : cupicef.Univ (cupicef.type s6) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s6 s7 B) => + x : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s6) B) => + y : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s6) B) => + Coq__Init__Datatypes.pair + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) (cupicef.axiom (cupicef.type s4)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_8 + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (pointwise__relation s0 s6 s7 s1 s4 cstr_1 cstr_2 cstr_11 cstr_12 + A B R x y) + (__ : + cupicef.Term (cupicef.type s4) + (pointwise__relation s0 s6 s7 s1 s4 cstr_1 cstr_2 cstr_11 + cstr_12 A B R x y) => + respectful s0 s5 s6 s7 s1 s4 cstr_1 cstr_2 cstr_10 cstr_11 + cstr_12 A B + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s5))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s5 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_3 A))) + R x y))) + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) (cupicef.axiom (cupicef.type s4)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_9 + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (respectful s0 s5 s6 s7 s1 s4 cstr_1 cstr_2 cstr_10 cstr_11 + cstr_12 A B + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s5))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s5 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_3 A))) + R x y) + (__ : + cupicef.Term (cupicef.type s4) + (respectful s0 s5 s6 s7 s1 s4 cstr_1 cstr_2 cstr_10 cstr_11 + cstr_12 A B + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s5))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s5 A) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_3 A))) + R x y) => + pointwise__relation s0 s6 s7 s1 s4 cstr_1 cstr_2 cstr_11 cstr_12 + A B R x y))) + (cupicef.cast (cupicef.rule (cupicef.type s4) (cupicef.type s4)) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (pointwise__relation s0 s6 s7 s1 s4 cstr_1 cstr_2 cstr_11 cstr_12 + A B R x y) + (__ : + cupicef.Term (cupicef.type s4) + (pointwise__relation s0 s6 s7 s1 s4 cstr_1 cstr_2 cstr_11 + cstr_12 A B R x y) => + respectful s0 s5 s6 s7 s1 s4 cstr_1 cstr_2 cstr_10 cstr_11 + cstr_12 A B + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s5))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s5 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_3 A))) + R x y)) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (pointwise__relation s0 s6 s7 s1 s4 cstr_1 cstr_2 cstr_11 cstr_12 + A B R x y) + (__ : + cupicef.Term (cupicef.type s4) + (pointwise__relation s0 s6 s7 s1 s4 cstr_1 cstr_2 cstr_11 + cstr_12 A B R x y) => + respectful s0 s5 s6 s7 s1 s4 cstr_1 cstr_2 cstr_10 cstr_11 + cstr_12 A B + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s5))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s5 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_3 A))) + R x y)) + cupicef.I + (cupicef.cast + (cupicef.rule (cupicef.type s4) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule cupicef.prop (cupicef.type s7))))) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) + (cupicef.prod (cupicef.type s4) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule cupicef.prop (cupicef.type s7)))) + (cupicef.rule (cupicef.type s4) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule cupicef.prop (cupicef.type s7))))) + cupicef.I + (pointwise__relation s0 s6 s7 s1 s4 cstr_1 cstr_2 cstr_11 + cstr_12 A B R x y) + (X : + cupicef.Term (cupicef.type s4) + (pointwise__relation s0 s6 s7 s1 s4 cstr_1 cstr_2 cstr_11 + cstr_12 A B R x y) => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule cupicef.prop (cupicef.type s7))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule cupicef.prop (cupicef.type s7)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule cupicef.prop (cupicef.type s7)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule cupicef.prop (cupicef.type s7))) + cupicef.I A + (y0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod cupicef.prop (cupicef.type s7) + (cupicef.rule cupicef.prop (cupicef.type s7)) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_3 A) + x0 y0) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_3 A) + x0 y0) => + R (x x0) (y y0)))))) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (pointwise__relation s0 s6 s7 s1 s4 cstr_1 cstr_2 cstr_11 + cstr_12 A B R x y) + (__ : + cupicef.Term (cupicef.type s4) + (pointwise__relation s0 s6 s7 s1 s4 cstr_1 cstr_2 cstr_11 + cstr_12 A B R x y) => + respectful s0 s5 s6 s7 s1 s4 cstr_1 cstr_2 cstr_10 cstr_11 + cstr_12 A B + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s5))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s5 A) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_3 A))) + R x y)) + cupicef.I + (X : + cupicef.Term (cupicef.type s4) + (pointwise__relation s0 s6 s7 s1 s4 cstr_1 cstr_2 cstr_11 + cstr_12 A B R x y) => + x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_3 A) + x0 y0) => + cupicef.cast (cupicef.type s7) (cupicef.type s7) + (R (x x0) (y y0)) (R (x x0) (y y0)) cupicef.I + (cupicef.cast (cupicef.type s7) (cupicef.type s7) + ((x1 : cupicef.Term (cupicef.type s0) A => R (x x1) (y y0)) + x0) + (R (x x0) (y y0)) cupicef.I + (Coq__Init__Logic.eq__rect__r + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_4 A) + y0 + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + cupicef.I A + (x1 : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I)) + (cupicef.prod (cupicef.type s0) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cstr_13 + (x1 : cupicef.Term (cupicef.type s0) A => + R (x x1) (y y0))) + (cupicef.cast (cupicef.type s7) (cupicef.type s7) + (R (x y0) (y y0)) (R (x y0) (y y0)) cupicef.I ( + X y0)) + x0 H))))) + (X : + cupicef.Term (cupicef.type s4) + (respectful s0 s5 s6 s7 s1 s4 cstr_1 cstr_2 cstr_10 cstr_11 cstr_12 + A B + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s5))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s5 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_3 A))) + R x y) => + cupicef.cast (cupicef.rule (cupicef.type s0) (cupicef.type s7)) + (cupicef.type s4) + (cupicef.prod (cupicef.type s0) (cupicef.type s7) + (cupicef.rule (cupicef.type s0) (cupicef.type s7)) cupicef.I A + (a : cupicef.Term (cupicef.type s0) A => R (x a) (y a))) + (pointwise__relation s0 s6 s7 s1 s4 cstr_1 cstr_2 cstr_11 cstr_12 A B + R x y) + cupicef.I + (a : cupicef.Term (cupicef.type s0) A => + (X0 : + (y0 : cupicef.Term (cupicef.type s0) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_3 A) + a y0) -> + cupicef.Term (cupicef.type s7) (R (x a) (y y0))) => + (X1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_3 A) + a a) -> + cupicef.Term (cupicef.type s7) (R (x a) (y a))) => + (X2 : cupicef.Term (cupicef.type s7) (R (x a) (y a)) => X2) + (X1 + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_3 A) + a))) + (X0 a)) + (X a))). + +thm subrelation__id__proper : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s1)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) -> + A : cupicef.Univ (cupicef.type s0) -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (Coq__Classes__CRelationClasses.crelation s0 s2 A) -> + RA_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (Coq__Classes__CRelationClasses.crelation s0 s3 A) -> + H : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s2) (cupicef.type s3))) + (Coq__Classes__CRelationClasses.subrelation s0 s2 s3 A RA RA_27_) -> + cupicef.Term (cupicef.type s1) + (Proper s0 s1 + (cupicef.prod (cupicef.type s0) (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s0)) cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => A)) + (respectful s0 s2 s0 s3 s0 s1 cupicef.I cstr_0 cstr_2 cupicef.I cstr_3 + A A RA RA_27_) + (Coq__Init__Datatypes.id + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_1 A))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) (cupicef.type (cupicef.s cupicef.z))) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s1)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) => + A : cupicef.Univ (cupicef.type s0) => + RA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (Coq__Classes__CRelationClasses.crelation s0 s2 A) => + RA_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (Coq__Classes__CRelationClasses.crelation s0 s3 A) => + H : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s2) (cupicef.type s3))) + (Coq__Classes__CRelationClasses.subrelation s0 s2 s3 A RA RA_27_) => + H. + +#TYPE_CLASS_INSTANCE subrelation__id__proper. + +thm subrelation__respectful : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + s7 : cupicef.Nat -> + s8 : cupicef.Nat -> + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s3)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s2)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s3)) -> + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s1)) -> + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s3)) -> + A : cupicef.Univ (cupicef.type s0) -> + RA_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s6))) + (Coq__Classes__CRelationClasses.crelation s0 s6 A) -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) -> + subl : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s4) (cupicef.type s6))) + (Coq__Classes__CRelationClasses.subrelation s0 s6 s4 A RA_27_ RA) -> + B : cupicef.Univ (cupicef.type s7) -> + RB : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type s5)) (cupicef.type s7)) + (Coq__Classes__CRelationClasses.crelation s7 s5 B) -> + RB_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s7) (cupicef.axiom (cupicef.type s8))) + (Coq__Classes__CRelationClasses.crelation s7 s8 B) -> + subr : + cupicef.Term + (cupicef.sup (cupicef.type s5) + (cupicef.sup (cupicef.type s7) (cupicef.type s8))) + (Coq__Classes__CRelationClasses.subrelation s7 s5 s8 B RB RB_27_) -> + cupicef.Term + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) (cupicef.type s3))) + (Coq__Classes__CRelationClasses.subrelation s1 s2 s3 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s7))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s7)) + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) (cupicef.type s1))) + cstr_6 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s7) + (cupicef.rule (cupicef.type s0) (cupicef.type s7)) cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => B))) + (respectful s0 s4 s7 s5 s1 s2 cstr_0 cstr_1 cstr_3 cstr_6 cstr_4 A B + RA RB) + (respectful s0 s6 s7 s8 s1 s3 cstr_0 cstr_2 cstr_5 cstr_6 cstr_7 A B + RA_27_ RB_27_)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + s7 : cupicef.Nat => + s8 : cupicef.Nat => + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s3)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s2)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s3)) => + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s1)) => + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s3)) => + A : cupicef.Univ (cupicef.type s0) => + RA_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s6))) + (Coq__Classes__CRelationClasses.crelation s0 s6 A) => + RA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) => + subl : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s4) (cupicef.type s6))) + (Coq__Classes__CRelationClasses.subrelation s0 s6 s4 A RA_27_ RA) => + B : cupicef.Univ (cupicef.type s7) => + RB : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type s5)) (cupicef.type s7)) + (Coq__Classes__CRelationClasses.crelation s7 s5 B) => + RB_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s7) (cupicef.axiom (cupicef.type s8))) + (Coq__Classes__CRelationClasses.crelation s7 s8 B) => + subr : + cupicef.Term + (cupicef.sup (cupicef.type s5) + (cupicef.sup (cupicef.type s7) (cupicef.type s8))) + (Coq__Classes__CRelationClasses.subrelation s7 s5 s8 B RB RB_27_) => + cupicef.cast + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) (cupicef.type s3))) + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) (cupicef.type s3))) + (Coq__Classes__CRelationClasses.subrelation s1 s2 s3 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s7))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s7)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) (cupicef.type s1))) + cstr_6 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s7) + (cupicef.rule (cupicef.type s0) (cupicef.type s7)) cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => B))) + (respectful s0 s4 s7 s5 s1 s2 cstr_0 cstr_1 cstr_3 cstr_6 cstr_4 A B + RA RB) + (respectful s0 s6 s7 s8 s1 s3 cstr_0 cstr_2 cstr_5 cstr_6 cstr_7 A B + RA_27_ RB_27_)) + (Coq__Classes__CRelationClasses.subrelation s1 s2 s3 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s7))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s7)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) (cupicef.type s1))) + cstr_6 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s7) + (cupicef.rule (cupicef.type s0) (cupicef.type s7)) cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => B))) + (respectful s0 s4 s7 s5 s1 s2 cstr_0 cstr_1 cstr_3 cstr_6 cstr_4 A B + RA RB) + (respectful s0 s6 s7 s8 s1 s3 cstr_0 cstr_2 cstr_5 cstr_6 cstr_7 A B + RA_27_ RB_27_)) + cupicef.I + (cupicef.cast + (cupicef.rule (cupicef.rule (cupicef.type s0) (cupicef.type s7)) + (cupicef.rule (cupicef.rule (cupicef.type s0) (cupicef.type s7)) + (cupicef.rule (cupicef.type s2) (cupicef.type s3)))) + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) (cupicef.type s3))) + (cupicef.prod (cupicef.rule (cupicef.type s0) (cupicef.type s7)) + (cupicef.rule (cupicef.rule (cupicef.type s0) (cupicef.type s7)) + (cupicef.rule (cupicef.type s2) (cupicef.type s3))) + (cupicef.rule (cupicef.rule (cupicef.type s0) (cupicef.type s7)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s7)) + (cupicef.rule (cupicef.type s2) (cupicef.type s3)))) + cupicef.I + (cupicef.prod (cupicef.type s0) (cupicef.type s7) + (cupicef.rule (cupicef.type s0) (cupicef.type s7)) cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => B)) + (x : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s7) B) => + cupicef.prod (cupicef.rule (cupicef.type s0) (cupicef.type s7)) + (cupicef.rule (cupicef.type s2) (cupicef.type s3)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s7)) + (cupicef.rule (cupicef.type s2) (cupicef.type s3))) + cupicef.I + (cupicef.prod (cupicef.type s0) (cupicef.type s7) + (cupicef.rule (cupicef.type s0) (cupicef.type s7)) cupicef.I + A (__ : cupicef.Term (cupicef.type s0) A => B)) + (y : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s7) B) => + cupicef.prod (cupicef.type s2) (cupicef.type s3) + (cupicef.rule (cupicef.type s2) (cupicef.type s3)) cupicef.I + (respectful s0 s4 s7 s5 s1 s2 cstr_0 cstr_1 cstr_3 cstr_6 + cstr_4 A B RA RB x y) + (X : + cupicef.Term (cupicef.type s2) + (respectful s0 s4 s7 s5 s1 s2 cstr_0 cstr_1 cstr_3 + cstr_6 cstr_4 A B RA RB x y) => + respectful s0 s6 s7 s8 s1 s3 cstr_0 cstr_2 cstr_5 cstr_6 + cstr_7 A B RA_27_ RB_27_ x y)))) + (Coq__Classes__CRelationClasses.subrelation s1 s2 s3 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s7)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) (cupicef.type s1))) + cstr_6 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s7) + (cupicef.rule (cupicef.type s0) (cupicef.type s7)) cupicef.I + A (__ : cupicef.Term (cupicef.type s0) A => B))) + (respectful s0 s4 s7 s5 s1 s2 cstr_0 cstr_1 cstr_3 cstr_6 cstr_4 A + B RA RB) + (respectful s0 s6 s7 s8 s1 s3 cstr_0 cstr_2 cstr_5 cstr_6 cstr_7 A + B RA_27_ RB_27_)) + cupicef.I + (x : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s7) B) => + y : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s7) B) => + X : + cupicef.Term (cupicef.type s2) + (respectful s0 s4 s7 s5 s1 s2 cstr_0 cstr_1 cstr_3 cstr_6 cstr_4 + A B RA RB x y) => + cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s8)))) + (cupicef.type s3) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s8))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s8)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s8)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s8))) + cupicef.I A + (y0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s6) (cupicef.type s8) + (cupicef.rule (cupicef.type s6) (cupicef.type s8)) + cupicef.I (RA_27_ x0 y0) + (X0 : cupicef.Term (cupicef.type s6) (RA_27_ x0 y0) => + RB_27_ (x x0) (y y0))))) + (respectful s0 s6 s7 s8 s1 s3 cstr_0 cstr_2 cstr_5 cstr_6 cstr_7 A + B RA_27_ RB_27_ x y) + cupicef.I + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + X0 : cupicef.Term (cupicef.type s6) (RA_27_ x0 y0) => + cupicef.cast (cupicef.type s8) (cupicef.type s8) + (RB_27_ (x x0) (y y0)) (RB_27_ (x x0) (y y0)) cupicef.I + (cupicef.cast (cupicef.type s8) (cupicef.type s8) + (RB_27_ (x x0) (y y0)) (RB_27_ (x x0) (y y0)) cupicef.I + (cupicef.cast (cupicef.type s8) (cupicef.type s8) + (RB_27_ (x x0) (y y0)) (RB_27_ (x x0) (y y0)) cupicef.I + (subr (x x0) (y y0) (X x0 y0 (subl x0 y0 X0)))))))). + +thm subrelation__refl : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Coq__Classes__CRelationClasses.subrelation s0 s1 s1 A R R) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) => + cupicef.cast (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Coq__Classes__CRelationClasses.subrelation s0 s1 s1 A R R) + (Coq__Classes__CRelationClasses.subrelation s0 s1 s1 A R R) cupicef.I + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))) + (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))) + cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) cupicef.I + (R x y) + (X : cupicef.Term (cupicef.type s1) (R x y) => R x y)))) + (Coq__Classes__CRelationClasses.subrelation s0 s1 s1 A R R) cupicef.I + (x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + X : cupicef.Term (cupicef.type s1) (R x y) => + cupicef.cast (cupicef.type s1) (cupicef.type s1) (R x y) (R x y) + cupicef.I + (cupicef.cast (cupicef.type s1) (cupicef.type s1) (R x y) ( + R x y) cupicef.I X))). + +thm subrelation__proper : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) + (cupicef.type (cupicef.s cupicef.z))) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) -> + m : cupicef.Term (cupicef.type s0) A -> + mor : cupicef.Term (cupicef.type s1) (Proper s0 s1 A R_27_ m) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) -> + unc : + cupicef.Term cupicef.set + (Coq__Classes__Init.Unconvertible + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) + (cupicef.type (cupicef.s cupicef.z)))) + cstr_0 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s0 s1 A)) + R R_27_) -> + sub : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Coq__Classes__CRelationClasses.subrelation s0 s1 s1 A R_27_ R) -> + cupicef.Term (cupicef.type s1) (Proper s0 s1 A R m) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) + (cupicef.type (cupicef.s cupicef.z))) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) => + m : cupicef.Term (cupicef.type s0) A => + mor : cupicef.Term (cupicef.type s1) (Proper s0 s1 A R_27_ m) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) => + unc : + cupicef.Term cupicef.set + (Coq__Classes__Init.Unconvertible + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) + (cupicef.type (cupicef.s cupicef.z)))) + cstr_0 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s0 s1 A)) + R R_27_) => + sub : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Coq__Classes__CRelationClasses.subrelation s0 s1 s1 A R_27_ R) => + sub m m mor. + +thm proper__subrelation__proper__arrow : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + s7 : cupicef.Nat -> + s8 : cupicef.Nat -> + s9 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) (cupicef.type s3)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) (cupicef.type s9)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s3)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s4)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s5)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s6)) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) -> + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_8 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) -> + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) -> + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) -> + cstr_11 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s1)) -> + cstr_12 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s2)) -> + cstr_13 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s4)) -> + cstr_14 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s6)) -> + cstr_15 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_16 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s6)) -> + cstr_17 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s5)) -> + A : cupicef.Univ (cupicef.type s0) -> + cupicef.Term (cupicef.type s2) + (Proper s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + (cupicef.trans (cupicef.type s0) + (cupicef.SortSome (cupicef.type s3)) cstr_2 + (cupicef.SortSome (cupicef.type s1)) cstr_8 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s1))) + (cupicef.trans (cupicef.axiom (cupicef.type s7)) + (cupicef.SortSome (cupicef.type s3)) cstr_0 + (cupicef.SortSome (cupicef.type s1)) cstr_8 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I (Coq__Classes__CRelationClasses.crelation s0 s7 A) + (__ : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s0 s7 A) => + cupicef.prod (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I)))) + (respectful s3 s4 s5 s6 s1 s2 cstr_8 cstr_9 cstr_10 cstr_11 cstr_12 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s3))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s3))) + cstr_0 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s0 s7 A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s5))) + cstr_4 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s5))) + (cupicef.trans (cupicef.axiom (cupicef.type s7)) + (cupicef.SortSome (cupicef.type s9)) cstr_1 + (cupicef.SortSome (cupicef.type s5)) cstr_17 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I))) + (cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7))))) + (cupicef.sup (cupicef.type s3) (cupicef.axiom (cupicef.type s4))) + (cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7)))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7))))) + cupicef.I (Coq__Classes__CRelationClasses.crelation s0 s7 A) + (R : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s0 s7 A) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7)))) + cupicef.I + (Coq__Classes__CRelationClasses.crelation s0 s7 A) + (R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s0 s7 A) => + cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.type s7)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7))) + cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s3 s4 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s3))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s3))) + cstr_0 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s0 s7 A))) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) (cupicef.type s4))) + cstr_13 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s4))) + cstr_3 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.subrelation s0 s7 s7 A)) + (respectful s0 s8 s9 s7 s5 s6 cstr_4 cstr_5 cstr_16 cstr_17 cstr_14 + A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s9)) + (cupicef.univ (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s9) + (cupicef.axiom (cupicef.type s9)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s8))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s8 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_6 A))) + (Coq__Classes__CRelationClasses.arrow s7 s7))) + (Proper s0 s7 A)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + s7 : cupicef.Nat => + s8 : cupicef.Nat => + s9 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) (cupicef.type s3)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) (cupicef.type s9)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s3)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s4)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s5)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s6)) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) => + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) (cupicef.type (cupicef.s cupicef.z))) => + cstr_8 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) => + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) => + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) => + cstr_11 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s1)) => + cstr_12 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s2)) => + cstr_13 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s4)) => + cstr_14 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s6)) => + cstr_15 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) => + cstr_16 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s6)) => + cstr_17 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s5)) => + A : cupicef.Univ (cupicef.type s0) => + cupicef.cast (cupicef.type s2) (cupicef.type s2) + (respectful s3 s4 s5 s6 s1 s2 cstr_8 cstr_9 cstr_10 cstr_11 cstr_12 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s3) (cupicef.axiom (cupicef.type s3)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s3))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s3))) + cstr_0 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s0 s7 A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s5) (cupicef.axiom (cupicef.type s5)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s5))) + cstr_4 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s5))) + (cupicef.trans (cupicef.axiom (cupicef.type s7)) + (cupicef.SortSome (cupicef.type s9)) cstr_1 + (cupicef.SortSome (cupicef.type s5)) cstr_17 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I))) + (cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7))))) + (cupicef.sup (cupicef.type s3) (cupicef.axiom (cupicef.type s4))) + (cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7)))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7))))) + cupicef.I (Coq__Classes__CRelationClasses.crelation s0 s7 A) + (R : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s0 s7 A) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7)))) + cupicef.I (Coq__Classes__CRelationClasses.crelation s0 s7 A) + (R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s0 s7 A) => + cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.type s7)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7))) + cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s3 s4 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s3))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s3))) + cstr_0 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s0 s7 A))) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) (cupicef.type s4))) + cstr_13 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s4))) + cstr_3 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.subrelation s0 s7 s7 A)) + (respectful s0 s8 s9 s7 s5 s6 cstr_4 cstr_5 cstr_16 cstr_17 cstr_14 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s9)) + (cupicef.univ (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.axiom (cupicef.type s7))) cupicef.I) + (cupicef.univ (cupicef.type s9) + (cupicef.axiom (cupicef.type s9)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s8))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s8 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_6 A))) + (Coq__Classes__CRelationClasses.arrow s7 s7)) + (Proper s0 s7 A) (Proper s0 s7 A)) + (Proper s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + (cupicef.trans (cupicef.type s0) + (cupicef.SortSome (cupicef.type s3)) cstr_2 + (cupicef.SortSome (cupicef.type s1)) cstr_8 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s1))) + (cupicef.trans (cupicef.axiom (cupicef.type s7)) + (cupicef.SortSome (cupicef.type s3)) cstr_0 + (cupicef.SortSome (cupicef.type s1)) cstr_8 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I (Coq__Classes__CRelationClasses.crelation s0 s7 A) + (__ : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s0 s7 A) => + cupicef.prod (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I)))) + (respectful s3 s4 s5 s6 s1 s2 cstr_8 cstr_9 cstr_10 cstr_11 cstr_12 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s3))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s3))) + cstr_0 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s0 s7 A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s5))) + cstr_4 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s5))) + (cupicef.trans (cupicef.axiom (cupicef.type s7)) + (cupicef.SortSome (cupicef.type s9)) cstr_1 + (cupicef.SortSome (cupicef.type s5)) cstr_17 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I))) + (cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7))))) + (cupicef.sup (cupicef.type s3) + (cupicef.axiom (cupicef.type s4))) + (cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7)))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7))))) + cupicef.I (Coq__Classes__CRelationClasses.crelation s0 s7 A) + (R : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s0 s7 A) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7)))) + cupicef.I + (Coq__Classes__CRelationClasses.crelation s0 s7 A) + (R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s0 s7 A) => + cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.type s7)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7))) + cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s3 s4 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s3))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s3))) + cstr_0 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s0 s7 A))) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) (cupicef.type s4))) + cstr_13 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s4))) + cstr_3 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.subrelation s0 s7 s7 A)) + (respectful s0 s8 s9 s7 s5 s6 cstr_4 cstr_5 cstr_16 cstr_17 + cstr_14 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s9)) + (cupicef.univ (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s9) + (cupicef.axiom (cupicef.type s9)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s8))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s8 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_6 A))) + (Coq__Classes__CRelationClasses.arrow s7 s7))) + (Proper s0 s7 A)) + cupicef.I + (cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.sup (cupicef.type s0) (cupicef.type s7)) + (cupicef.type s6)))) + (cupicef.type s2) + (cupicef.prod + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.sup (cupicef.type s0) (cupicef.type s7)) + (cupicef.type s6))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.type s7)) + (cupicef.type s6)))) + cupicef.I (Coq__Classes__CRelationClasses.crelation s0 s7 A) + (x : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s0 s7 A) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.sup (cupicef.type s0) (cupicef.type s7)) + (cupicef.type s6)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.type s7)) + (cupicef.type s6))) + cupicef.I (Coq__Classes__CRelationClasses.crelation s0 s7 A) + (y : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s0 s7 A) => + cupicef.prod (cupicef.sup (cupicef.type s0) (cupicef.type s7)) + (cupicef.type s6) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.type s7)) + (cupicef.type s6)) + cupicef.I + (Coq__Classes__CRelationClasses.subrelation s0 s7 s7 A x y) + (X : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.type s7)) + (Coq__Classes__CRelationClasses.subrelation s0 s7 s7 A + x y) => + respectful s0 s8 s9 s7 s5 s6 cstr_4 cstr_5 cstr_16 cstr_17 + cstr_14 A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s9)) + (cupicef.univ (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s9) + (cupicef.axiom (cupicef.type s9)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s8))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s8 A) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_6 A))) + (Coq__Classes__CRelationClasses.arrow s7 s7) + (Proper s0 s7 A x) (Proper s0 s7 A y))))) + (respectful s3 s4 s5 s6 s1 s2 cstr_8 cstr_9 cstr_10 cstr_11 cstr_12 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s3))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s3))) + cstr_0 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s0 s7 A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s5))) + cstr_4 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s5))) + (cupicef.trans (cupicef.axiom (cupicef.type s7)) + (cupicef.SortSome (cupicef.type s9)) cstr_1 + (cupicef.SortSome (cupicef.type s5)) cstr_17 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I))) + (cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7))))) + (cupicef.sup (cupicef.type s3) + (cupicef.axiom (cupicef.type s4))) + (cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7)))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7))))) + cupicef.I (Coq__Classes__CRelationClasses.crelation s0 s7 A) + (R : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s0 s7 A) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7)))) + cupicef.I + (Coq__Classes__CRelationClasses.crelation s0 s7 A) + (R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s0 s7 A) => + cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.type s7)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s7))) + cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s3 s4 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s3))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s3))) + cstr_0 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s0 s7 A))) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) (cupicef.type s4))) + cstr_13 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s4))) + cstr_3 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.subrelation s0 s7 s7 A)) + (respectful s0 s8 s9 s7 s5 s6 cstr_4 cstr_5 cstr_16 cstr_17 + cstr_14 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s9)) + (cupicef.univ (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s9) + (cupicef.axiom (cupicef.type s9)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s8))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s8 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_6 A))) + (Coq__Classes__CRelationClasses.arrow s7 s7)) + (Proper s0 s7 A) (Proper s0 s7 A)) + cupicef.I + (x : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s0 s7 A) => + y : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s0 s7 A) => + X : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s7)) + (Coq__Classes__CRelationClasses.subrelation s0 s7 s7 A x y) => + cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule cupicef.prop (cupicef.type s7)))) + (cupicef.type s6) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule cupicef.prop (cupicef.type s7))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule cupicef.prop (cupicef.type s7)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule cupicef.prop (cupicef.type s7)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule cupicef.prop (cupicef.type s7))) + cupicef.I A + (y0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod cupicef.prop (cupicef.type s7) + (cupicef.rule cupicef.prop (cupicef.type s7)) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_6 A) + x0 y0) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_6 A) + x0 y0) => + Coq__Classes__CRelationClasses.arrow s7 s7 + (Proper s0 s7 A x x0) (Proper s0 s7 A y y0))))) + (respectful s0 s8 s9 s7 s5 s6 cstr_4 cstr_5 cstr_16 cstr_17 + cstr_14 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s9)) + (cupicef.univ (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s9) + (cupicef.axiom (cupicef.type s9)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s8))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s8 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_6 A))) + (Coq__Classes__CRelationClasses.arrow s7 s7) (Proper s0 s7 A x) + (Proper s0 s7 A y)) + cupicef.I + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_6 A) + x0 y0) => + cupicef.cast (cupicef.rule (cupicef.type s7) (cupicef.type s7)) + (cupicef.type s7) + (cupicef.prod (cupicef.type s7) (cupicef.type s7) + (cupicef.rule (cupicef.type s7) (cupicef.type s7)) cupicef.I + (Proper s0 s7 A x x0) + (X0 : cupicef.Term (cupicef.type s7) (Proper s0 s7 A x x0) => + Proper s0 s7 A y y0)) + (Coq__Classes__CRelationClasses.arrow s7 s7 + (Proper s0 s7 A x x0) (Proper s0 s7 A y y0)) + cupicef.I + (X0 : cupicef.Term (cupicef.type s7) (Proper s0 s7 A x x0) => + cupicef.cast (cupicef.type s7) (cupicef.type s7) (y y0 y0) + (Proper s0 s7 A y y0) cupicef.I + (Coq__Init__Logic.eq__rect__r + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_7 A) + y0 + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + cupicef.I A + (x1 : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I)) + (cupicef.prod (cupicef.type s0) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cstr_15 + (x1 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s7) (cupicef.type s7) + (cupicef.rule (cupicef.type s7) (cupicef.type s7)) + cupicef.I (Proper s0 s7 A x x1) + (X1 : + cupicef.Term (cupicef.type s7) + (Proper s0 s7 A x x1) => + y y0 y0))) + (X1 : + cupicef.Term (cupicef.type s7) (Proper s0 s7 A x y0) => + (X2 : + (y1 : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s7) (x y0 y1) -> + cupicef.Term (cupicef.type s7) (y y0 y1)) => + (X3 : + (__ : cupicef.Term (cupicef.type s7) (x y0 y0) -> + cupicef.Term (cupicef.type s7) (y y0 y0)) => + (X4 : cupicef.Term (cupicef.type s7) (y y0 y0) => X4) + (X3 X1)) + (X2 y0)) + (X y0)) + x0 H X0))))). + +#TYPE_CLASS_INSTANCE proper__subrelation__proper__arrow. + +thm pointwise__subrelation : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s3)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s1)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s3)) -> + A : cupicef.Univ (cupicef.type s0) -> + B : cupicef.Univ (cupicef.type s5) -> + R : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type s4)) (cupicef.type s5)) + (Coq__Classes__CRelationClasses.crelation s5 s4 B) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s5) (cupicef.axiom (cupicef.type s6))) + (Coq__Classes__CRelationClasses.crelation s5 s6 B) -> + sub : + cupicef.Term + (cupicef.sup (cupicef.type s4) + (cupicef.sup (cupicef.type s5) (cupicef.type s6))) + (Coq__Classes__CRelationClasses.subrelation s5 s4 s6 B R R_27_) -> + cupicef.Term + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) (cupicef.type s3))) + (Coq__Classes__CRelationClasses.subrelation s1 s2 s3 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s5)) + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s1))) + cstr_4 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s5) + (cupicef.rule (cupicef.type s0) (cupicef.type s5)) cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => B))) + (pointwise__relation s0 s5 s4 s1 s2 cstr_0 cstr_1 cstr_4 cstr_3 A B R) + (pointwise__relation s0 s5 s6 s1 s3 cstr_0 cstr_2 cstr_4 cstr_5 A B + R_27_)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s3)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s1)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s3)) => + A : cupicef.Univ (cupicef.type s0) => + B : cupicef.Univ (cupicef.type s5) => + R : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type s4)) (cupicef.type s5)) + (Coq__Classes__CRelationClasses.crelation s5 s4 B) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s5) (cupicef.axiom (cupicef.type s6))) + (Coq__Classes__CRelationClasses.crelation s5 s6 B) => + sub : + cupicef.Term + (cupicef.sup (cupicef.type s4) + (cupicef.sup (cupicef.type s5) (cupicef.type s6))) + (Coq__Classes__CRelationClasses.subrelation s5 s4 s6 B R R_27_) => + cupicef.cast + (cupicef.rule (cupicef.rule (cupicef.type s0) (cupicef.type s5)) + (cupicef.rule (cupicef.rule (cupicef.type s0) (cupicef.type s5)) + (cupicef.rule (cupicef.type s2) (cupicef.type s3)))) + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) (cupicef.type s3))) + (cupicef.prod (cupicef.rule (cupicef.type s0) (cupicef.type s5)) + (cupicef.rule (cupicef.rule (cupicef.type s0) (cupicef.type s5)) + (cupicef.rule (cupicef.type s2) (cupicef.type s3))) + (cupicef.rule (cupicef.rule (cupicef.type s0) (cupicef.type s5)) + (cupicef.rule (cupicef.rule (cupicef.type s0) (cupicef.type s5)) + (cupicef.rule (cupicef.type s2) (cupicef.type s3)))) + cupicef.I + (cupicef.prod (cupicef.type s0) (cupicef.type s5) + (cupicef.rule (cupicef.type s0) (cupicef.type s5)) cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => B)) + (x : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s5) B) => + cupicef.prod (cupicef.rule (cupicef.type s0) (cupicef.type s5)) + (cupicef.rule (cupicef.type s2) (cupicef.type s3)) + (cupicef.rule (cupicef.rule (cupicef.type s0) (cupicef.type s5)) + (cupicef.rule (cupicef.type s2) (cupicef.type s3))) + cupicef.I + (cupicef.prod (cupicef.type s0) (cupicef.type s5) + (cupicef.rule (cupicef.type s0) (cupicef.type s5)) cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => B)) + (y : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s5) B) => + cupicef.prod (cupicef.type s2) (cupicef.type s3) + (cupicef.rule (cupicef.type s2) (cupicef.type s3)) cupicef.I + (pointwise__relation s0 s5 s4 s1 s2 cstr_0 cstr_1 cstr_4 cstr_3 + A B R x y) + (X : + cupicef.Term (cupicef.type s2) + (pointwise__relation s0 s5 s4 s1 s2 cstr_0 cstr_1 cstr_4 + cstr_3 A B R x y) => + pointwise__relation s0 s5 s6 s1 s3 cstr_0 cstr_2 cstr_4 cstr_5 + A B R_27_ x y)))) + (Coq__Classes__CRelationClasses.subrelation s1 s2 s3 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s1))) + cstr_4 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s5) + (cupicef.rule (cupicef.type s0) (cupicef.type s5)) cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => B))) + (pointwise__relation s0 s5 s4 s1 s2 cstr_0 cstr_1 cstr_4 cstr_3 A B R) + (pointwise__relation s0 s5 s6 s1 s3 cstr_0 cstr_2 cstr_4 cstr_5 A B + R_27_)) + cupicef.I + (x : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s5) B) => + y : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s5) B) => + X : + cupicef.Term (cupicef.type s2) + (pointwise__relation s0 s5 s4 s1 s2 cstr_0 cstr_1 cstr_4 cstr_3 A B + R x y) => + cupicef.cast (cupicef.rule (cupicef.type s0) (cupicef.type s6)) + (cupicef.type s3) + (cupicef.prod (cupicef.type s0) (cupicef.type s6) + (cupicef.rule (cupicef.type s0) (cupicef.type s6)) cupicef.I A + (a : cupicef.Term (cupicef.type s0) A => R_27_ (x a) (y a))) + (pointwise__relation s0 s5 s6 s1 s3 cstr_0 cstr_2 cstr_4 cstr_5 A B + R_27_ x y) + cupicef.I + (a : cupicef.Term (cupicef.type s0) A => + cupicef.cast (cupicef.type s6) (cupicef.type s6) (R_27_ (x a) (y a)) + (R_27_ (x a) (y a)) cupicef.I (sub (x a) (y a) (X a)))). + +#TYPE_CLASS_INSTANCE pointwise__subrelation. + +thm forall__subrelation : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s3)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s4)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s3)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s4)) -> + A : cupicef.Univ (cupicef.type s0) -> + P : + (__ : cupicef.Term (cupicef.type s0) A -> cupicef.Univ (cupicef.type s1)) -> + R : + (x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s5))) + (Coq__Classes__CRelationClasses.crelation s1 s5 (P x))) -> + S : + (x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s6))) + (Coq__Classes__CRelationClasses.crelation s1 s6 (P x))) -> + __ : + (a : cupicef.Term (cupicef.type s0) A -> + cupicef.Term + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s5) (cupicef.type s6))) + (Coq__Classes__CRelationClasses.subrelation s1 s5 s6 (P a) (R a) (S a))) -> + cupicef.Term + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s3) (cupicef.type s4))) + (Coq__Classes__CRelationClasses.subrelation s2 s3 s4 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) (cupicef.axiom (cupicef.type s2)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s2))) + cstr_3 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s2))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => P x))) + (forall__relation s0 s1 s5 s2 s3 cstr_0 cstr_1 cstr_3 cstr_4 A P R) + (forall__relation s0 s1 s6 s2 s4 cstr_0 cstr_2 cstr_3 cstr_5 A P S)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s3)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s4)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s3)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s4)) => + A : cupicef.Univ (cupicef.type s0) => + P : + (__ : cupicef.Term (cupicef.type s0) A -> cupicef.Univ (cupicef.type s1)) => + R : + (x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s5))) + (Coq__Classes__CRelationClasses.crelation s1 s5 (P x))) => + S : + (x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s6))) + (Coq__Classes__CRelationClasses.crelation s1 s6 (P x))) => + cupicef.cast + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s5) (cupicef.type s6)))) + (cupicef.rule (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule (cupicef.type s3) (cupicef.type s4))))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s5) (cupicef.type s6)))) + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s3) (cupicef.type s4)))) + (cupicef.prod + (cupicef.rule (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s5) (cupicef.type s6)))) + (cupicef.rule (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule (cupicef.type s3) (cupicef.type s4)))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s5) (cupicef.type s6)))) + (cupicef.rule (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule (cupicef.type s3) (cupicef.type s4))))) + cupicef.I + (cupicef.prod (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s5) (cupicef.type s6))) + (cupicef.rule (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s5) (cupicef.type s6)))) + cupicef.I A + (a : cupicef.Term (cupicef.type s0) A => + Coq__Classes__CRelationClasses.subrelation s1 s5 s6 (P a) ( + R a) (S a))) + (X : + (a : cupicef.Term (cupicef.type s0) A -> + cupicef.Term + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s5) (cupicef.type s6))) + (Coq__Classes__CRelationClasses.subrelation s1 s5 s6 (P a) + (R a) (S a))) => + cupicef.prod (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule (cupicef.type s3) (cupicef.type s4))) + (cupicef.rule (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule (cupicef.type s3) (cupicef.type s4)))) + cupicef.I + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => P x)) + (x : + (x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) (P x)) => + cupicef.prod (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule (cupicef.type s3) (cupicef.type s4)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule (cupicef.type s3) (cupicef.type s4))) + cupicef.I + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I + A (x0 : cupicef.Term (cupicef.type s0) A => P x0)) + (y : + (x0 : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) (P x0)) => + cupicef.prod (cupicef.type s3) (cupicef.type s4) + (cupicef.rule (cupicef.type s3) (cupicef.type s4)) cupicef.I + (forall__relation s0 s1 s5 s2 s3 cstr_0 cstr_1 cstr_3 cstr_4 + A P R x y) + (X0 : + cupicef.Term (cupicef.type s3) + (forall__relation s0 s1 s5 s2 s3 cstr_0 cstr_1 cstr_3 + cstr_4 A P R x y) => + forall__relation s0 s1 s6 s2 s4 cstr_0 cstr_2 cstr_3 cstr_5 + A P S x y))))) + (cupicef.prod + (cupicef.rule (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s5) (cupicef.type s6)))) + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s3) (cupicef.type s4))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s5) (cupicef.type s6)))) + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s3) (cupicef.type s4)))) + cupicef.I + (cupicef.prod (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s5) (cupicef.type s6))) + (cupicef.rule (cupicef.type s0) + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s5) (cupicef.type s6)))) + cupicef.I A + (a : cupicef.Term (cupicef.type s0) A => + Coq__Classes__CRelationClasses.subrelation s1 s5 s6 (P a) ( + R a) (S a))) + (__ : + (a : cupicef.Term (cupicef.type s0) A -> + cupicef.Term + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s5) (cupicef.type s6))) + (Coq__Classes__CRelationClasses.subrelation s1 s5 s6 (P a) + (R a) (S a))) => + Coq__Classes__CRelationClasses.subrelation s2 s3 s4 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s2))) + cstr_3 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s2))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I + A (x : cupicef.Term (cupicef.type s0) A => P x))) + (forall__relation s0 s1 s5 s2 s3 cstr_0 cstr_1 cstr_3 cstr_4 A P R) + (forall__relation s0 s1 s6 s2 s4 cstr_0 cstr_2 cstr_3 cstr_5 A P S))) + cupicef.I + (X : + (a : cupicef.Term (cupicef.type s0) A -> + cupicef.Term + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s5) (cupicef.type s6))) + (Coq__Classes__CRelationClasses.subrelation s1 s5 s6 (P a) ( + R a) (S a))) => + x : + (x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) (P x)) => + y : + (x0 : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) (P x0)) => + X0 : + cupicef.Term (cupicef.type s3) + (forall__relation s0 s1 s5 s2 s3 cstr_0 cstr_1 cstr_3 cstr_4 A P R + x y) => + cupicef.cast (cupicef.rule (cupicef.type s0) (cupicef.type s6)) + (cupicef.type s4) + (cupicef.prod (cupicef.type s0) (cupicef.type s6) + (cupicef.rule (cupicef.type s0) (cupicef.type s6)) cupicef.I A + (a : cupicef.Term (cupicef.type s0) A => S a (x a) (y a))) + (forall__relation s0 s1 s6 s2 s4 cstr_0 cstr_2 cstr_3 cstr_5 A P S x + y) + cupicef.I + (a : cupicef.Term (cupicef.type s0) A => + (X1 : + cupicef.Term + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s5) (cupicef.type s6))) + (Coq__Classes__CRelationClasses.subrelation s1 s5 s6 (P a) + (R a) (S a)) => + (X2 : cupicef.Term (cupicef.type s5) (R a (x a) (y a)) => + (X3 : + (y0 : cupicef.Term (cupicef.type s1) (P a) -> + __ : cupicef.Term (cupicef.type s5) (R a (x a) y0) -> + cupicef.Term (cupicef.type s6) (S a (x a) y0)) => + X1 (x a) (y a) X2) (X1 (x a))) (X0 a)) + (X a))). + +def apply__subrelation : + cupicef.Univ cupicef.prop. + +def do__subrelation : + cupicef.Term cupicef.prop apply__subrelation. + +thm iff__impl__subrelation : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + cstr_0 : + cupicef.eps (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0)) -> + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (Coq__Classes__CRelationClasses.subrelation s0 s1 s2 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s1 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + cupicef.I Coq__Init__Logic.iff) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s2 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + cupicef.I Coq__Program__Basics.impl)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0)) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (x : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (y : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.iff x y) + (H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + Coq__Program__Basics.impl x y)))) + (Coq__Classes__CRelationClasses.subrelation s0 s1 s2 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s1 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + cupicef.I Coq__Init__Logic.iff) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s2 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + cupicef.I Coq__Program__Basics.impl)) + cupicef.I + (x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I x + (H0 : cupicef.Term cupicef.prop x => y)) + (Coq__Program__Basics.impl x y) cupicef.I + (H0 : cupicef.Term cupicef.prop x => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I x + (__ : cupicef.Term cupicef.prop x => y)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I y + (__ : cupicef.Term cupicef.prop y => x)) + y + (H1 : + (__ : cupicef.Term cupicef.prop x -> + cupicef.Term cupicef.prop y) => + H2 : + (__ : cupicef.Term cupicef.prop y -> + cupicef.Term cupicef.prop x) => + (H3 : cupicef.Term cupicef.prop y => H3) (H1 H0)) + H)). + +#TYPE_CLASS_INSTANCE iff__impl__subrelation. + +thm iff__flip__impl__subrelation : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + cstr_0 : + cupicef.eps (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0)) -> + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (Coq__Classes__CRelationClasses.subrelation s0 s1 s2 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s1 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + cupicef.I Coq__Init__Logic.iff) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (x : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (y : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s2 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + cupicef.I + (Coq__Classes__CRelationClasses.flip s0 s0 s0 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Program__Basics.impl))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0)) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (x : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (y : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.iff x y) + (H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + Coq__Classes__CRelationClasses.flip s0 s0 s0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Program__Basics.impl x y)))) + (Coq__Classes__CRelationClasses.subrelation s0 s1 s2 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s1 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + cupicef.I Coq__Init__Logic.iff) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (x : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (y : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s2 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + cupicef.I + (Coq__Classes__CRelationClasses.flip s0 s0 s0 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Program__Basics.impl))) + cupicef.I + (x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I y + (H0 : cupicef.Term cupicef.prop y => x)) + (Coq__Classes__CRelationClasses.flip s0 s0 s0 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl x y) + cupicef.I + (H0 : cupicef.Term cupicef.prop y => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I x + (__ : cupicef.Term cupicef.prop x => y)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I y + (__ : cupicef.Term cupicef.prop y => x)) + x + (H1 : + (__ : cupicef.Term cupicef.prop x -> + cupicef.Term cupicef.prop y) => + H2 : + (__ : cupicef.Term cupicef.prop y -> + cupicef.Term cupicef.prop x) => + (H3 : cupicef.Term cupicef.prop x => H3) (H2 H0)) + H)). + +#TYPE_CLASS_INSTANCE iff__flip__impl__subrelation. + +thm iffT__arrow__subrelation : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s0)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_3 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Coq__Classes__CRelationClasses.subrelation s0 s1 s1 + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I)) + (Coq__Classes__CRelationClasses.iffT s1 s1 cstr_1 cstr_2 cstr_1 cstr_2) + (Coq__Classes__CRelationClasses.arrow s1 s1)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s0)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) => + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) => + cstr_3 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) => + cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))) + (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))) + cupicef.I + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (x : cupicef.Univ (cupicef.type s1) => + cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))) + cupicef.I + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (y : cupicef.Univ (cupicef.type s1) => + cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) cupicef.I + (Coq__Classes__CRelationClasses.iffT s1 s1 cstr_1 cstr_2 cstr_1 + cstr_2 x y) + (X : + cupicef.Term (cupicef.type s1) + (Coq__Classes__CRelationClasses.iffT s1 s1 cstr_1 cstr_2 + cstr_1 cstr_2 x y) => + Coq__Classes__CRelationClasses.arrow s1 s1 x y)))) + (Coq__Classes__CRelationClasses.subrelation s0 s1 s1 + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I)) + (Coq__Classes__CRelationClasses.iffT s1 s1 cstr_1 cstr_2 cstr_1 + cstr_2) + (Coq__Classes__CRelationClasses.arrow s1 s1)) + cupicef.I + (x : cupicef.Univ (cupicef.type s1) => + y : cupicef.Univ (cupicef.type s1) => + X : + cupicef.Term (cupicef.type s1) + (Coq__Classes__CRelationClasses.iffT s1 s1 cstr_1 cstr_2 cstr_1 + cstr_2 x y) => + cupicef.cast (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) cupicef.I x + (X0 : cupicef.Term (cupicef.type s1) x => + (X1 : + cupicef.Term (cupicef.type s1) + (Coq__Init__Datatypes.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I x + (__ : cupicef.Term (cupicef.type s1) x => y)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => x))) => + y) X)) + (Coq__Classes__CRelationClasses.arrow s1 s1 x y) cupicef.I + (X0 : cupicef.Term (cupicef.type s1) x => + Coq__Init__Datatypes.prod__rect + (cupicef.cast (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_1 + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) cupicef.I + x (__ : cupicef.Term (cupicef.type s1) x => y))) + (cupicef.cast (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_2 + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) cupicef.I + y (__ : cupicef.Term (cupicef.type s1) y => x))) + (cupicef.cast + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s1))) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I x + (__ : cupicef.Term (cupicef.type s1) x => y)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => x))) + (X1 : + cupicef.Term (cupicef.type s1) + (Coq__Init__Datatypes.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I x + (__ : cupicef.Term (cupicef.type s1) x => y)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => x))) => + cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + (cupicef.prod (cupicef.type s1) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I x + (__ : cupicef.Term (cupicef.type s1) x => y)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => x))) + (p : + cupicef.Term (cupicef.type s1) + (Coq__Init__Datatypes.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I x + (__ : cupicef.Term (cupicef.type s1) x => y)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => x))) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cstr_3 + (X1 : + cupicef.Term (cupicef.type s1) + (Coq__Init__Datatypes.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I x + (__ : cupicef.Term (cupicef.type s1) x => y)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => x))) => + y)) + (a : + (__ : cupicef.Term (cupicef.type s1) x -> + cupicef.Term (cupicef.type s1) y) => + b : + (__ : cupicef.Term (cupicef.type s1) y -> + cupicef.Term (cupicef.type s1) x) => + (X1 : cupicef.Term (cupicef.type s1) y => X1) (a X0)) + X)). + +#TYPE_CLASS_INSTANCE iffT__arrow__subrelation. + +thm iffT__flip__arrow__subrelation : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s0)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s2)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) -> + cstr_3 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) -> + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s3))) + (Coq__Classes__CRelationClasses.subrelation s0 s1 s3 + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I)) + (Coq__Classes__CRelationClasses.iffT s1 s1 cstr_3 cstr_4 cstr_3 cstr_4) + (Coq__Classes__CRelationClasses.flip s0 s0 s2 + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s1)))) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s3)))) + (cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s1)))) + cupicef.I + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (A : cupicef.Univ (cupicef.type s1) => + cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s1))) + cupicef.I + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (B : cupicef.Univ (cupicef.type s1) => + cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I))) + (cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s3)))) + cupicef.I + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (__ : cupicef.Univ (cupicef.type s1) => + cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s3)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s3))) + cupicef.I + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (__0 : cupicef.Univ (cupicef.type s1) => + cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I))) + cstr_2 (Coq__Classes__CRelationClasses.arrow s1 s1)))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s0)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s2)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) => + cstr_3 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) => + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) => + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) => + cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) (cupicef.type s3)))) + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s3))) + (cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) (cupicef.type s3))) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) (cupicef.type s3)))) + cupicef.I + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (x : cupicef.Univ (cupicef.type s1) => + cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) (cupicef.type s3)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) (cupicef.type s3))) + cupicef.I + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (y : cupicef.Univ (cupicef.type s1) => + cupicef.prod (cupicef.type s1) (cupicef.type s3) + (cupicef.rule (cupicef.type s1) (cupicef.type s3)) cupicef.I + (Coq__Classes__CRelationClasses.iffT s1 s1 cstr_3 cstr_4 cstr_3 + cstr_4 x y) + (X : + cupicef.Term (cupicef.type s1) + (Coq__Classes__CRelationClasses.iffT s1 s1 cstr_3 cstr_4 + cstr_3 cstr_4 x y) => + Coq__Classes__CRelationClasses.flip s0 s0 s2 + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s1)))) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s3)))) + (cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s1)))) + cupicef.I + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (A : cupicef.Univ (cupicef.type s1) => + cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s1))) + cupicef.I + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (B : cupicef.Univ (cupicef.type s1) => + cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I))) + (cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s3)))) + cupicef.I + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (__ : cupicef.Univ (cupicef.type s1) => + cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s3)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s3))) + cupicef.I + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (__0 : cupicef.Univ (cupicef.type s1) => + cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I))) + cstr_2 (Coq__Classes__CRelationClasses.arrow s1 s1)) + x y)))) + (Coq__Classes__CRelationClasses.subrelation s0 s1 s3 + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I)) + (Coq__Classes__CRelationClasses.iffT s1 s1 cstr_3 cstr_4 cstr_3 + cstr_4) + (Coq__Classes__CRelationClasses.flip s0 s0 s2 + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s1)))) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s3)))) + (cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s1)))) + cupicef.I + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (A : cupicef.Univ (cupicef.type s1) => + cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s1))) + cupicef.I + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (B : cupicef.Univ (cupicef.type s1) => + cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I))) + (cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s3)))) + cupicef.I + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (__ : cupicef.Univ (cupicef.type s1) => + cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s3)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s3))) + cupicef.I + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (__0 : cupicef.Univ (cupicef.type s1) => + cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I))) + cstr_2 (Coq__Classes__CRelationClasses.arrow s1 s1)))) + cupicef.I + (x : cupicef.Univ (cupicef.type s1) => + y : cupicef.Univ (cupicef.type s1) => + X : + cupicef.Term (cupicef.type s1) + (Coq__Classes__CRelationClasses.iffT s1 s1 cstr_3 cstr_4 cstr_3 + cstr_4 x y) => + cupicef.cast (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + (cupicef.type s3) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) cupicef.I y + (X0 : cupicef.Term (cupicef.type s1) y => + (X1 : + cupicef.Term (cupicef.type s1) + (Coq__Init__Datatypes.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I x + (__ : cupicef.Term (cupicef.type s1) x => y)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => x))) => + x) X)) + (Coq__Classes__CRelationClasses.flip s0 s0 s2 + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s1)))) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s3)))) + (cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s1)))) + cupicef.I + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (A : cupicef.Univ (cupicef.type s1) => + cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s1))) + cupicef.I + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (B : cupicef.Univ (cupicef.type s1) => + cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I))) + (cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s3)))) + cupicef.I + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (__ : cupicef.Univ (cupicef.type s1) => + cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s3)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type s3))) + cupicef.I + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (__0 : cupicef.Univ (cupicef.type s1) => + cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I))) + cstr_2 (Coq__Classes__CRelationClasses.arrow s1 s1)) + x y) + cupicef.I + (X0 : cupicef.Term (cupicef.type s1) y => + Coq__Init__Datatypes.prod__rect + (cupicef.cast (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_3 + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) cupicef.I + x (__ : cupicef.Term (cupicef.type s1) x => y))) + (cupicef.cast (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_4 + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) cupicef.I + y (__ : cupicef.Term (cupicef.type s1) y => x))) + (cupicef.cast + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s1))) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I x + (__ : cupicef.Term (cupicef.type s1) x => y)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => x))) + (X1 : + cupicef.Term (cupicef.type s1) + (Coq__Init__Datatypes.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I x + (__ : cupicef.Term (cupicef.type s1) x => y)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => x))) => + cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + (cupicef.prod (cupicef.type s1) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I x + (__ : cupicef.Term (cupicef.type s1) x => y)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => x))) + (p : + cupicef.Term (cupicef.type s1) + (Coq__Init__Datatypes.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I x + (__ : cupicef.Term (cupicef.type s1) x => y)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => x))) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cstr_5 + (X1 : + cupicef.Term (cupicef.type s1) + (Coq__Init__Datatypes.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I x + (__ : cupicef.Term (cupicef.type s1) x => y)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => x))) => + x)) + (a : + (__ : cupicef.Term (cupicef.type s1) x -> + cupicef.Term (cupicef.type s1) y) => + b : + (__ : cupicef.Term (cupicef.type s1) y -> + cupicef.Term (cupicef.type s1) x) => + (X1 : cupicef.Term (cupicef.type s1) x => X1) (b X0)) + X)). + +#TYPE_CLASS_INSTANCE iffT__flip__arrow__subrelation. + +def respectful__per__obligation__1 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s3)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s3)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PER s0 s4 A R) -> + B : cupicef.Univ (cupicef.type s1) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s5))) + (Coq__Classes__CRelationClasses.crelation s1 s5 B) -> + H0 : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s5)) + (Coq__Classes__CRelationClasses.PER s1 s5 B R_27_) -> + cupicef.Term (cupicef.sup (cupicef.type s2) (cupicef.type s3)) + (Coq__Classes__CRelationClasses.Symmetric s2 s3 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) (cupicef.axiom (cupicef.type s2)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s2))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s2))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => B))) + (respectful s0 s4 s1 s5 s2 s3 cstr_0 cstr_1 cstr_3 cstr_2 cstr_4 A B R + R_27_)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s3)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s3)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PER s0 s4 A R) => + B : cupicef.Univ (cupicef.type s1) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s5))) + (Coq__Classes__CRelationClasses.crelation s1 s5 B) => + H0 : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s5)) + (Coq__Classes__CRelationClasses.PER s1 s5 B R_27_) => + x : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) B) => + y : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) B) => + X : + cupicef.Term (cupicef.type s3) + (respectful s0 s4 s1 s5 s2 s3 cstr_0 cstr_1 cstr_3 cstr_2 cstr_4 A B R + R_27_ x y) => + x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + X0 : cupicef.Term (cupicef.type s4) (R x0 y0) => + Coq__Classes__CRelationClasses.match____PER s0 s4 (cupicef.type s5) A R + (p : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PER s0 s4 A R) => + R_27_ (y x0) (x y0)) + (PER__Symmetric : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Symmetric s0 s4 A R) => + PER__Transitive : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Transitive s0 s4 A R) => + Coq__Classes__CRelationClasses.match____PER s1 s5 (cupicef.type s5) B + R_27_ + (p : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s5)) + (Coq__Classes__CRelationClasses.PER s1 s5 B R_27_) => + R_27_ (y x0) (x y0)) + (PER__Symmetric0 : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s5)) + (Coq__Classes__CRelationClasses.Symmetric s1 s5 B R_27_) => + PER__Transitive0 : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s5)) + (Coq__Classes__CRelationClasses.Transitive s1 s5 B R_27_) => + PER__Symmetric0 (x y0) (y x0) (X y0 x0 (PER__Symmetric x0 y0 X0))) + H0) + H. + +def let__X2 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s3)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s3)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PER s0 s4 A R) -> + B : cupicef.Univ (cupicef.type s1) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s5))) + (Coq__Classes__CRelationClasses.crelation s1 s5 B) -> + H0 : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s5)) + (Coq__Classes__CRelationClasses.PER s1 s5 B R_27_) -> + x : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) B) -> + y : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) B) -> + z : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) B) -> + X : + cupicef.Term (cupicef.type s3) + (respectful s0 s4 s1 s5 s2 s3 cstr_0 cstr_1 cstr_3 cstr_2 cstr_4 A B R + R_27_ x y) -> + X0 : + cupicef.Term (cupicef.type s3) + (respectful s0 s4 s1 s5 s2 s3 cstr_0 cstr_1 cstr_3 cstr_2 cstr_4 A B R + R_27_ y z) -> + x0 : cupicef.Term (cupicef.type s0) A -> + y0 : cupicef.Term (cupicef.type s0) A -> + X1 : cupicef.Term (cupicef.type s4) (R x0 y0) -> + cupicef.Term (cupicef.type s4) (R x0 x0) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s3)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s3)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PER s0 s4 A R) => + B : cupicef.Univ (cupicef.type s1) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s5))) + (Coq__Classes__CRelationClasses.crelation s1 s5 B) => + H0 : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s5)) + (Coq__Classes__CRelationClasses.PER s1 s5 B R_27_) => + x : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) B) => + y : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) B) => + z : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) B) => + X : + cupicef.Term (cupicef.type s3) + (respectful s0 s4 s1 s5 s2 s3 cstr_0 cstr_1 cstr_3 cstr_2 cstr_4 A B R + R_27_ x y) => + X0 : + cupicef.Term (cupicef.type s3) + (respectful s0 s4 s1 s5 s2 s3 cstr_0 cstr_1 cstr_3 cstr_2 cstr_4 A B R + R_27_ y z) => + x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + X1 : cupicef.Term (cupicef.type s4) (R x0 y0) => + Coq__Classes__CRelationClasses.transitivity s0 s4 A R + (Coq__Classes__CRelationClasses.PER__Transitive s0 s4 A R H) x0 y0 x0 X1 + (Coq__Classes__CRelationClasses.symmetry s0 s4 A R + (Coq__Classes__CRelationClasses.PER__Symmetric s0 s4 A R H) x0 y0 X1). + +thm respectful__per__obligation__2 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s3)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s3)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PER s0 s4 A R) -> + B : cupicef.Univ (cupicef.type s1) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s5))) + (Coq__Classes__CRelationClasses.crelation s1 s5 B) -> + H0 : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s5)) + (Coq__Classes__CRelationClasses.PER s1 s5 B R_27_) -> + cupicef.Term (cupicef.sup (cupicef.type s2) (cupicef.type s3)) + (Coq__Classes__CRelationClasses.Transitive s2 s3 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) (cupicef.axiom (cupicef.type s2)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s2))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s2))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => B))) + (respectful s0 s4 s1 s5 s2 s3 cstr_0 cstr_1 cstr_3 cstr_2 cstr_4 A B R + R_27_)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s3)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s3)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PER s0 s4 A R) => + B : cupicef.Univ (cupicef.type s1) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s5))) + (Coq__Classes__CRelationClasses.crelation s1 s5 B) => + H0 : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s5)) + (Coq__Classes__CRelationClasses.PER s1 s5 B R_27_) => + x : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) B) => + y : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) B) => + z : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) B) => + X : + cupicef.Term (cupicef.type s3) + (respectful s0 s4 s1 s5 s2 s3 cstr_0 cstr_1 cstr_3 cstr_2 cstr_4 A B R + R_27_ x y) => + X0 : + cupicef.Term (cupicef.type s3) + (respectful s0 s4 s1 s5 s2 s3 cstr_0 cstr_1 cstr_3 cstr_2 cstr_4 A B R + R_27_ y z) => + x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + X1 : cupicef.Term (cupicef.type s4) (R x0 y0) => + Coq__Classes__CRelationClasses.transitivity s1 s5 B R_27_ + (Coq__Classes__CRelationClasses.PER__Transitive s1 s5 B R_27_ H0) ( + x x0) (y x0) (z y0) + (X x0 x0 + (let__X2 s0 s1 s2 s3 s4 s5 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 A R H B + R_27_ H0 x y z X X0 x0 y0 X1)) + (X0 x0 y0 X1). + +def respectful__per : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s3)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s3)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PER s0 s4 A R) -> + B : cupicef.Univ (cupicef.type s1) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s5))) + (Coq__Classes__CRelationClasses.crelation s1 s5 B) -> + H0 : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s5)) + (Coq__Classes__CRelationClasses.PER s1 s5 B R_27_) -> + cupicef.Term (cupicef.sup (cupicef.type s2) (cupicef.type s3)) + (Coq__Classes__CRelationClasses.PER s2 s3 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) (cupicef.axiom (cupicef.type s2)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s2))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s2))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => B))) + (respectful s0 s4 s1 s5 s2 s3 cstr_0 cstr_1 cstr_3 cstr_2 cstr_4 A B R + R_27_)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s3)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s3)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PER s0 s4 A R) => + B : cupicef.Univ (cupicef.type s1) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s5))) + (Coq__Classes__CRelationClasses.crelation s1 s5 B) => + H0 : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s5)) + (Coq__Classes__CRelationClasses.PER s1 s5 B R_27_) => + Coq__Classes__CRelationClasses.Build__PER s2 s3 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) (cupicef.axiom (cupicef.type s2)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s2))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s2))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => B))) + (respectful s0 s4 s1 s5 s2 s3 cstr_0 cstr_1 cstr_3 cstr_2 cstr_4 A B R + R_27_) + (respectful__per__obligation__1 s0 s1 s2 s3 s4 s5 cstr_0 cstr_1 cstr_2 + cstr_3 cstr_4 A R H B R_27_ H0) + (respectful__per__obligation__2 s0 s1 s2 s3 s4 s5 cstr_0 cstr_1 cstr_2 + cstr_3 cstr_4 A R H B R_27_ H0). + +thm flip__proper__obligation__1 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + s7 : cupicef.Nat -> + s8 : cupicef.Nat -> + s9 : cupicef.Nat -> + s10 : cupicef.Nat -> + s11 : cupicef.Nat -> + s12 : cupicef.Nat -> + s13 : cupicef.Nat -> + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s3)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s4)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s10)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s11)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s5)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s6)) -> + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s7)) -> + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s8)) -> + cstr_8 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s5)) -> + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s10)) -> + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s3)) -> + cstr_11 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s4)) -> + cstr_12 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s6)) -> + cstr_13 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s8)) -> + cstr_14 : cupicef.eps (cupicef.Cumul (cupicef.type s10) (cupicef.type s7)) -> + cstr_15 : cupicef.eps (cupicef.Cumul (cupicef.type s11) (cupicef.type s8)) -> + cstr_16 : cupicef.eps (cupicef.Cumul (cupicef.type s12) (cupicef.type s4)) -> + cstr_17 : cupicef.eps (cupicef.Cumul (cupicef.type s12) (cupicef.type s11)) -> + cstr_18 : cupicef.eps (cupicef.Cumul (cupicef.type s13) (cupicef.type s6)) -> + cstr_19 : cupicef.eps (cupicef.Cumul (cupicef.type s13) (cupicef.type s11)) -> + A : cupicef.Univ (cupicef.type s0) -> + B : cupicef.Univ (cupicef.type s1) -> + C : cupicef.Univ (cupicef.type s2) -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s12))) + (Coq__Classes__CRelationClasses.crelation s0 s12 A) -> + RB : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s9))) + (Coq__Classes__CRelationClasses.crelation s1 s9 B) -> + RC : + cupicef.Term + (cupicef.sup (cupicef.type s2) (cupicef.axiom (cupicef.type s13))) + (Coq__Classes__CRelationClasses.crelation s2 s13 C) -> + f : + (__ : cupicef.Term (cupicef.type s0) A -> + __0 : cupicef.Term (cupicef.type s1) B -> + cupicef.Term (cupicef.type s2) C) -> + mor : + cupicef.Term (cupicef.type s4) + (Proper s3 s4 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))) + cupicef.I) + (cupicef.univ (cupicef.type s3) (cupicef.axiom (cupicef.type s3)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s3))) + (cupicef.trans (cupicef.type s2) + (cupicef.SortSome (cupicef.type s5)) cstr_8 + (cupicef.SortSome (cupicef.type s3)) cstr_10 + cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s3))) + (cupicef.trans (cupicef.type s1) + (cupicef.SortSome (cupicef.type s5)) cstr_4 + (cupicef.SortSome (cupicef.type s3)) cstr_10 + cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s3))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s2)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s2))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s1) (cupicef.type s2) + (cupicef.rule (cupicef.type s1) (cupicef.type s2)) + cupicef.I B (__0 : cupicef.Term (cupicef.type s1) B => C)))) + (respectful s0 s12 s5 s6 s3 s4 cstr_0 cstr_1 cstr_16 cstr_10 cstr_11 + A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ + (cupicef.sup (cupicef.type s1) (cupicef.type s2)) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s5))) + cstr_8 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s5))) + cstr_4 cupicef.BoolNone) + (cupicef.prod (cupicef.type s1) (cupicef.type s2) + (cupicef.rule (cupicef.type s1) (cupicef.type s2)) + cupicef.I B (__ : cupicef.Term (cupicef.type s1) B => C))) + RA + (respectful s1 s9 s2 s13 s5 s6 cstr_4 cstr_5 cstr_12 cstr_8 + cstr_18 B C RB RC)) + f) -> + cupicef.Term (cupicef.type s8) + (Proper s7 s8 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))) + (cupicef.axiom (cupicef.type s7)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))) + cupicef.I) + (cupicef.univ (cupicef.type s7) (cupicef.axiom (cupicef.type s7)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s7))) + (cupicef.trans (cupicef.type s2) + (cupicef.SortSome (cupicef.type s10)) cstr_9 + (cupicef.SortSome (cupicef.type s7)) cstr_14 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s7))) + cstr_6 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s7))) + (cupicef.trans (cupicef.type s0) + (cupicef.SortSome (cupicef.type s10)) cstr_2 + (cupicef.SortSome (cupicef.type s7)) cstr_14 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s2))) + cupicef.I B + (__ : cupicef.Term (cupicef.type s1) B => + cupicef.prod (cupicef.type s0) (cupicef.type s2) + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) cupicef.I + A (__0 : cupicef.Term (cupicef.type s0) A => C)))) + (respectful s1 s9 s10 s11 s7 s8 cstr_6 cstr_7 cstr_13 cstr_14 cstr_15 + B + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s2))) + (cupicef.axiom (cupicef.type s10)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s10) + (cupicef.axiom (cupicef.type s10)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s10))) + cstr_9 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s10))) + cstr_2 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s2) + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) cupicef.I + A (__ : cupicef.Term (cupicef.type s0) A => C))) + RB + (respectful s0 s12 s2 s13 s10 s11 cstr_2 cstr_3 cstr_17 cstr_9 + cstr_19 A C RA RC)) + (Coq__Classes__CRelationClasses.flip s0 s1 s2 A B C f)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + s7 : cupicef.Nat => + s8 : cupicef.Nat => + s9 : cupicef.Nat => + s10 : cupicef.Nat => + s11 : cupicef.Nat => + s12 : cupicef.Nat => + s13 : cupicef.Nat => + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s3)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s4)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s10)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s11)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s5)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s6)) => + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s7)) => + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s8)) => + cstr_8 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s5)) => + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s10)) => + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s3)) => + cstr_11 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s4)) => + cstr_12 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s6)) => + cstr_13 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s8)) => + cstr_14 : cupicef.eps (cupicef.Cumul (cupicef.type s10) (cupicef.type s7)) => + cstr_15 : cupicef.eps (cupicef.Cumul (cupicef.type s11) (cupicef.type s8)) => + cstr_16 : cupicef.eps (cupicef.Cumul (cupicef.type s12) (cupicef.type s4)) => + cstr_17 : + cupicef.eps (cupicef.Cumul (cupicef.type s12) (cupicef.type s11)) => + cstr_18 : cupicef.eps (cupicef.Cumul (cupicef.type s13) (cupicef.type s6)) => + cstr_19 : + cupicef.eps (cupicef.Cumul (cupicef.type s13) (cupicef.type s11)) => + A : cupicef.Univ (cupicef.type s0) => + B : cupicef.Univ (cupicef.type s1) => + C : cupicef.Univ (cupicef.type s2) => + RA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s12))) + (Coq__Classes__CRelationClasses.crelation s0 s12 A) => + RB : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s9))) + (Coq__Classes__CRelationClasses.crelation s1 s9 B) => + RC : + cupicef.Term + (cupicef.sup (cupicef.type s2) (cupicef.axiom (cupicef.type s13))) + (Coq__Classes__CRelationClasses.crelation s2 s13 C) => + f : + (__ : cupicef.Term (cupicef.type s0) A -> + __0 : cupicef.Term (cupicef.type s1) B -> + cupicef.Term (cupicef.type s2) C) => + mor : + cupicef.Term (cupicef.type s4) + (Proper s3 s4 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s3))) + (cupicef.trans (cupicef.type s2) + (cupicef.SortSome (cupicef.type s5)) cstr_8 + (cupicef.SortSome (cupicef.type s3)) cstr_10 + cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s3))) + (cupicef.trans (cupicef.type s1) + (cupicef.SortSome (cupicef.type s5)) cstr_4 + (cupicef.SortSome (cupicef.type s3)) cstr_10 + cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s3))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s2)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s2))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s1) (cupicef.type s2) + (cupicef.rule (cupicef.type s1) (cupicef.type s2)) + cupicef.I B (__0 : cupicef.Term (cupicef.type s1) B => C)))) + (respectful s0 s12 s5 s6 s3 s4 cstr_0 cstr_1 cstr_16 cstr_10 + cstr_11 A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ + (cupicef.sup (cupicef.type s1) (cupicef.type s2)) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s5))) + cstr_8 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s5))) + cstr_4 cupicef.BoolNone) + (cupicef.prod (cupicef.type s1) (cupicef.type s2) + (cupicef.rule (cupicef.type s1) (cupicef.type s2)) + cupicef.I B (__ : cupicef.Term (cupicef.type s1) B => C))) + RA + (respectful s1 s9 s2 s13 s5 s6 cstr_4 cstr_5 cstr_12 cstr_8 + cstr_18 B C RB RC)) + f) => + x : cupicef.Term (cupicef.type s1) B => + y : cupicef.Term (cupicef.type s1) B => + X : cupicef.Term (cupicef.type s9) (RB x y) => + x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + X0 : cupicef.Term (cupicef.type s12) (RA x0 y0) => + mor x0 y0 X0 x y X. + +def flip__proper : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + s7 : cupicef.Nat -> + s8 : cupicef.Nat -> + s9 : cupicef.Nat -> + s10 : cupicef.Nat -> + s11 : cupicef.Nat -> + s12 : cupicef.Nat -> + s13 : cupicef.Nat -> + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s3)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s4)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s10)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s11)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s5)) -> + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s6)) -> + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s7)) -> + cstr_8 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s8)) -> + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s3)) -> + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s5)) -> + cstr_11 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s10)) -> + cstr_12 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s3)) -> + cstr_13 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s4)) -> + cstr_14 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s6)) -> + cstr_15 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s8)) -> + cstr_16 : cupicef.eps (cupicef.Cumul (cupicef.type s10) (cupicef.type s7)) -> + cstr_17 : cupicef.eps (cupicef.Cumul (cupicef.type s11) (cupicef.type s8)) -> + cstr_18 : cupicef.eps (cupicef.Cumul (cupicef.type s12) (cupicef.type s4)) -> + cstr_19 : cupicef.eps (cupicef.Cumul (cupicef.type s12) (cupicef.type s11)) -> + cstr_20 : cupicef.eps (cupicef.Cumul (cupicef.type s13) (cupicef.type s6)) -> + cstr_21 : cupicef.eps (cupicef.Cumul (cupicef.type s13) (cupicef.type s11)) -> + A : cupicef.Univ (cupicef.type s0) -> + B : cupicef.Univ (cupicef.type s1) -> + C : cupicef.Univ (cupicef.type s2) -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s12))) + (Coq__Classes__CRelationClasses.crelation s0 s12 A) -> + RB : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s9))) + (Coq__Classes__CRelationClasses.crelation s1 s9 B) -> + RC : + cupicef.Term + (cupicef.sup (cupicef.type s2) (cupicef.axiom (cupicef.type s13))) + (Coq__Classes__CRelationClasses.crelation s2 s13 C) -> + f : + (__ : cupicef.Term (cupicef.type s0) A -> + __0 : cupicef.Term (cupicef.type s1) B -> + cupicef.Term (cupicef.type s2) C) -> + mor : + cupicef.Term (cupicef.type s4) + (Proper s3 s4 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))) + cupicef.I) + (cupicef.univ (cupicef.type s3) (cupicef.axiom (cupicef.type s3)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s3))) + cstr_9 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s3))) + cstr_4 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s3))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s2)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s2))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s1) (cupicef.type s2) + (cupicef.rule (cupicef.type s1) (cupicef.type s2)) + cupicef.I B (__0 : cupicef.Term (cupicef.type s1) B => C)))) + (respectful s0 s12 s5 s6 s3 s4 cstr_0 cstr_1 cstr_18 cstr_12 cstr_13 + A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ + (cupicef.sup (cupicef.type s1) (cupicef.type s2)) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s5))) + cstr_10 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s5))) + cstr_5 cupicef.BoolNone) + (cupicef.prod (cupicef.type s1) (cupicef.type s2) + (cupicef.rule (cupicef.type s1) (cupicef.type s2)) + cupicef.I B (__ : cupicef.Term (cupicef.type s1) B => C))) + RA + (respectful s1 s9 s2 s13 s5 s6 cstr_5 cstr_6 cstr_14 cstr_10 + cstr_20 B C RB RC)) + f) -> + cupicef.Term (cupicef.type s8) + (Proper s7 s8 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))) + (cupicef.axiom (cupicef.type s7)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))) + cupicef.I) + (cupicef.univ (cupicef.type s7) (cupicef.axiom (cupicef.type s7)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s7))) + (cupicef.trans (cupicef.type s2) + (cupicef.SortSome (cupicef.type s10)) cstr_11 + (cupicef.SortSome (cupicef.type s7)) cstr_16 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s7))) + cstr_7 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s7))) + (cupicef.trans (cupicef.type s0) + (cupicef.SortSome (cupicef.type s10)) cstr_2 + (cupicef.SortSome (cupicef.type s7)) cstr_16 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s2))) + cupicef.I B + (__ : cupicef.Term (cupicef.type s1) B => + cupicef.prod (cupicef.type s0) (cupicef.type s2) + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) cupicef.I + A (__0 : cupicef.Term (cupicef.type s0) A => C)))) + (respectful s1 s9 s10 s11 s7 s8 cstr_7 cstr_8 cstr_15 cstr_16 cstr_17 + B + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s2))) + (cupicef.axiom (cupicef.type s10)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s10) + (cupicef.axiom (cupicef.type s10)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s10))) + cstr_11 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s10))) + cstr_2 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s2) + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) cupicef.I + A (__ : cupicef.Term (cupicef.type s0) A => C))) + RB + (respectful s0 s12 s2 s13 s10 s11 cstr_2 cstr_3 cstr_19 cstr_11 + cstr_21 A C RA RC)) + (Coq__Classes__CRelationClasses.flip s0 s1 s2 A B C f)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + s7 : cupicef.Nat => + s8 : cupicef.Nat => + s9 : cupicef.Nat => + s10 : cupicef.Nat => + s11 : cupicef.Nat => + s12 : cupicef.Nat => + s13 : cupicef.Nat => + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s3)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s4)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s10)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s11)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s5)) => + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s6)) => + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s7)) => + cstr_8 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s8)) => + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s3)) => + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s5)) => + cstr_11 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s10)) => + cstr_12 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s3)) => + cstr_13 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s4)) => + cstr_14 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s6)) => + cstr_15 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s8)) => + cstr_16 : cupicef.eps (cupicef.Cumul (cupicef.type s10) (cupicef.type s7)) => + cstr_17 : cupicef.eps (cupicef.Cumul (cupicef.type s11) (cupicef.type s8)) => + cstr_18 : cupicef.eps (cupicef.Cumul (cupicef.type s12) (cupicef.type s4)) => + cstr_19 : + cupicef.eps (cupicef.Cumul (cupicef.type s12) (cupicef.type s11)) => + cstr_20 : cupicef.eps (cupicef.Cumul (cupicef.type s13) (cupicef.type s6)) => + cstr_21 : + cupicef.eps (cupicef.Cumul (cupicef.type s13) (cupicef.type s11)) => + A : cupicef.Univ (cupicef.type s0) => + B : cupicef.Univ (cupicef.type s1) => + C : cupicef.Univ (cupicef.type s2) => + RA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s12))) + (Coq__Classes__CRelationClasses.crelation s0 s12 A) => + RB : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s9))) + (Coq__Classes__CRelationClasses.crelation s1 s9 B) => + RC : + cupicef.Term + (cupicef.sup (cupicef.type s2) (cupicef.axiom (cupicef.type s13))) + (Coq__Classes__CRelationClasses.crelation s2 s13 C) => + f : + (__ : cupicef.Term (cupicef.type s0) A -> + __0 : cupicef.Term (cupicef.type s1) B -> + cupicef.Term (cupicef.type s2) C) => + mor : + cupicef.Term (cupicef.type s4) + (Proper s3 s4 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s3))) + cstr_9 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s3))) + cstr_4 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s3))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s2)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s2))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s1) (cupicef.type s2) + (cupicef.rule (cupicef.type s1) (cupicef.type s2)) + cupicef.I B (__0 : cupicef.Term (cupicef.type s1) B => C)))) + (respectful s0 s12 s5 s6 s3 s4 cstr_0 cstr_1 cstr_18 cstr_12 + cstr_13 A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ + (cupicef.sup (cupicef.type s1) (cupicef.type s2)) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s5))) + cstr_10 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s5))) + cstr_5 cupicef.BoolNone) + (cupicef.prod (cupicef.type s1) (cupicef.type s2) + (cupicef.rule (cupicef.type s1) (cupicef.type s2)) + cupicef.I B (__ : cupicef.Term (cupicef.type s1) B => C))) + RA + (respectful s1 s9 s2 s13 s5 s6 cstr_5 cstr_6 cstr_14 cstr_10 + cstr_20 B C RB RC)) + f) => + flip__proper__obligation__1 s0 s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 s12 s13 + cstr_0 cstr_1 cstr_2 cstr_3 cstr_5 cstr_6 cstr_7 cstr_8 cstr_10 cstr_11 + cstr_12 cstr_13 cstr_14 cstr_15 cstr_16 cstr_17 cstr_18 cstr_19 cstr_20 + cstr_21 A B C RA RB RC f mor. + +thm trans__contra__co__type__morphism__obligation__1 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Transitive s0 s4 A R) -> + cupicef.Term (cupicef.type s2) + (respectful s0 s4 s1 s2 s1 s2 cstr_1 cstr_2 cstr_4 cupicef.I cupicef.I A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s4)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) + (cupicef.type s1))) + (cupicef.trans (cupicef.axiom (cupicef.type s4)) + (cupicef.SortSome (cupicef.type s3)) cstr_0 + (cupicef.SortSome (cupicef.type s1)) cstr_3 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.axiom (cupicef.type s4)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s4))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I))) + (Coq__Classes__CRelationClasses.flip s0 s0 s3 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + (respectful s0 s4 s3 s4 s1 s2 cstr_1 cstr_2 cstr_4 cstr_3 cstr_4 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R (Coq__Classes__CRelationClasses.arrow s4 s4)) + R R) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Transitive s0 s4 A R) => + x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + X : cupicef.Term (cupicef.type s4) (R y x) => + x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + X0 : cupicef.Term (cupicef.type s4) (R x0 y0) => + X1 : cupicef.Term (cupicef.type s4) (R x x0) => + Coq__Classes__CRelationClasses.transitivity s0 s4 A R H y x y0 X + (Coq__Classes__CRelationClasses.transitivity s0 s4 A R H x x0 y0 X1 X0). + +def trans__contra__co__type__morphism : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Transitive s0 s4 A R) -> + cupicef.Term (cupicef.type s2) + (Proper s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s4)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) + (cupicef.type s1))) + (cupicef.trans (cupicef.axiom (cupicef.type s4)) + (cupicef.SortSome (cupicef.type s3)) cstr_0 + (cupicef.SortSome (cupicef.type s1)) cstr_3 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s4)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.axiom (cupicef.type s4)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s4))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)))) + (respectful s0 s4 s1 s2 s1 s2 cstr_1 cstr_2 cstr_4 cupicef.I cupicef.I + A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s4)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s4)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) + (cupicef.type s1))) + (cupicef.trans (cupicef.axiom (cupicef.type s4)) + (cupicef.SortSome (cupicef.type s3)) cstr_0 + (cupicef.SortSome (cupicef.type s1)) cstr_3 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) + (cupicef.axiom (cupicef.type s4)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s4))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I))) + (Coq__Classes__CRelationClasses.flip s0 s0 s3 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + (respectful s0 s4 s3 s4 s1 s2 cstr_1 cstr_2 cstr_4 cstr_3 cstr_4 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R (Coq__Classes__CRelationClasses.arrow s4 s4))) + R) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Transitive s0 s4 A R) => + trans__contra__co__type__morphism__obligation__1 s0 s1 s2 s3 s4 cstr_0 + cstr_1 cstr_2 cstr_3 cstr_4 A R H. + +#TYPE_CLASS_INSTANCE trans__contra__co__type__morphism. + +thm trans__contra__inv__impl__type__morphism__obligation__1 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s5)) (cupicef.type s4)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s6)) (cupicef.type s3)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s2)) -> + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s5)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s6))) + (Coq__Classes__CRelationClasses.crelation s0 s6 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s6)) + (Coq__Classes__CRelationClasses.Transitive s0 s6 A R) -> + x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s2) + (respectful s0 s6 s3 s5 s1 s2 cstr_2 cstr_3 + (cupicef.trans (cupicef.type s6) (cupicef.SortSome (cupicef.type s5)) + cstr_6 (cupicef.SortSome (cupicef.type s2)) cstr_5 cupicef.SortNone) + cstr_4 cstr_5 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) cupicef.I) + (cupicef.univ (cupicef.type s3) (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s6) (cupicef.axiom (cupicef.type s6)) + cupicef.I)) + (Coq__Classes__CRelationClasses.flip s0 s0 s3 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I)) + R) + (Coq__Classes__CRelationClasses.flip s3 s3 s4 + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I)) + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I)) + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom (cupicef.axiom (cupicef.type s5))) cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s6)))) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.prod (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s6))) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s6)))) + cupicef.I + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I) + (A0 : cupicef.Univ (cupicef.type s6) => + cupicef.prod (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s6))) + cupicef.I + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I) + (B : cupicef.Univ (cupicef.type s6) => + cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I))) + (cupicef.prod (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s5))) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s5)))) + cupicef.I + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I) + (__ : cupicef.Univ (cupicef.type s6) => + cupicef.prod (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s5)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s5))) + cupicef.I + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I) + (__0 : cupicef.Univ (cupicef.type s6) => + cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I))) + cstr_6 (Coq__Classes__CRelationClasses.arrow s6 s6))) + (R x) (R x)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s5)) (cupicef.type s4)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s6)) (cupicef.type s3)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s2)) => + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s5)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s6))) + (Coq__Classes__CRelationClasses.crelation s0 s6 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s6)) + (Coq__Classes__CRelationClasses.Transitive s0 s6 A R) => + x : cupicef.Term (cupicef.type s0) A => + x0 : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + X : cupicef.Term (cupicef.type s6) (R y x0) => + X0 : cupicef.Term (cupicef.type s6) (R x y) => + Coq__Classes__CRelationClasses.transitivity s0 s6 A R H x y x0 X0 X. + +def trans__contra__inv__impl__type__morphism : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s5)) (cupicef.type s4)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s6)) (cupicef.type s3)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s2)) -> + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s5)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s6))) + (Coq__Classes__CRelationClasses.crelation s0 s6 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s6)) + (Coq__Classes__CRelationClasses.Transitive s0 s6 A R) -> + x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s2) + (Proper s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s6)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s6)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s6)) + (cupicef.type s1))) + (cupicef.trans (cupicef.axiom (cupicef.type s6)) + (cupicef.SortSome (cupicef.type s3)) cstr_1 + (cupicef.SortSome (cupicef.type s1)) cstr_4 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s6))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I))) + (respectful s0 s6 s3 s5 s1 s2 cstr_2 cstr_3 + (cupicef.trans (cupicef.type s6) + (cupicef.SortSome (cupicef.type s5)) cstr_6 + (cupicef.SortSome (cupicef.type s2)) cstr_5 cupicef.SortNone) + cstr_4 cstr_5 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I)) + (Coq__Classes__CRelationClasses.flip s0 s0 s3 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I)) + R) + (Coq__Classes__CRelationClasses.flip s3 s3 s4 + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I)) + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I)) + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s6)))) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.prod (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s6))) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s6)))) + cupicef.I + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I) + (A0 : cupicef.Univ (cupicef.type s6) => + cupicef.prod (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s6))) + cupicef.I + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I) + (B : cupicef.Univ (cupicef.type s6) => + cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I))) + (cupicef.prod (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s5))) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s5)))) + cupicef.I + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I) + (__ : cupicef.Univ (cupicef.type s6) => + cupicef.prod (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s5)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s5))) + cupicef.I + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I) + (__0 : cupicef.Univ (cupicef.type s6) => + cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I))) + cstr_6 (Coq__Classes__CRelationClasses.arrow s6 s6)))) + (R x)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s5)) (cupicef.type s4)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s6)) (cupicef.type s3)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s2)) => + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s5)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s6))) + (Coq__Classes__CRelationClasses.crelation s0 s6 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s6)) + (Coq__Classes__CRelationClasses.Transitive s0 s6 A R) => + x : cupicef.Term (cupicef.type s0) A => + trans__contra__inv__impl__type__morphism__obligation__1 s0 s1 s2 s3 s4 s5 + s6 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 cstr_6 A R H x. + +#TYPE_CLASS_INSTANCE trans__contra__inv__impl__type__morphism. + +thm trans__co__impl__type__morphism__obligation__1 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Transitive s0 s4 A R) -> + x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s2) + (respectful s0 s4 s3 s4 s1 s2 cstr_1 cstr_2 cstr_4 cstr_3 cstr_4 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) cupicef.I) + (cupicef.univ (cupicef.type s3) (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) (cupicef.axiom (cupicef.type s4)) + cupicef.I)) + R (Coq__Classes__CRelationClasses.arrow s4 s4) (R x) (R x)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Transitive s0 s4 A R) => + x : cupicef.Term (cupicef.type s0) A => + x0 : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + X : cupicef.Term (cupicef.type s4) (R x0 y) => + X0 : cupicef.Term (cupicef.type s4) (R x x0) => + Coq__Classes__CRelationClasses.transitivity s0 s4 A R H x x0 y X0 X. + +def trans__co__impl__type__morphism : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Transitive s0 s4 A R) -> + x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s2) + (Proper s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s4)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) + (cupicef.type s1))) + (cupicef.trans (cupicef.axiom (cupicef.type s4)) + (cupicef.SortSome (cupicef.type s3)) cstr_0 + (cupicef.SortSome (cupicef.type s1)) cstr_3 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.axiom (cupicef.type s4)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s4))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I))) + (respectful s0 s4 s3 s4 s1 s2 cstr_1 cstr_2 cstr_4 cstr_3 cstr_4 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R (Coq__Classes__CRelationClasses.arrow s4 s4)) + (R x)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Transitive s0 s4 A R) => + x : cupicef.Term (cupicef.type s0) A => + trans__co__impl__type__morphism__obligation__1 s0 s1 s2 s3 s4 cstr_0 + cstr_1 cstr_2 cstr_3 cstr_4 A R H x. + +#TYPE_CLASS_INSTANCE trans__co__impl__type__morphism. + +thm trans__sym__co__inv__impl__type__morphism__obligation__1 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s5)) (cupicef.type s4)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s6)) (cupicef.type s3)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s2)) -> + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s5)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s6))) + (Coq__Classes__CRelationClasses.crelation s0 s6 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s6)) + (Coq__Classes__CRelationClasses.PER s0 s6 A R) -> + x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s2) + (respectful s0 s6 s3 s5 s1 s2 cstr_2 cstr_3 + (cupicef.trans (cupicef.type s6) (cupicef.SortSome (cupicef.type s5)) + cstr_6 (cupicef.SortSome (cupicef.type s2)) cstr_5 cupicef.SortNone) + cstr_4 cstr_5 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) cupicef.I) + (cupicef.univ (cupicef.type s3) (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s6) (cupicef.axiom (cupicef.type s6)) + cupicef.I)) + R + (Coq__Classes__CRelationClasses.flip s3 s3 s4 + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I)) + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I)) + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom (cupicef.axiom (cupicef.type s5))) cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s6)))) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.prod (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s6))) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s6)))) + cupicef.I + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I) + (A0 : cupicef.Univ (cupicef.type s6) => + cupicef.prod (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s6))) + cupicef.I + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I) + (B : cupicef.Univ (cupicef.type s6) => + cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I))) + (cupicef.prod (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s5))) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s5)))) + cupicef.I + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I) + (__ : cupicef.Univ (cupicef.type s6) => + cupicef.prod (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s5)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s5))) + cupicef.I + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I) + (__0 : cupicef.Univ (cupicef.type s6) => + cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I))) + cstr_6 (Coq__Classes__CRelationClasses.arrow s6 s6))) + (R x) (R x)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s5)) (cupicef.type s4)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s6)) (cupicef.type s3)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s2)) => + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s5)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s6))) + (Coq__Classes__CRelationClasses.crelation s0 s6 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s6)) + (Coq__Classes__CRelationClasses.PER s0 s6 A R) => + x : cupicef.Term (cupicef.type s0) A => + x0 : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + X : cupicef.Term (cupicef.type s6) (R x0 y) => + X0 : cupicef.Term (cupicef.type s6) (R x y) => + Coq__Classes__CRelationClasses.transitivity s0 s6 A R + (Coq__Classes__CRelationClasses.PER__Transitive s0 s6 A R H) x y x0 X0 + (Coq__Classes__CRelationClasses.symmetry s0 s6 A R + (Coq__Classes__CRelationClasses.PER__Symmetric s0 s6 A R H) x0 y X). + +def trans__sym__co__inv__impl__type__morphism : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s5)) (cupicef.type s4)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s6)) (cupicef.type s3)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s2)) -> + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s5)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s6))) + (Coq__Classes__CRelationClasses.crelation s0 s6 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s6)) + (Coq__Classes__CRelationClasses.PER s0 s6 A R) -> + x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s2) + (Proper s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s6)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s6)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s6)) + (cupicef.type s1))) + (cupicef.trans (cupicef.axiom (cupicef.type s6)) + (cupicef.SortSome (cupicef.type s3)) cstr_1 + (cupicef.SortSome (cupicef.type s1)) cstr_4 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s6))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I))) + (respectful s0 s6 s3 s5 s1 s2 cstr_2 cstr_3 + (cupicef.trans (cupicef.type s6) + (cupicef.SortSome (cupicef.type s5)) cstr_6 + (cupicef.SortSome (cupicef.type s2)) cstr_5 cupicef.SortNone) + cstr_4 cstr_5 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I)) + R + (Coq__Classes__CRelationClasses.flip s3 s3 s4 + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I)) + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I)) + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s6)))) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s5)))) + (cupicef.prod (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s6))) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s6)))) + cupicef.I + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I) + (A0 : cupicef.Univ (cupicef.type s6) => + cupicef.prod (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s6))) + cupicef.I + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I) + (B : cupicef.Univ (cupicef.type s6) => + cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I))) + (cupicef.prod (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s5))) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s5)))) + cupicef.I + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I) + (__ : cupicef.Univ (cupicef.type s6) => + cupicef.prod (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s5)) + (cupicef.rule (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s5))) + cupicef.I + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I) + (__0 : cupicef.Univ (cupicef.type s6) => + cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I))) + cstr_6 (Coq__Classes__CRelationClasses.arrow s6 s6)))) + (R x)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s5)) (cupicef.type s4)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s6)) (cupicef.type s3)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s2)) => + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s5)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s6))) + (Coq__Classes__CRelationClasses.crelation s0 s6 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s6)) + (Coq__Classes__CRelationClasses.PER s0 s6 A R) => + x : cupicef.Term (cupicef.type s0) A => + trans__sym__co__inv__impl__type__morphism__obligation__1 s0 s1 s2 s3 s4 s5 + s6 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 cstr_6 A R H x. + +#TYPE_CLASS_INSTANCE trans__sym__co__inv__impl__type__morphism. + +thm trans__sym__contra__arrow__morphism__obligation__1 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PER s0 s4 A R) -> + x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s2) + (respectful s0 s4 s3 s4 s1 s2 cstr_1 cstr_2 cstr_4 cstr_3 cstr_4 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) cupicef.I) + (cupicef.univ (cupicef.type s3) (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) (cupicef.axiom (cupicef.type s4)) + cupicef.I)) + (Coq__Classes__CRelationClasses.flip s0 s0 s3 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + (Coq__Classes__CRelationClasses.arrow s4 s4) (R x) (R x)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PER s0 s4 A R) => + x : cupicef.Term (cupicef.type s0) A => + x0 : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + X : cupicef.Term (cupicef.type s4) (R y x0) => + X0 : cupicef.Term (cupicef.type s4) (R x x0) => + Coq__Classes__CRelationClasses.transitivity s0 s4 A R + (Coq__Classes__CRelationClasses.PER__Transitive s0 s4 A R H) x x0 y X0 + (Coq__Classes__CRelationClasses.symmetry s0 s4 A R + (Coq__Classes__CRelationClasses.PER__Symmetric s0 s4 A R H) y x0 X). + +def trans__sym__contra__arrow__morphism : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PER s0 s4 A R) -> + x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s2) + (Proper s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s4)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) + (cupicef.type s1))) + (cupicef.trans (cupicef.axiom (cupicef.type s4)) + (cupicef.SortSome (cupicef.type s3)) cstr_0 + (cupicef.SortSome (cupicef.type s1)) cstr_3 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.axiom (cupicef.type s4)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s4))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I))) + (respectful s0 s4 s3 s4 s1 s2 cstr_1 cstr_2 cstr_4 cstr_3 cstr_4 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + (Coq__Classes__CRelationClasses.flip s0 s0 s3 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + (Coq__Classes__CRelationClasses.arrow s4 s4)) + (R x)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PER s0 s4 A R) => + x : cupicef.Term (cupicef.type s0) A => + trans__sym__contra__arrow__morphism__obligation__1 s0 s1 s2 s3 s4 cstr_0 + cstr_1 cstr_2 cstr_3 cstr_4 A R H x. + +#TYPE_CLASS_INSTANCE trans__sym__contra__arrow__morphism. + +thm per__partial__app__type__morphism__obligation__1 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) -> + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PER s0 s4 A R) -> + x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s2) + (respectful s0 s4 s3 s4 s1 s2 cstr_1 cstr_2 cstr_4 cstr_3 cstr_4 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) cupicef.I) + (cupicef.univ (cupicef.type s3) (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) (cupicef.axiom (cupicef.type s4)) + cupicef.I)) + R + (Coq__Classes__CRelationClasses.iffT s4 s4 cstr_5 cstr_6 cstr_5 cstr_6) + (R x) (R x)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) => + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PER s0 s4 A R) => + x : cupicef.Term (cupicef.type s0) A => + x0 : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + X : cupicef.Term (cupicef.type s4) (R x0 y) => + Coq__Init__Datatypes.pair + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) (cupicef.axiom (cupicef.type s4)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_5 + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (R x x0) (__ : cupicef.Term (cupicef.type s4) (R x x0) => R x y))) + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) (cupicef.axiom (cupicef.type s4)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_6 + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (R x y) (__ : cupicef.Term (cupicef.type s4) (R x y) => R x x0))) + (X0 : cupicef.Term (cupicef.type s4) (R x x0) => + Coq__Classes__CRelationClasses.transitivity s0 s4 A R + (Coq__Classes__CRelationClasses.PER__Transitive s0 s4 A R H) x x0 y + X0 X) + (X0 : cupicef.Term (cupicef.type s4) (R x y) => + Coq__Classes__CRelationClasses.transitivity s0 s4 A R + (Coq__Classes__CRelationClasses.PER__Transitive s0 s4 A R H) x y x0 + X0 + (Coq__Classes__CRelationClasses.symmetry s0 s4 A R + (Coq__Classes__CRelationClasses.PER__Symmetric s0 s4 A R H) x0 y X)). + +def per__partial__app__type__morphism : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) -> + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PER s0 s4 A R) -> + x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s2) + (Proper s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s4)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) + (cupicef.type s1))) + (cupicef.trans (cupicef.axiom (cupicef.type s4)) + (cupicef.SortSome (cupicef.type s3)) cstr_0 + (cupicef.SortSome (cupicef.type s1)) cstr_3 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.axiom (cupicef.type s4)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s4))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I))) + (respectful s0 s4 s3 s4 s1 s2 cstr_1 cstr_2 cstr_4 cstr_3 cstr_4 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R + (Coq__Classes__CRelationClasses.iffT s4 s4 cstr_5 cstr_6 cstr_5 + cstr_6)) + (R x)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) => + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PER s0 s4 A R) => + x : cupicef.Term (cupicef.type s0) A => + per__partial__app__type__morphism__obligation__1 s0 s1 s2 s3 s4 cstr_0 + cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 cstr_6 A R H x. + +#TYPE_CLASS_INSTANCE per__partial__app__type__morphism. + +thm trans__co__eq__inv__arrow__morphism__obligation__1 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + s7 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s6)) (cupicef.type s5)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) (cupicef.type s4)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) -> + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) -> + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s1)) -> + cstr_8 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s2)) -> + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s6)) -> + cstr_10 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s0 s7 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s7)) + (Coq__Classes__CRelationClasses.Transitive s0 s7 A R) -> + cupicef.Term (cupicef.type s2) + (respectful s0 s7 s1 s2 s1 s2 cstr_2 cstr_3 + (cupicef.trans (cupicef.type s7) (cupicef.SortSome (cupicef.type s6)) + cstr_9 (cupicef.SortSome (cupicef.type s2)) cstr_8 cupicef.SortNone) + cupicef.I cupicef.I A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s1))) + (cupicef.trans (cupicef.axiom (cupicef.type s7)) + (cupicef.SortSome (cupicef.type s4)) cstr_1 + (cupicef.SortSome (cupicef.type s1)) cstr_7 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I))) + R + (respectful s0 s3 s4 s6 s1 s2 cstr_2 cstr_3 cstr_6 cstr_7 cstr_8 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.axiom (cupicef.type s7))) cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s3 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_4 A))) + (Coq__Classes__CRelationClasses.flip s4 s4 s5 + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I)) + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I)) + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.rule (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type s6)))) + (cupicef.prod (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + (A0 : cupicef.Univ (cupicef.type s7) => + cupicef.prod (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type s7))) + cupicef.I + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + (B : cupicef.Univ (cupicef.type s7) => + cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I))) + (cupicef.prod (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type s6))) + (cupicef.rule (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type s6)))) + cupicef.I + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + (__ : cupicef.Univ (cupicef.type s7) => + cupicef.prod (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type s6))) + cupicef.I + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + (__0 : cupicef.Univ (cupicef.type s7) => + cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I))) + cstr_9 (Coq__Classes__CRelationClasses.arrow s7 s7)))) + R R) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + s7 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s6)) (cupicef.type s5)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) (cupicef.type s4)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) => + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) (cupicef.type (cupicef.s cupicef.z))) => + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) => + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s1)) => + cstr_8 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s2)) => + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s6)) => + cstr_10 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s0 s7 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s7)) + (Coq__Classes__CRelationClasses.Transitive s0 s7 A R) => + x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + X : cupicef.Term (cupicef.type s7) (R x y) => + x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_4 A) + x0 y0) => + X0 : cupicef.Term (cupicef.type s7) (R y y0) => + Coq__Init__Logic.eq__rect__r + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_5 A) + y0 + (cupicef.cast + (cupicef.rule (cupicef.type s0) (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type s0) (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.type s0) (cupicef.axiom (cupicef.type s7))) + cupicef.I A + (x1 : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s7) (cupicef.axiom (cupicef.type s7)) + cupicef.I)) + (cupicef.prod (cupicef.type s0) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cstr_10 (x1 : cupicef.Term (cupicef.type s0) A => R x x1)) + (Coq__Classes__CRelationClasses.transitivity s0 s7 A R H x y y0 X X0) x0 + H0. + +def trans__co__eq__inv__arrow__morphism : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + s7 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s6)) (cupicef.type s5)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) (cupicef.type s4)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) -> + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) -> + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s1)) -> + cstr_8 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s2)) -> + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s6)) -> + cstr_10 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s0 s7 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s7)) + (Coq__Classes__CRelationClasses.Transitive s0 s7 A R) -> + cupicef.Term (cupicef.type s2) + (Proper s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s1))) + (cupicef.trans (cupicef.axiom (cupicef.type s7)) + (cupicef.SortSome (cupicef.type s4)) cstr_1 + (cupicef.SortSome (cupicef.type s1)) cstr_7 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I)))) + (respectful s0 s7 s1 s2 s1 s2 cstr_2 cstr_3 + (cupicef.trans (cupicef.type s7) + (cupicef.SortSome (cupicef.type s6)) cstr_9 + (cupicef.SortSome (cupicef.type s2)) cstr_8 cupicef.SortNone) + cupicef.I cupicef.I A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s1))) + (cupicef.trans (cupicef.axiom (cupicef.type s7)) + (cupicef.SortSome (cupicef.type s4)) cstr_1 + (cupicef.SortSome (cupicef.type s1)) cstr_7 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) + (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s7))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I))) + R + (respectful s0 s3 s4 s6 s1 s2 cstr_2 cstr_3 cstr_6 cstr_7 cstr_8 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s3 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_4 A))) + (Coq__Classes__CRelationClasses.flip s4 s4 s5 + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I)) + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I)) + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.rule (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type s6)))) + (cupicef.prod (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + (A0 : cupicef.Univ (cupicef.type s7) => + cupicef.prod (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type s7))) + cupicef.I + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + (B : cupicef.Univ (cupicef.type s7) => + cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I))) + (cupicef.prod (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type s6))) + (cupicef.rule (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type s6)))) + cupicef.I + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + (__ : cupicef.Univ (cupicef.type s7) => + cupicef.prod (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type s6)) + (cupicef.rule (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type s6))) + cupicef.I + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + (__0 : cupicef.Univ (cupicef.type s7) => + cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I))) + cstr_9 (Coq__Classes__CRelationClasses.arrow s7 s7))))) + R) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + s7 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s6)) (cupicef.type s5)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) (cupicef.type s4)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) => + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) (cupicef.type (cupicef.s cupicef.z))) => + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) => + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s1)) => + cstr_8 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s2)) => + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s6)) => + cstr_10 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s0 s7 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s7)) + (Coq__Classes__CRelationClasses.Transitive s0 s7 A R) => + trans__co__eq__inv__arrow__morphism__obligation__1 s0 s1 s2 s3 s4 s5 s6 s7 + cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 cstr_6 cstr_7 cstr_8 cstr_9 + cstr_10 A R H. + +#TYPE_CLASS_INSTANCE trans__co__eq__inv__arrow__morphism. + +thm PER__type__morphism__obligation__1 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) -> + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PER s0 s4 A R) -> + cupicef.Term (cupicef.type s2) + (respectful s0 s4 s1 s2 s1 s2 cstr_1 cstr_2 cstr_4 cupicef.I cupicef.I A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s4)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) + (cupicef.type s1))) + (cupicef.trans (cupicef.axiom (cupicef.type s4)) + (cupicef.SortSome (cupicef.type s3)) cstr_0 + (cupicef.SortSome (cupicef.type s1)) cstr_3 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.axiom (cupicef.type s4)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s4))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I))) + R + (respectful s0 s4 s3 s4 s1 s2 cstr_1 cstr_2 cstr_4 cstr_3 cstr_4 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R + (Coq__Classes__CRelationClasses.iffT s4 s4 cstr_5 cstr_6 cstr_5 + cstr_6)) + R R) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) => + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PER s0 s4 A R) => + x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + X : cupicef.Term (cupicef.type s4) (R x y) => + x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + X0 : cupicef.Term (cupicef.type s4) (R x0 y0) => + Coq__Init__Datatypes.pair + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) (cupicef.axiom (cupicef.type s4)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_5 + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (R x x0) (__ : cupicef.Term (cupicef.type s4) (R x x0) => R y y0))) + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) (cupicef.axiom (cupicef.type s4)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_6 + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (R y y0) (__ : cupicef.Term (cupicef.type s4) (R y y0) => R x x0))) + (X1 : cupicef.Term (cupicef.type s4) (R x x0) => + Coq__Classes__CRelationClasses.transitivity s0 s4 A R + (Coq__Classes__CRelationClasses.PER__Transitive s0 s4 A R H) y x0 y0 + (Coq__Classes__CRelationClasses.transitivity s0 s4 A R + (Coq__Classes__CRelationClasses.PER__Transitive s0 s4 A R H) y x + x0 + (Coq__Classes__CRelationClasses.symmetry s0 s4 A R + (Coq__Classes__CRelationClasses.PER__Symmetric s0 s4 A R H) x y + X) + X1) + X0) + (X1 : cupicef.Term (cupicef.type s4) (R y y0) => + Coq__Classes__CRelationClasses.transitivity s0 s4 A R + (Coq__Classes__CRelationClasses.PER__Transitive s0 s4 A R H) x y x0 X + (Coq__Classes__CRelationClasses.transitivity s0 s4 A R + (Coq__Classes__CRelationClasses.PER__Transitive s0 s4 A R H) y y0 + x0 X1 + (Coq__Classes__CRelationClasses.symmetry s0 s4 A R + (Coq__Classes__CRelationClasses.PER__Symmetric s0 s4 A R H) x0 + y0 X0))). + +def PER__type__morphism : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) -> + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PER s0 s4 A R) -> + cupicef.Term (cupicef.type s2) + (Proper s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s4)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) + (cupicef.type s1))) + (cupicef.trans (cupicef.axiom (cupicef.type s4)) + (cupicef.SortSome (cupicef.type s3)) cstr_0 + (cupicef.SortSome (cupicef.type s1)) cstr_3 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s4)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.axiom (cupicef.type s4)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s4))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)))) + (respectful s0 s4 s1 s2 s1 s2 cstr_1 cstr_2 cstr_4 cupicef.I cupicef.I + A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s4)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s4)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) + (cupicef.type s1))) + (cupicef.trans (cupicef.axiom (cupicef.type s4)) + (cupicef.SortSome (cupicef.type s3)) cstr_0 + (cupicef.SortSome (cupicef.type s1)) cstr_3 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) + (cupicef.axiom (cupicef.type s4)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s4))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I))) + R + (respectful s0 s4 s3 s4 s1 s2 cstr_1 cstr_2 cstr_4 cstr_3 cstr_4 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R + (Coq__Classes__CRelationClasses.iffT s4 s4 cstr_5 cstr_6 cstr_5 + cstr_6))) + R) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) => + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PER s0 s4 A R) => + PER__type__morphism__obligation__1 s0 s1 s2 s3 s4 cstr_0 cstr_1 cstr_2 + cstr_3 cstr_4 cstr_5 cstr_6 A R H. + +#TYPE_CLASS_INSTANCE PER__type__morphism. + +thm symmetric__equiv__flip : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s1)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s4)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) -> + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (Coq__Classes__CRelationClasses.crelation s0 s3 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (Coq__Classes__CRelationClasses.Symmetric s0 s3 A R) -> + cupicef.Term (cupicef.type s2) + (Coq__Classes__CRelationClasses.relation__equivalence s0 s1 s2 s3 cstr_0 + cstr_2 cstr_3 cstr_4 cstr_5 cstr_6 A R + (Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s1)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s4)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) => + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (Coq__Classes__CRelationClasses.crelation s0 s3 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (Coq__Classes__CRelationClasses.Symmetric s0 s3 A R) => + cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3))) + (cupicef.type s2) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) (cupicef.type s3) + (cupicef.rule (cupicef.type s0) (cupicef.type s3)) cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + Coq__Init__Datatypes.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (R x y) + (__ : cupicef.Term (cupicef.type s3) (R x y) => + Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R x y)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R x y) + (__ : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R x y) => + R x y))))) + (Coq__Classes__CRelationClasses.relation__equivalence s0 s1 s2 s3 cstr_0 + cstr_2 cstr_3 cstr_4 cstr_5 cstr_6 A R + (Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R)) + cupicef.I + (x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + Coq__Init__Datatypes.pair + (cupicef.cast (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s3) (cupicef.axiom (cupicef.type s3)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_5 + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (R x y) + (__ : cupicef.Term (cupicef.type s3) (R x y) => + Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R x y))) + (cupicef.cast (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s3) (cupicef.axiom (cupicef.type s3)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_6 + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R x y) + (__ : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R x y) => + R x y))) + (X : cupicef.Term (cupicef.type s3) (R x y) => + (X0 : + (y0 : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s3) (R x y0) -> + cupicef.Term (cupicef.type s3) (R y0 x)) => + (X1 : + (__ : cupicef.Term (cupicef.type s3) (R x y) -> + cupicef.Term (cupicef.type s3) (R y x)) => + (X2 : cupicef.Term (cupicef.type s3) (R y x) => X2) (X1 X)) ( + X0 y)) + (H x)) + (X : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R x y) => + (X0 : + (y0 : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s3) (R y y0) -> + cupicef.Term (cupicef.type s3) (R y0 y)) => + (X1 : + (__ : cupicef.Term (cupicef.type s3) (R y y) -> + cupicef.Term (cupicef.type s3) (R y y)) => + (X2 : + (__ : cupicef.Term (cupicef.type s3) (R y x) -> + cupicef.Term (cupicef.type s3) (R x y)) => + (X3 : cupicef.Term (cupicef.type s3) (R x y) => X3) (X2 X)) + (X0 x)) + (X0 y)) + (H y))). + +thm compose__proper__obligation__1 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + s7 : cupicef.Nat -> + s8 : cupicef.Nat -> + s9 : cupicef.Nat -> + s10 : cupicef.Nat -> + s11 : cupicef.Nat -> + s12 : cupicef.Nat -> + s13 : cupicef.Nat -> + s14 : cupicef.Nat -> + s15 : cupicef.Nat -> + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s9)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s10)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s11)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s12)) -> + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s5)) -> + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s6)) -> + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s9)) -> + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s5)) -> + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s11)) -> + cstr_11 : + cupicef.eps + (cupicef.Cumul (cupicef.type s2) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_12 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s3)) -> + cstr_13 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s4)) -> + cstr_14 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s4)) -> + cstr_15 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s3)) -> + cstr_16 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s4)) -> + cstr_17 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s7)) -> + cstr_18 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s8)) -> + cstr_19 : cupicef.eps (cupicef.Cumul (cupicef.type s10) (cupicef.type s8)) -> + cstr_20 : cupicef.eps (cupicef.Cumul (cupicef.type s11) (cupicef.type s7)) -> + cstr_21 : cupicef.eps (cupicef.Cumul (cupicef.type s12) (cupicef.type s8)) -> + cstr_22 : cupicef.eps (cupicef.Cumul (cupicef.type s13) (cupicef.type s6)) -> + cstr_23 : cupicef.eps (cupicef.Cumul (cupicef.type s13) (cupicef.type s10)) -> + cstr_24 : cupicef.eps (cupicef.Cumul (cupicef.type s14) (cupicef.type s10)) -> + cstr_25 : cupicef.eps (cupicef.Cumul (cupicef.type s14) (cupicef.type s12)) -> + cstr_26 : cupicef.eps (cupicef.Cumul (cupicef.type s15) (cupicef.type s6)) -> + cstr_27 : cupicef.eps (cupicef.Cumul (cupicef.type s15) (cupicef.type s12)) -> + A : cupicef.Univ (cupicef.type s0) -> + B : cupicef.Univ (cupicef.type s1) -> + C : cupicef.Univ (cupicef.type s2) -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s14))) + (Coq__Classes__CRelationClasses.crelation s0 s14 A) -> + RB : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s13))) + (Coq__Classes__CRelationClasses.crelation s1 s13 B) -> + RC : + cupicef.Term + (cupicef.sup (cupicef.type s2) (cupicef.axiom (cupicef.type s15))) + (Coq__Classes__CRelationClasses.crelation s2 s15 C) -> + cupicef.Term (cupicef.type s4) + (respectful s5 s6 s7 s8 s3 s4 cstr_12 cstr_13 cstr_14 cstr_15 cstr_16 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ (cupicef.sup (cupicef.type s1) (cupicef.type s2)) + (cupicef.axiom (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s5) (cupicef.axiom (cupicef.type s5)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s5))) + cstr_9 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s5))) + cstr_5 cupicef.BoolNone) + (cupicef.prod (cupicef.type s1) (cupicef.type s2) + (cupicef.rule (cupicef.type s1) (cupicef.type s2)) cupicef.I B + (__ : cupicef.Term (cupicef.type s1) B => C))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))) + (cupicef.axiom (cupicef.type s7)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))) + cupicef.I) + (cupicef.univ (cupicef.type s7) (cupicef.axiom (cupicef.type s7)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s7))) + (cupicef.trans (cupicef.type s2) + (cupicef.SortSome (cupicef.type s11)) cstr_10 + (cupicef.SortSome (cupicef.type s7)) cstr_20 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s7))) + (cupicef.trans (cupicef.type s1) + (cupicef.SortSome (cupicef.type s9)) cstr_7 + (cupicef.SortSome (cupicef.type s7)) cstr_17 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s7))) + (cupicef.trans (cupicef.type s0) + (cupicef.SortSome (cupicef.type s9)) cstr_0 + (cupicef.SortSome (cupicef.type s7)) cstr_17 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.rule (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule (cupicef.type s0) (cupicef.type s2))) + cupicef.I + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I + A (__ : cupicef.Term (cupicef.type s0) A => B)) + (__ : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) B) => + cupicef.prod (cupicef.type s0) (cupicef.type s2) + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) cupicef.I + A (__0 : cupicef.Term (cupicef.type s0) A => C)))) + (respectful s1 s13 s2 s15 s5 s6 cstr_5 cstr_6 cstr_22 cstr_9 cstr_26 B + C RB RC) + (respectful s9 s10 s11 s12 s7 s8 cstr_17 cstr_18 cstr_19 cstr_20 + cstr_21 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + (cupicef.axiom (cupicef.type s9)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s9) + (cupicef.axiom (cupicef.type s9)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s9))) + cstr_7 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s9))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I + A (__ : cupicef.Term (cupicef.type s0) A => B))) + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s2))) + (cupicef.axiom (cupicef.type s11)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s11) + (cupicef.axiom (cupicef.type s11)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s11))) + cstr_10 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s11))) + cstr_2 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s2) + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) cupicef.I + A (__ : cupicef.Term (cupicef.type s0) A => C))) + (respectful s0 s14 s1 s13 s9 s10 cstr_0 cstr_1 cstr_24 cstr_7 + cstr_23 A B RA RB) + (respectful s0 s14 s2 s15 s11 s12 cstr_2 cstr_3 cstr_25 cstr_10 + cstr_27 A C RA RC)) + (Coq__Program__Basics.compose + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_4 A) + (cupicef.cast (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_8 B) + (cupicef.cast (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_11 C)) + (Coq__Program__Basics.compose + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_4 A) + (cupicef.cast (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_8 B) + (cupicef.cast (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_11 C))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + s7 : cupicef.Nat => + s8 : cupicef.Nat => + s9 : cupicef.Nat => + s10 : cupicef.Nat => + s11 : cupicef.Nat => + s12 : cupicef.Nat => + s13 : cupicef.Nat => + s14 : cupicef.Nat => + s15 : cupicef.Nat => + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s9)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s10)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s11)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s12)) => + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) (cupicef.type (cupicef.s cupicef.z))) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s5)) => + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s6)) => + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s9)) => + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) => + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s5)) => + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s11)) => + cstr_11 : + cupicef.eps + (cupicef.Cumul (cupicef.type s2) (cupicef.type (cupicef.s cupicef.z))) => + cstr_12 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s3)) => + cstr_13 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s4)) => + cstr_14 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s4)) => + cstr_15 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s3)) => + cstr_16 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s4)) => + cstr_17 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s7)) => + cstr_18 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s8)) => + cstr_19 : cupicef.eps (cupicef.Cumul (cupicef.type s10) (cupicef.type s8)) => + cstr_20 : cupicef.eps (cupicef.Cumul (cupicef.type s11) (cupicef.type s7)) => + cstr_21 : cupicef.eps (cupicef.Cumul (cupicef.type s12) (cupicef.type s8)) => + cstr_22 : cupicef.eps (cupicef.Cumul (cupicef.type s13) (cupicef.type s6)) => + cstr_23 : + cupicef.eps (cupicef.Cumul (cupicef.type s13) (cupicef.type s10)) => + cstr_24 : + cupicef.eps (cupicef.Cumul (cupicef.type s14) (cupicef.type s10)) => + cstr_25 : + cupicef.eps (cupicef.Cumul (cupicef.type s14) (cupicef.type s12)) => + cstr_26 : cupicef.eps (cupicef.Cumul (cupicef.type s15) (cupicef.type s6)) => + cstr_27 : + cupicef.eps (cupicef.Cumul (cupicef.type s15) (cupicef.type s12)) => + A : cupicef.Univ (cupicef.type s0) => + B : cupicef.Univ (cupicef.type s1) => + C : cupicef.Univ (cupicef.type s2) => + RA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s14))) + (Coq__Classes__CRelationClasses.crelation s0 s14 A) => + RB : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s13))) + (Coq__Classes__CRelationClasses.crelation s1 s13 B) => + RC : + cupicef.Term + (cupicef.sup (cupicef.type s2) (cupicef.axiom (cupicef.type s15))) + (Coq__Classes__CRelationClasses.crelation s2 s15 C) => + x : + (__ : cupicef.Term (cupicef.type s1) B -> + cupicef.Term (cupicef.type s2) C) => + y : + (__ : cupicef.Term (cupicef.type s1) B -> + cupicef.Term (cupicef.type s2) C) => + X : + cupicef.Term (cupicef.type s6) + (respectful s1 s13 s2 s15 s5 s6 cstr_5 cstr_6 cstr_22 cstr_9 cstr_26 B + C RB RC x y) => + x0 : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) B) => + y0 : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) B) => + X0 : + cupicef.Term (cupicef.type s10) + (respectful s0 s14 s1 s13 s9 s10 cstr_0 cstr_1 cstr_24 cstr_7 cstr_23 + A B RA RB x0 y0) => + x1 : cupicef.Term (cupicef.type s0) A => + y1 : cupicef.Term (cupicef.type s0) A => + X1 : cupicef.Term (cupicef.type s14) (RA x1 y1) => + X (x0 x1) (y0 y1) (X0 x1 y1 X1). + +def compose__proper : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + s7 : cupicef.Nat -> + s8 : cupicef.Nat -> + s9 : cupicef.Nat -> + s10 : cupicef.Nat -> + s11 : cupicef.Nat -> + s12 : cupicef.Nat -> + s13 : cupicef.Nat -> + s14 : cupicef.Nat -> + s15 : cupicef.Nat -> + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s9)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s10)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s11)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s12)) -> + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s5)) -> + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s6)) -> + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s9)) -> + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s5)) -> + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s11)) -> + cstr_11 : + cupicef.eps + (cupicef.Cumul (cupicef.type s2) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_12 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s3)) -> + cstr_13 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s4)) -> + cstr_14 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s4)) -> + cstr_15 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s3)) -> + cstr_16 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s4)) -> + cstr_17 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s7)) -> + cstr_18 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s8)) -> + cstr_19 : cupicef.eps (cupicef.Cumul (cupicef.type s10) (cupicef.type s8)) -> + cstr_20 : cupicef.eps (cupicef.Cumul (cupicef.type s11) (cupicef.type s7)) -> + cstr_21 : cupicef.eps (cupicef.Cumul (cupicef.type s12) (cupicef.type s8)) -> + cstr_22 : cupicef.eps (cupicef.Cumul (cupicef.type s13) (cupicef.type s6)) -> + cstr_23 : cupicef.eps (cupicef.Cumul (cupicef.type s13) (cupicef.type s10)) -> + cstr_24 : cupicef.eps (cupicef.Cumul (cupicef.type s14) (cupicef.type s10)) -> + cstr_25 : cupicef.eps (cupicef.Cumul (cupicef.type s14) (cupicef.type s12)) -> + cstr_26 : cupicef.eps (cupicef.Cumul (cupicef.type s15) (cupicef.type s6)) -> + cstr_27 : cupicef.eps (cupicef.Cumul (cupicef.type s15) (cupicef.type s12)) -> + A : cupicef.Univ (cupicef.type s0) -> + B : cupicef.Univ (cupicef.type s1) -> + C : cupicef.Univ (cupicef.type s2) -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s14))) + (Coq__Classes__CRelationClasses.crelation s0 s14 A) -> + RB : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s13))) + (Coq__Classes__CRelationClasses.crelation s1 s13 B) -> + RC : + cupicef.Term + (cupicef.sup (cupicef.type s2) (cupicef.axiom (cupicef.type s15))) + (Coq__Classes__CRelationClasses.crelation s2 s15 C) -> + cupicef.Term (cupicef.type s4) + (Proper s3 s4 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))) + cupicef.I) + (cupicef.univ (cupicef.type s3) (cupicef.axiom (cupicef.type s3)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s3))) + (cupicef.trans (cupicef.type s2) + (cupicef.SortSome (cupicef.type s5)) cstr_9 + (cupicef.SortSome (cupicef.type s3)) cstr_12 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s3))) + (cupicef.trans (cupicef.type s1) + (cupicef.SortSome (cupicef.type s5)) cstr_5 + (cupicef.SortSome (cupicef.type s3)) cstr_12 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s3))) + (cupicef.trans (cupicef.type s0) + (cupicef.SortSome (cupicef.type s9)) cstr_0 + (cupicef.SortSome (cupicef.type s7)) cstr_17 + (cupicef.SortSome (cupicef.type s3)) cstr_15 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.rule (cupicef.type s1) (cupicef.type s2)) + (cupicef.rule (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule (cupicef.type s0) (cupicef.type s2))) + (cupicef.rule (cupicef.rule (cupicef.type s1) (cupicef.type s2)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule (cupicef.type s0) (cupicef.type s2)))) + cupicef.I + (cupicef.prod (cupicef.type s1) (cupicef.type s2) + (cupicef.rule (cupicef.type s1) (cupicef.type s2)) cupicef.I + B (__ : cupicef.Term (cupicef.type s1) B => C)) + (__ : + (__ : cupicef.Term (cupicef.type s1) B -> + cupicef.Term (cupicef.type s2) C) => + cupicef.prod (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule (cupicef.type s0) (cupicef.type s2))) + cupicef.I + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + cupicef.I A (__0 : cupicef.Term (cupicef.type s0) A => B)) + (__0 : + (__0 : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) B) => + cupicef.prod (cupicef.type s0) (cupicef.type s2) + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + cupicef.I A (__1 : cupicef.Term (cupicef.type s0) A => C))))) + (respectful s5 s6 s7 s8 s3 s4 cstr_12 cstr_13 cstr_14 cstr_15 cstr_16 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ (cupicef.sup (cupicef.type s1) (cupicef.type s2)) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s5))) + cstr_9 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s5))) + cstr_5 cupicef.BoolNone) + (cupicef.prod (cupicef.type s1) (cupicef.type s2) + (cupicef.rule (cupicef.type s1) (cupicef.type s2)) cupicef.I + B (__ : cupicef.Term (cupicef.type s1) B => C))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))) + (cupicef.axiom (cupicef.type s7)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))) + cupicef.I) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s7))) + (cupicef.trans (cupicef.type s2) + (cupicef.SortSome (cupicef.type s11)) cstr_10 + (cupicef.SortSome (cupicef.type s7)) cstr_20 + cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s7))) + (cupicef.trans (cupicef.type s1) + (cupicef.SortSome (cupicef.type s9)) cstr_7 + (cupicef.SortSome (cupicef.type s7)) cstr_17 + cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s7))) + (cupicef.trans (cupicef.type s0) + (cupicef.SortSome (cupicef.type s9)) cstr_0 + (cupicef.SortSome (cupicef.type s7)) cstr_17 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + (cupicef.rule + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule (cupicef.type s0) (cupicef.type s2))) + cupicef.I + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + cupicef.I A (__ : cupicef.Term (cupicef.type s0) A => B)) + (__ : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) B) => + cupicef.prod (cupicef.type s0) (cupicef.type s2) + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + cupicef.I A (__0 : cupicef.Term (cupicef.type s0) A => C)))) + (respectful s1 s13 s2 s15 s5 s6 cstr_5 cstr_6 cstr_22 cstr_9 + cstr_26 B C RB RC) + (respectful s9 s10 s11 s12 s7 s8 cstr_17 cstr_18 cstr_19 cstr_20 + cstr_21 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + (cupicef.axiom (cupicef.type s9)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s9) + (cupicef.axiom (cupicef.type s9)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s9))) + cstr_7 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s9))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + cupicef.I A (__ : cupicef.Term (cupicef.type s0) A => B))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2))) + (cupicef.axiom (cupicef.type s11)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s11) + (cupicef.axiom (cupicef.type s11)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s11))) + cstr_10 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s11))) + cstr_2 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s2) + (cupicef.rule (cupicef.type s0) (cupicef.type s2)) + cupicef.I A (__ : cupicef.Term (cupicef.type s0) A => C))) + (respectful s0 s14 s1 s13 s9 s10 cstr_0 cstr_1 cstr_24 cstr_7 + cstr_23 A B RA RB) + (respectful s0 s14 s2 s15 s11 s12 cstr_2 cstr_3 cstr_25 cstr_10 + cstr_27 A C RA RC))) + (Coq__Program__Basics.compose + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_4 A) + (cupicef.cast (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_8 B) + (cupicef.cast (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_11 C))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + s7 : cupicef.Nat => + s8 : cupicef.Nat => + s9 : cupicef.Nat => + s10 : cupicef.Nat => + s11 : cupicef.Nat => + s12 : cupicef.Nat => + s13 : cupicef.Nat => + s14 : cupicef.Nat => + s15 : cupicef.Nat => + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s9)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s10)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s11)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s12)) => + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) (cupicef.type (cupicef.s cupicef.z))) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s5)) => + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s6)) => + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s9)) => + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) => + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s5)) => + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s11)) => + cstr_11 : + cupicef.eps + (cupicef.Cumul (cupicef.type s2) (cupicef.type (cupicef.s cupicef.z))) => + cstr_12 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s3)) => + cstr_13 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s4)) => + cstr_14 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s4)) => + cstr_15 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s3)) => + cstr_16 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s4)) => + cstr_17 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s7)) => + cstr_18 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s8)) => + cstr_19 : cupicef.eps (cupicef.Cumul (cupicef.type s10) (cupicef.type s8)) => + cstr_20 : cupicef.eps (cupicef.Cumul (cupicef.type s11) (cupicef.type s7)) => + cstr_21 : cupicef.eps (cupicef.Cumul (cupicef.type s12) (cupicef.type s8)) => + cstr_22 : cupicef.eps (cupicef.Cumul (cupicef.type s13) (cupicef.type s6)) => + cstr_23 : + cupicef.eps (cupicef.Cumul (cupicef.type s13) (cupicef.type s10)) => + cstr_24 : + cupicef.eps (cupicef.Cumul (cupicef.type s14) (cupicef.type s10)) => + cstr_25 : + cupicef.eps (cupicef.Cumul (cupicef.type s14) (cupicef.type s12)) => + cstr_26 : cupicef.eps (cupicef.Cumul (cupicef.type s15) (cupicef.type s6)) => + cstr_27 : + cupicef.eps (cupicef.Cumul (cupicef.type s15) (cupicef.type s12)) => + A : cupicef.Univ (cupicef.type s0) => + B : cupicef.Univ (cupicef.type s1) => + C : cupicef.Univ (cupicef.type s2) => + RA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s14))) + (Coq__Classes__CRelationClasses.crelation s0 s14 A) => + RB : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s13))) + (Coq__Classes__CRelationClasses.crelation s1 s13 B) => + RC : + cupicef.Term + (cupicef.sup (cupicef.type s2) (cupicef.axiom (cupicef.type s15))) + (Coq__Classes__CRelationClasses.crelation s2 s15 C) => + compose__proper__obligation__1 s0 s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 s12 + s13 s14 s15 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 cstr_6 cstr_7 + cstr_8 cstr_9 cstr_10 cstr_11 cstr_12 cstr_13 cstr_14 cstr_15 cstr_16 + cstr_17 cstr_18 cstr_19 cstr_20 cstr_21 cstr_22 cstr_23 cstr_24 cstr_25 + cstr_26 cstr_27 A B C RA RB RC. + +#TYPE_CLASS_INSTANCE compose__proper. + +thm reflexive__eq__dom__reflexive : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) -> + cstr_3 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s1)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s2)) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s5) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) -> + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.type s5) (cupicef.type (cupicef.s cupicef.z))) -> + B : cupicef.Univ (cupicef.type s0) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 B) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Reflexive s0 s4 B R_27_) -> + A : cupicef.Univ (cupicef.type s5) -> + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s2)) + (Coq__Classes__CRelationClasses.Reflexive s1 s2 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s5)) + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s1))) + cstr_4 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s5) (cupicef.type s0) + (cupicef.rule (cupicef.type s5) (cupicef.type s0)) cupicef.I A + (__ : cupicef.Term (cupicef.type s5) A => B))) + (respectful s5 s3 s0 s4 s1 s2 cstr_4 cstr_5 cstr_1 cstr_0 cstr_2 A B + (cupicef.cast + (cupicef.rule (cupicef.type s5) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.axiom (cupicef.type s3)) (cupicef.type s5)) + (cupicef.prod (cupicef.type s5) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s5) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s5) A => + cupicef.prod (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s5) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s5 s3 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_6 A))) + R_27_)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) => + cstr_3 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s1)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s2)) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s5) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) => + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.type s5) (cupicef.type (cupicef.s cupicef.z))) => + B : cupicef.Univ (cupicef.type s0) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 B) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Reflexive s0 s4 B R_27_) => + cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type s5)) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (cupicef.rule (cupicef.axiom (cupicef.type s5)) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (cupicef.prod (cupicef.axiom (cupicef.type s5)) + (cupicef.sup (cupicef.type s1) (cupicef.type s2)) + (cupicef.rule (cupicef.axiom (cupicef.type s5)) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + cupicef.I + (cupicef.univ (cupicef.type s5) (cupicef.axiom (cupicef.type s5)) + cupicef.I) + (A : cupicef.Univ (cupicef.type s5) => + Coq__Classes__CRelationClasses.Reflexive s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s1))) + cstr_4 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s5) (cupicef.type s0) + (cupicef.rule (cupicef.type s5) (cupicef.type s0)) cupicef.I + A (__ : cupicef.Term (cupicef.type s5) A => B))) + (respectful s5 s3 s0 s4 s1 s2 cstr_4 cstr_5 cstr_1 cstr_0 cstr_2 A + B + (cupicef.cast + (cupicef.rule (cupicef.type s5) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.axiom (cupicef.type s3)) + (cupicef.type s5)) + (cupicef.prod (cupicef.type s5) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s5) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s5) A => + cupicef.prod (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s5) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s5 s3 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_6 A))) + R_27_))) + (cupicef.prod (cupicef.axiom (cupicef.type s5)) + (cupicef.sup (cupicef.type s1) (cupicef.type s2)) + (cupicef.rule (cupicef.axiom (cupicef.type s5)) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + cupicef.I + (cupicef.univ (cupicef.type s5) (cupicef.axiom (cupicef.type s5)) + cupicef.I) + (A : cupicef.Univ (cupicef.type s5) => + Coq__Classes__CRelationClasses.Reflexive s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s1))) + cstr_4 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s5) (cupicef.type s0) + (cupicef.rule (cupicef.type s5) (cupicef.type s0)) cupicef.I + A (__ : cupicef.Term (cupicef.type s5) A => B))) + (respectful s5 s3 s0 s4 s1 s2 cstr_4 cstr_5 cstr_1 cstr_0 cstr_2 A + B + (cupicef.cast + (cupicef.rule (cupicef.type s5) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.axiom (cupicef.type s3)) + (cupicef.type s5)) + (cupicef.prod (cupicef.type s5) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s5) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s5) A => + cupicef.prod (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s5) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s5 s3 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_6 A))) + R_27_))) + cupicef.I + (cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type s5)) + (cupicef.rule (cupicef.rule (cupicef.type s5) (cupicef.type s0)) + (cupicef.type s2))) + (cupicef.rule (cupicef.axiom (cupicef.type s5)) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (cupicef.prod (cupicef.axiom (cupicef.type s5)) + (cupicef.rule (cupicef.rule (cupicef.type s5) (cupicef.type s0)) + (cupicef.type s2)) + (cupicef.rule (cupicef.axiom (cupicef.type s5)) + (cupicef.rule + (cupicef.rule (cupicef.type s5) (cupicef.type s0)) + (cupicef.type s2))) + cupicef.I + (cupicef.univ (cupicef.type s5) (cupicef.axiom (cupicef.type s5)) + cupicef.I) + (A : cupicef.Univ (cupicef.type s5) => + cupicef.prod (cupicef.rule (cupicef.type s5) (cupicef.type s0)) + (cupicef.type s2) + (cupicef.rule + (cupicef.rule (cupicef.type s5) (cupicef.type s0)) + (cupicef.type s2)) + cupicef.I + (cupicef.prod (cupicef.type s5) (cupicef.type s0) + (cupicef.rule (cupicef.type s5) (cupicef.type s0)) cupicef.I + A (__ : cupicef.Term (cupicef.type s5) A => B)) + (x : + (__ : cupicef.Term (cupicef.type s5) A -> + cupicef.Term (cupicef.type s0) B) => + respectful s5 s3 s0 s4 s1 s2 cstr_4 cstr_5 cstr_1 cstr_0 + cstr_2 A B + (cupicef.cast + (cupicef.rule (cupicef.type s5) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.axiom (cupicef.type s3)) + (cupicef.type s5)) + (cupicef.prod (cupicef.type s5) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s5) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s5) A => + cupicef.prod (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s5) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s5 s3 A) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_6 A))) + R_27_ x x))) + (cupicef.prod (cupicef.axiom (cupicef.type s5)) + (cupicef.sup (cupicef.type s1) (cupicef.type s2)) + (cupicef.rule (cupicef.axiom (cupicef.type s5)) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + cupicef.I + (cupicef.univ (cupicef.type s5) (cupicef.axiom (cupicef.type s5)) + cupicef.I) + (A : cupicef.Univ (cupicef.type s5) => + Coq__Classes__CRelationClasses.Reflexive s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s1))) + cstr_4 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s5) (cupicef.type s0) + (cupicef.rule (cupicef.type s5) (cupicef.type s0)) + cupicef.I A (__ : cupicef.Term (cupicef.type s5) A => B))) + (respectful s5 s3 s0 s4 s1 s2 cstr_4 cstr_5 cstr_1 cstr_0 + cstr_2 A B + (cupicef.cast + (cupicef.rule (cupicef.type s5) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.axiom (cupicef.type s3)) + (cupicef.type s5)) + (cupicef.prod (cupicef.type s5) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s5) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s5) A => + cupicef.prod (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s5) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s5 s3 A) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_6 A))) + R_27_))) + cupicef.I + (A : cupicef.Univ (cupicef.type s5) => + x : + (__ : cupicef.Term (cupicef.type s5) A -> + cupicef.Term (cupicef.type s0) B) => + cupicef.cast + (cupicef.rule (cupicef.type s5) + (cupicef.rule (cupicef.type s5) + (cupicef.rule cupicef.prop (cupicef.type s4)))) + (cupicef.type s2) + (cupicef.prod (cupicef.type s5) + (cupicef.rule (cupicef.type s5) + (cupicef.rule cupicef.prop (cupicef.type s4))) + (cupicef.rule (cupicef.type s5) + (cupicef.rule (cupicef.type s5) + (cupicef.rule cupicef.prop (cupicef.type s4)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s5) A => + cupicef.prod (cupicef.type s5) + (cupicef.rule cupicef.prop (cupicef.type s4)) + (cupicef.rule (cupicef.type s5) + (cupicef.rule cupicef.prop (cupicef.type s4))) + cupicef.I A + (y : cupicef.Term (cupicef.type s5) A => + cupicef.prod cupicef.prop (cupicef.type s4) + (cupicef.rule cupicef.prop (cupicef.type s4)) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_6 A) + x0 y) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_6 A) + x0 y) => + R_27_ (x x0) (x y))))) + (respectful s5 s3 s0 s4 s1 s2 cstr_4 cstr_5 cstr_1 cstr_0 cstr_2 A + B + (cupicef.cast + (cupicef.rule (cupicef.type s5) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.axiom (cupicef.type s3)) + (cupicef.type s5)) + (cupicef.prod (cupicef.type s5) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s5) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s5) A => + cupicef.prod (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s5) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s5 s3 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_6 A))) + R_27_ x x) + cupicef.I + (x0 : cupicef.Term (cupicef.type s5) A => + y : cupicef.Term (cupicef.type s5) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_6 A) + x0 y) => + cupicef.cast (cupicef.type s4) (cupicef.type s4) + (R_27_ (x x0) (x y)) (R_27_ (x x0) (x y)) cupicef.I + (cupicef.cast (cupicef.type s4) (cupicef.type s4) + ((x1 : cupicef.Term (cupicef.type s5) A => + R_27_ (x x1) (x y)) x0) + (R_27_ (x x0) (x y)) cupicef.I + (Coq__Init__Logic.eq__rect__r + (cupicef.cast (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_7 A) + y + (cupicef.cast + (cupicef.rule (cupicef.type s5) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule (cupicef.type s5) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type s5) + (cupicef.axiom (cupicef.type s4)) + (cupicef.rule (cupicef.type s5) + (cupicef.axiom (cupicef.type s4))) + cupicef.I A + (x1 : cupicef.Term (cupicef.type s5) A => + cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + (cupicef.prod (cupicef.type s5) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s5) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s5) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cstr_3 + (x1 : cupicef.Term (cupicef.type s5) A => + R_27_ (x x1) (x y))) + (cupicef.cast (cupicef.type s4) (cupicef.type s4) + (R_27_ (x y) (x y)) (R_27_ (x y) (x y)) cupicef.I + (H (x y))) + x0 H0))))). + +#TYPE_CLASS_INSTANCE reflexive__eq__dom__reflexive. + +thm respectful__morphism : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + s7 : cupicef.Nat -> + s8 : cupicef.Nat -> + s9 : cupicef.Nat -> + s10 : cupicef.Nat -> + s11 : cupicef.Nat -> + s12 : cupicef.Nat -> + s13 : cupicef.Nat -> + s14 : cupicef.Nat -> + s15 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) (cupicef.type s2)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s13)) (cupicef.type s8)) -> + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s15)) (cupicef.type s10)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s0)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s1)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) -> + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s0)) -> + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s1)) -> + cstr_8 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s2)) -> + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s3)) -> + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s14)) -> + cstr_11 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s15)) -> + cstr_12 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s3)) -> + cstr_13 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s15)) -> + cstr_14 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_15 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_16 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_17 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_18 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s4)) -> + cstr_19 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s5)) -> + cstr_20 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s5)) -> + cstr_21 : cupicef.eps (cupicef.Cumul (cupicef.type s10) (cupicef.type s4)) -> + cstr_22 : cupicef.eps (cupicef.Cumul (cupicef.type s11) (cupicef.type s5)) -> + cstr_23 : cupicef.eps (cupicef.Cumul (cupicef.type s12) (cupicef.type s8)) -> + cstr_24 : cupicef.eps (cupicef.Cumul (cupicef.type s12) (cupicef.type s9)) -> + cstr_25 : cupicef.eps (cupicef.Cumul (cupicef.type s12) (cupicef.type s14)) -> + cstr_26 : cupicef.eps (cupicef.Cumul (cupicef.type s13) (cupicef.type s9)) -> + cstr_27 : cupicef.eps (cupicef.Cumul (cupicef.type s13) (cupicef.type s15)) -> + cstr_28 : + cupicef.eps + (cupicef.Cumul (cupicef.type s13) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_29 : + cupicef.eps + (cupicef.Cumul (cupicef.type s13) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_30 : + cupicef.eps + (cupicef.Cumul (cupicef.type s13) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_31 : + cupicef.eps + (cupicef.Cumul (cupicef.type s13) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_32 : cupicef.eps (cupicef.Cumul (cupicef.type s14) (cupicef.type s10)) -> + cstr_33 : cupicef.eps (cupicef.Cumul (cupicef.type s14) (cupicef.type s11)) -> + cstr_34 : cupicef.eps (cupicef.Cumul (cupicef.type s15) (cupicef.type s11)) -> + cstr_35 : + cupicef.eps + (cupicef.Cumul (cupicef.type s15) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_36 : + cupicef.eps + (cupicef.Cumul (cupicef.type s15) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s6) -> + B : cupicef.Univ (cupicef.type s12) -> + cupicef.Term (cupicef.type s1) + (Proper s0 s1 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.axiom (cupicef.type s7)) + (cupicef.sup (cupicef.type s12) + (cupicef.sup (cupicef.axiom (cupicef.type s13)) + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15)))))))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.axiom (cupicef.type s7)) + (cupicef.sup (cupicef.type s12) + (cupicef.sup (cupicef.axiom (cupicef.type s13)) + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15))))))) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.axiom (cupicef.type s7)) + (cupicef.sup (cupicef.type s12) + (cupicef.sup (cupicef.axiom (cupicef.type s13)) + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15)))))))) + cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s14) (cupicef.type s0))) + (cupicef.trans (cupicef.type s14) + (cupicef.SortSome (cupicef.type s10)) cstr_32 + (cupicef.SortSome (cupicef.type s4)) cstr_21 + (cupicef.SortSome (cupicef.type s0)) cstr_6 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s12) (cupicef.type s0))) + (cupicef.trans (cupicef.type s12) + (cupicef.SortSome (cupicef.type s8)) cstr_23 + (cupicef.SortSome (cupicef.type s4)) cstr_18 + (cupicef.SortSome (cupicef.type s0)) cstr_6 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) (cupicef.type s0))) + (cupicef.trans (cupicef.type s6) + (cupicef.SortSome (cupicef.type s2)) cstr_8 + (cupicef.SortSome (cupicef.type s0)) cstr_3 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s15)) + (cupicef.type s0))) + (cupicef.trans (cupicef.axiom (cupicef.type s15)) + (cupicef.SortSome (cupicef.type s10)) cstr_2 + (cupicef.SortSome (cupicef.type s4)) cstr_21 + (cupicef.SortSome (cupicef.type s0)) cstr_6 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s13)) + (cupicef.type s0))) + (cupicef.trans (cupicef.axiom (cupicef.type s13)) + (cupicef.SortSome (cupicef.type s8)) cstr_1 + (cupicef.SortSome (cupicef.type s4)) cstr_18 + (cupicef.SortSome (cupicef.type s0)) cstr_6 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s0))) + (cupicef.trans (cupicef.axiom (cupicef.type s7)) + (cupicef.SortSome (cupicef.type s2)) cstr_0 + (cupicef.SortSome (cupicef.type s0)) cstr_3 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15)))) + (cupicef.rule + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15))))) + cupicef.I (Coq__Classes__CRelationClasses.crelation s6 s7 A) + (__ : + cupicef.Term + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s6 s7 A) => + cupicef.prod + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15))) + (cupicef.rule + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15)))) + cupicef.I + (Coq__Classes__CRelationClasses.crelation s12 s13 B) + (__0 : + cupicef.Term + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (Coq__Classes__CRelationClasses.crelation s12 s13 B) => + Coq__Classes__CRelationClasses.crelation s14 s15 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) (cupicef.type s12))) + (cupicef.axiom (cupicef.type s14)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) (cupicef.type s12)) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) (cupicef.type s12))) + cupicef.I) + (cupicef.univ (cupicef.type s14) + (cupicef.axiom (cupicef.type s14)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s12) + (cupicef.type s14))) + cstr_25 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) + (cupicef.type s14))) + cstr_10 cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) (cupicef.type s12) + (cupicef.rule (cupicef.type s6) (cupicef.type s12)) + cupicef.I A + (__1 : cupicef.Term (cupicef.type s6) A => B))))))) + (respectful s2 s3 s4 s5 s0 s1 cstr_3 cstr_4 cstr_5 cstr_6 cstr_7 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) (cupicef.type s2))) + cstr_8 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s2))) + cstr_0 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s6 s7 A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s12) + (cupicef.sup (cupicef.axiom (cupicef.type s13)) + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15)))))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ + (cupicef.sup (cupicef.type s12) + (cupicef.sup (cupicef.axiom (cupicef.type s13)) + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15))))) + (cupicef.axiom + (cupicef.sup (cupicef.type s12) + (cupicef.sup (cupicef.axiom (cupicef.type s13)) + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15)))))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s14) (cupicef.type s4))) + (cupicef.trans (cupicef.type s14) + (cupicef.SortSome (cupicef.type s10)) cstr_32 + (cupicef.SortSome (cupicef.type s4)) cstr_21 + cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s12) (cupicef.type s4))) + (cupicef.trans (cupicef.type s12) + (cupicef.SortSome (cupicef.type s8)) cstr_23 + (cupicef.SortSome (cupicef.type s4)) cstr_18 + cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s15)) + (cupicef.type s4))) + (cupicef.trans (cupicef.axiom (cupicef.type s15)) + (cupicef.SortSome (cupicef.type s10)) cstr_2 + (cupicef.SortSome (cupicef.type s4)) cstr_21 + cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s13)) + (cupicef.type s4))) + (cupicef.trans (cupicef.axiom (cupicef.type s13)) + (cupicef.SortSome (cupicef.type s8)) cstr_1 + (cupicef.SortSome (cupicef.type s4)) cstr_18 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15))) + (cupicef.rule + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15)))) + cupicef.I + (Coq__Classes__CRelationClasses.crelation s12 s13 B) + (__ : + cupicef.Term + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (Coq__Classes__CRelationClasses.crelation s12 s13 B) => + Coq__Classes__CRelationClasses.crelation s14 s15 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) (cupicef.type s12))) + (cupicef.axiom (cupicef.type s14)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) (cupicef.type s12)) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) (cupicef.type s12))) + cupicef.I) + (cupicef.univ (cupicef.type s14) + (cupicef.axiom (cupicef.type s14)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s12) + (cupicef.type s14))) + cstr_25 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) + (cupicef.type s14))) + cstr_10 cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) (cupicef.type s12) + (cupicef.rule (cupicef.type s6) (cupicef.type s12)) + cupicef.I A + (__0 : cupicef.Term (cupicef.type s6) A => B)))))) + (Coq__Classes__CRelationClasses.relation__equivalence s6 s2 s3 s7 + cstr_0 cstr_8 cstr_9 cstr_12 cstr_14 cstr_15 A) + (respectful s8 s9 s10 s11 s4 s5 cstr_18 cstr_19 cstr_20 cstr_21 + cstr_22 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13)))) + (cupicef.axiom (cupicef.type s8)) + (cupicef.univ + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (cupicef.axiom + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13)))) + cupicef.I) + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s12) (cupicef.type s8))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s13)) + (cupicef.type s8))) + cstr_1 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s12 s13 B)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15)))) + (cupicef.axiom (cupicef.type s10)) + (cupicef.univ + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15))) + (cupicef.axiom + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15)))) + cupicef.I) + (cupicef.univ (cupicef.type s10) + (cupicef.axiom (cupicef.type s10)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s14) (cupicef.type s10))) + cstr_32 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s15)) + (cupicef.type s10))) + cstr_2 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s14 s15 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) (cupicef.type s12))) + (cupicef.axiom (cupicef.type s14)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) (cupicef.type s12)) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) (cupicef.type s12))) + cupicef.I) + (cupicef.univ (cupicef.type s14) + (cupicef.axiom (cupicef.type s14)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s12) + (cupicef.type s14))) + cstr_25 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) + (cupicef.type s14))) + cstr_10 cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) (cupicef.type s12) + (cupicef.rule (cupicef.type s6) (cupicef.type s12)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => B))))) + (Coq__Classes__CRelationClasses.relation__equivalence s12 s8 s9 + s13 cstr_1 cstr_23 cstr_24 cstr_26 cstr_28 cstr_29 B) + (Coq__Classes__CRelationClasses.relation__equivalence s14 s10 + s11 s15 cstr_2 cstr_32 cstr_33 cstr_34 cstr_35 cstr_36 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) (cupicef.type s12))) + (cupicef.axiom (cupicef.type s14)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) (cupicef.type s12)) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) (cupicef.type s12))) + cupicef.I) + (cupicef.univ (cupicef.type s14) + (cupicef.axiom (cupicef.type s14)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s12) (cupicef.type s14))) + cstr_25 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) (cupicef.type s14))) + cstr_10 cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) (cupicef.type s12) + (cupicef.rule (cupicef.type s6) (cupicef.type s12)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => B)))))) + (respectful s6 s7 s12 s13 s14 s15 cstr_10 cstr_11 cstr_13 cstr_25 + cstr_27 A B)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + s7 : cupicef.Nat => + s8 : cupicef.Nat => + s9 : cupicef.Nat => + s10 : cupicef.Nat => + s11 : cupicef.Nat => + s12 : cupicef.Nat => + s13 : cupicef.Nat => + s14 : cupicef.Nat => + s15 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) (cupicef.type s2)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s13)) (cupicef.type s8)) => + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s15)) (cupicef.type s10)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s0)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s1)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) => + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s0)) => + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s1)) => + cstr_8 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s2)) => + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s3)) => + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s14)) => + cstr_11 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s15)) => + cstr_12 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s3)) => + cstr_13 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s15)) => + cstr_14 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) => + cstr_15 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) => + cstr_16 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) => + cstr_17 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) => + cstr_18 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s4)) => + cstr_19 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s5)) => + cstr_20 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s5)) => + cstr_21 : cupicef.eps (cupicef.Cumul (cupicef.type s10) (cupicef.type s4)) => + cstr_22 : cupicef.eps (cupicef.Cumul (cupicef.type s11) (cupicef.type s5)) => + cstr_23 : cupicef.eps (cupicef.Cumul (cupicef.type s12) (cupicef.type s8)) => + cstr_24 : cupicef.eps (cupicef.Cumul (cupicef.type s12) (cupicef.type s9)) => + cstr_25 : + cupicef.eps (cupicef.Cumul (cupicef.type s12) (cupicef.type s14)) => + cstr_26 : cupicef.eps (cupicef.Cumul (cupicef.type s13) (cupicef.type s9)) => + cstr_27 : + cupicef.eps (cupicef.Cumul (cupicef.type s13) (cupicef.type s15)) => + cstr_28 : + cupicef.eps + (cupicef.Cumul (cupicef.type s13) (cupicef.type (cupicef.s cupicef.z))) => + cstr_29 : + cupicef.eps + (cupicef.Cumul (cupicef.type s13) (cupicef.type (cupicef.s cupicef.z))) => + cstr_30 : + cupicef.eps + (cupicef.Cumul (cupicef.type s13) (cupicef.type (cupicef.s cupicef.z))) => + cstr_31 : + cupicef.eps + (cupicef.Cumul (cupicef.type s13) (cupicef.type (cupicef.s cupicef.z))) => + cstr_32 : + cupicef.eps (cupicef.Cumul (cupicef.type s14) (cupicef.type s10)) => + cstr_33 : + cupicef.eps (cupicef.Cumul (cupicef.type s14) (cupicef.type s11)) => + cstr_34 : + cupicef.eps (cupicef.Cumul (cupicef.type s15) (cupicef.type s11)) => + cstr_35 : + cupicef.eps + (cupicef.Cumul (cupicef.type s15) (cupicef.type (cupicef.s cupicef.z))) => + cstr_36 : + cupicef.eps + (cupicef.Cumul (cupicef.type s15) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s6) => + B : cupicef.Univ (cupicef.type s12) => + cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)))) + (cupicef.type s1) + (cupicef.prod + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.type s3) (cupicef.type s5))) + (cupicef.rule + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)))) + cupicef.I (Coq__Classes__CRelationClasses.crelation s6 s7 A) + (R : + cupicef.Term + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s6 s7 A) => + cupicef.prod + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + (cupicef.rule + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.type s3) (cupicef.type s5))) + cupicef.I (Coq__Classes__CRelationClasses.crelation s6 s7 A) + (R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s6 s7 A) => + cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) cupicef.I + (Coq__Classes__CRelationClasses.relation__equivalence s6 s2 s3 + s7 cstr_0 cstr_8 cstr_9 cstr_12 cstr_14 cstr_15 A R R_27_) + (HRR_27_ : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.relation__equivalence s6 + s2 s3 s7 cstr_0 cstr_8 cstr_9 cstr_12 cstr_14 cstr_15 A + R R_27_) => + respectful s8 s9 s10 s11 s4 s5 cstr_18 cstr_19 cstr_20 cstr_21 + cstr_22 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13)))) + (cupicef.axiom (cupicef.type s8)) + (cupicef.univ + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (cupicef.axiom + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13)))) + cupicef.I) + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s12) (cupicef.type s8))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s13)) + (cupicef.type s8))) + cstr_1 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s12 s13 B)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15)))) + (cupicef.axiom (cupicef.type s10)) + (cupicef.univ + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15))) + (cupicef.axiom + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15)))) + cupicef.I) + (cupicef.univ (cupicef.type s10) + (cupicef.axiom (cupicef.type s10)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s14) + (cupicef.type s10))) + cstr_32 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s15)) + (cupicef.type s10))) + cstr_2 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s14 s15 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.type s12))) + (cupicef.axiom (cupicef.type s14)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) + (cupicef.type s12)) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.type s12))) + cupicef.I) + (cupicef.univ (cupicef.type s14) + (cupicef.axiom (cupicef.type s14)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s12) + (cupicef.type s14))) + cstr_25 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) + (cupicef.type s14))) + cstr_10 cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) (cupicef.type s12) + (cupicef.rule (cupicef.type s6) + (cupicef.type s12)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => B))))) + (Coq__Classes__CRelationClasses.relation__equivalence s12 s8 + s9 s13 cstr_1 cstr_23 cstr_24 cstr_26 cstr_28 cstr_29 B) + (Coq__Classes__CRelationClasses.relation__equivalence s14 + s10 s11 s15 cstr_2 cstr_32 cstr_33 cstr_34 cstr_35 + cstr_36 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) (cupicef.type s12))) + (cupicef.axiom (cupicef.type s14)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) (cupicef.type s12)) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.type s12))) + cupicef.I) + (cupicef.univ (cupicef.type s14) + (cupicef.axiom (cupicef.type s14)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s12) + (cupicef.type s14))) + cstr_25 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) + (cupicef.type s14))) + cstr_10 cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) (cupicef.type s12) + (cupicef.rule (cupicef.type s6) (cupicef.type s12)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => B)))) + (respectful s6 s7 s12 s13 s14 s15 cstr_10 cstr_11 cstr_13 + cstr_25 cstr_27 A B R) + (respectful s6 s7 s12 s13 s14 s15 cstr_10 cstr_11 cstr_13 + cstr_25 cstr_27 A B R_27_))))) + (Proper s0 s1 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.axiom (cupicef.type s7)) + (cupicef.sup (cupicef.type s12) + (cupicef.sup (cupicef.axiom (cupicef.type s13)) + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15)))))))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.axiom (cupicef.type s7)) + (cupicef.sup (cupicef.type s12) + (cupicef.sup (cupicef.axiom (cupicef.type s13)) + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15))))))) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.axiom (cupicef.type s7)) + (cupicef.sup (cupicef.type s12) + (cupicef.sup (cupicef.axiom (cupicef.type s13)) + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15)))))))) + cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s14) (cupicef.type s0))) + (cupicef.trans (cupicef.type s14) + (cupicef.SortSome (cupicef.type s10)) cstr_32 + (cupicef.SortSome (cupicef.type s4)) cstr_21 + (cupicef.SortSome (cupicef.type s0)) cstr_6 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s12) (cupicef.type s0))) + (cupicef.trans (cupicef.type s12) + (cupicef.SortSome (cupicef.type s8)) cstr_23 + (cupicef.SortSome (cupicef.type s4)) cstr_18 + (cupicef.SortSome (cupicef.type s0)) cstr_6 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) (cupicef.type s0))) + (cupicef.trans (cupicef.type s6) + (cupicef.SortSome (cupicef.type s2)) cstr_8 + (cupicef.SortSome (cupicef.type s0)) cstr_3 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s15)) + (cupicef.type s0))) + (cupicef.trans (cupicef.axiom (cupicef.type s15)) + (cupicef.SortSome (cupicef.type s10)) cstr_2 + (cupicef.SortSome (cupicef.type s4)) cstr_21 + (cupicef.SortSome (cupicef.type s0)) cstr_6 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s13)) + (cupicef.type s0))) + (cupicef.trans (cupicef.axiom (cupicef.type s13)) + (cupicef.SortSome (cupicef.type s8)) cstr_1 + (cupicef.SortSome (cupicef.type s4)) cstr_18 + (cupicef.SortSome (cupicef.type s0)) cstr_6 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s0))) + (cupicef.trans (cupicef.axiom (cupicef.type s7)) + (cupicef.SortSome (cupicef.type s2)) cstr_0 + (cupicef.SortSome (cupicef.type s0)) cstr_3 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15)))) + (cupicef.rule + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15))))) + cupicef.I (Coq__Classes__CRelationClasses.crelation s6 s7 A) + (__ : + cupicef.Term + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s6 s7 A) => + cupicef.prod + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15))) + (cupicef.rule + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15)))) + cupicef.I + (Coq__Classes__CRelationClasses.crelation s12 s13 B) + (__0 : + cupicef.Term + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (Coq__Classes__CRelationClasses.crelation s12 s13 B) => + Coq__Classes__CRelationClasses.crelation s14 s15 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) (cupicef.type s12))) + (cupicef.axiom (cupicef.type s14)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) (cupicef.type s12)) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.type s12))) + cupicef.I) + (cupicef.univ (cupicef.type s14) + (cupicef.axiom (cupicef.type s14)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s12) + (cupicef.type s14))) + cstr_25 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) + (cupicef.type s14))) + cstr_10 cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) (cupicef.type s12) + (cupicef.rule (cupicef.type s6) (cupicef.type s12)) + cupicef.I A + (__1 : cupicef.Term (cupicef.type s6) A => B))))))) + (respectful s2 s3 s4 s5 s0 s1 cstr_3 cstr_4 cstr_5 cstr_6 cstr_7 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) (cupicef.type s2))) + cstr_8 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s2))) + cstr_0 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s6 s7 A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s12) + (cupicef.sup (cupicef.axiom (cupicef.type s13)) + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15)))))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ + (cupicef.sup (cupicef.type s12) + (cupicef.sup (cupicef.axiom (cupicef.type s13)) + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15))))) + (cupicef.axiom + (cupicef.sup (cupicef.type s12) + (cupicef.sup (cupicef.axiom (cupicef.type s13)) + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15)))))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s14) (cupicef.type s4))) + (cupicef.trans (cupicef.type s14) + (cupicef.SortSome (cupicef.type s10)) cstr_32 + (cupicef.SortSome (cupicef.type s4)) cstr_21 + cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s12) (cupicef.type s4))) + (cupicef.trans (cupicef.type s12) + (cupicef.SortSome (cupicef.type s8)) cstr_23 + (cupicef.SortSome (cupicef.type s4)) cstr_18 + cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s15)) + (cupicef.type s4))) + (cupicef.trans (cupicef.axiom (cupicef.type s15)) + (cupicef.SortSome (cupicef.type s10)) cstr_2 + (cupicef.SortSome (cupicef.type s4)) cstr_21 + cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s13)) + (cupicef.type s4))) + (cupicef.trans (cupicef.axiom (cupicef.type s13)) + (cupicef.SortSome (cupicef.type s8)) cstr_1 + (cupicef.SortSome (cupicef.type s4)) cstr_18 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15))) + (cupicef.rule + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15)))) + cupicef.I + (Coq__Classes__CRelationClasses.crelation s12 s13 B) + (__ : + cupicef.Term + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (Coq__Classes__CRelationClasses.crelation s12 s13 B) => + Coq__Classes__CRelationClasses.crelation s14 s15 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) (cupicef.type s12))) + (cupicef.axiom (cupicef.type s14)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) (cupicef.type s12)) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.type s12))) + cupicef.I) + (cupicef.univ (cupicef.type s14) + (cupicef.axiom (cupicef.type s14)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s12) + (cupicef.type s14))) + cstr_25 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) + (cupicef.type s14))) + cstr_10 cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) (cupicef.type s12) + (cupicef.rule (cupicef.type s6) (cupicef.type s12)) + cupicef.I A + (__0 : cupicef.Term (cupicef.type s6) A => B)))))) + (Coq__Classes__CRelationClasses.relation__equivalence s6 s2 s3 s7 + cstr_0 cstr_8 cstr_9 cstr_12 cstr_14 cstr_15 A) + (respectful s8 s9 s10 s11 s4 s5 cstr_18 cstr_19 cstr_20 cstr_21 + cstr_22 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13)))) + (cupicef.axiom (cupicef.type s8)) + (cupicef.univ + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (cupicef.axiom + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13)))) + cupicef.I) + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s12) (cupicef.type s8))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s13)) + (cupicef.type s8))) + cstr_1 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s12 s13 B)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15)))) + (cupicef.axiom (cupicef.type s10)) + (cupicef.univ + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15))) + (cupicef.axiom + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15)))) + cupicef.I) + (cupicef.univ (cupicef.type s10) + (cupicef.axiom (cupicef.type s10)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s14) (cupicef.type s10))) + cstr_32 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s15)) + (cupicef.type s10))) + cstr_2 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s14 s15 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) (cupicef.type s12))) + (cupicef.axiom (cupicef.type s14)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) (cupicef.type s12)) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.type s12))) + cupicef.I) + (cupicef.univ (cupicef.type s14) + (cupicef.axiom (cupicef.type s14)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s12) + (cupicef.type s14))) + cstr_25 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) + (cupicef.type s14))) + cstr_10 cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) (cupicef.type s12) + (cupicef.rule (cupicef.type s6) (cupicef.type s12)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => B))))) + (Coq__Classes__CRelationClasses.relation__equivalence s12 s8 s9 + s13 cstr_1 cstr_23 cstr_24 cstr_26 cstr_28 cstr_29 B) + (Coq__Classes__CRelationClasses.relation__equivalence s14 s10 + s11 s15 cstr_2 cstr_32 cstr_33 cstr_34 cstr_35 cstr_36 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) (cupicef.type s12))) + (cupicef.axiom (cupicef.type s14)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) (cupicef.type s12)) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) (cupicef.type s12))) + cupicef.I) + (cupicef.univ (cupicef.type s14) + (cupicef.axiom (cupicef.type s14)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s12) + (cupicef.type s14))) + cstr_25 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) (cupicef.type s14))) + cstr_10 cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) (cupicef.type s12) + (cupicef.rule (cupicef.type s6) (cupicef.type s12)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => B)))))) + (respectful s6 s7 s12 s13 s14 s15 cstr_10 cstr_11 cstr_13 cstr_25 + cstr_27 A B)) + cupicef.I + (R : + cupicef.Term + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s6 s7 A) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s6 s7 A) => + HRR_27_ : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.relation__equivalence s6 s2 s3 s7 + cstr_0 cstr_8 cstr_9 cstr_12 cstr_14 cstr_15 A R R_27_) => + cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type s12) (cupicef.axiom (cupicef.type s13))) + (cupicef.rule + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (cupicef.rule (cupicef.type s9) (cupicef.type s11)))) + (cupicef.type s5) + (cupicef.prod + (cupicef.sup (cupicef.type s12) (cupicef.axiom (cupicef.type s13))) + (cupicef.rule + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (cupicef.rule (cupicef.type s9) (cupicef.type s11))) + (cupicef.rule + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (cupicef.rule + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (cupicef.rule (cupicef.type s9) (cupicef.type s11)))) + cupicef.I (Coq__Classes__CRelationClasses.crelation s12 s13 B) + (S : + cupicef.Term + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (Coq__Classes__CRelationClasses.crelation s12 s13 B) => + cupicef.prod + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (cupicef.rule (cupicef.type s9) (cupicef.type s11)) + (cupicef.rule + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (cupicef.rule (cupicef.type s9) (cupicef.type s11))) + cupicef.I (Coq__Classes__CRelationClasses.crelation s12 s13 B) + (S_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (Coq__Classes__CRelationClasses.crelation s12 s13 B) => + cupicef.prod (cupicef.type s9) (cupicef.type s11) + (cupicef.rule (cupicef.type s9) (cupicef.type s11)) + cupicef.I + (Coq__Classes__CRelationClasses.relation__equivalence s12 s8 + s9 s13 cstr_1 cstr_23 cstr_24 cstr_26 cstr_28 cstr_29 B S + S_27_) + (HSS_27_ : + cupicef.Term (cupicef.type s9) + (Coq__Classes__CRelationClasses.relation__equivalence + s12 s8 s9 s13 cstr_1 cstr_23 cstr_24 cstr_26 cstr_28 + cstr_29 B S S_27_) => + Coq__Classes__CRelationClasses.relation__equivalence s14 + s10 s11 s15 cstr_2 cstr_32 cstr_33 cstr_34 cstr_35 + cstr_36 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) (cupicef.type s12))) + (cupicef.axiom (cupicef.type s14)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) (cupicef.type s12)) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.type s12))) + cupicef.I) + (cupicef.univ (cupicef.type s14) + (cupicef.axiom (cupicef.type s14)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s12) + (cupicef.type s14))) + cstr_25 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) + (cupicef.type s14))) + cstr_10 cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) (cupicef.type s12) + (cupicef.rule (cupicef.type s6) (cupicef.type s12)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => B))) + (respectful s6 s7 s12 s13 s14 s15 cstr_10 cstr_11 cstr_13 + cstr_25 cstr_27 A B R S) + (respectful s6 s7 s12 s13 s14 s15 cstr_10 cstr_11 cstr_13 + cstr_25 cstr_27 A B R_27_ S_27_))))) + (respectful s8 s9 s10 s11 s4 s5 cstr_18 cstr_19 cstr_20 cstr_21 + cstr_22 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13)))) + (cupicef.axiom (cupicef.type s8)) + (cupicef.univ + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (cupicef.axiom + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13)))) + cupicef.I) + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s12) (cupicef.type s8))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s13)) + (cupicef.type s8))) + cstr_1 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s12 s13 B)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15)))) + (cupicef.axiom (cupicef.type s10)) + (cupicef.univ + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15))) + (cupicef.axiom + (cupicef.sup (cupicef.type s14) + (cupicef.axiom (cupicef.type s15)))) + cupicef.I) + (cupicef.univ (cupicef.type s10) + (cupicef.axiom (cupicef.type s10)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s14) (cupicef.type s10))) + cstr_32 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s15)) + (cupicef.type s10))) + cstr_2 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s14 s15 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) (cupicef.type s12))) + (cupicef.axiom (cupicef.type s14)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) (cupicef.type s12)) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) (cupicef.type s12))) + cupicef.I) + (cupicef.univ (cupicef.type s14) + (cupicef.axiom (cupicef.type s14)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s12) + (cupicef.type s14))) + cstr_25 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) (cupicef.type s14))) + cstr_10 cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) (cupicef.type s12) + (cupicef.rule (cupicef.type s6) (cupicef.type s12)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => B))))) + (Coq__Classes__CRelationClasses.relation__equivalence s12 s8 s9 + s13 cstr_1 cstr_23 cstr_24 cstr_26 cstr_28 cstr_29 B) + (Coq__Classes__CRelationClasses.relation__equivalence s14 s10 s11 + s15 cstr_2 cstr_32 cstr_33 cstr_34 cstr_35 cstr_36 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) (cupicef.type s12))) + (cupicef.axiom (cupicef.type s14)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) (cupicef.type s12)) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) (cupicef.type s12))) + cupicef.I) + (cupicef.univ (cupicef.type s14) + (cupicef.axiom (cupicef.type s14)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s12) (cupicef.type s14))) + cstr_25 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) (cupicef.type s14))) + cstr_10 cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) (cupicef.type s12) + (cupicef.rule (cupicef.type s6) (cupicef.type s12)) + cupicef.I A (__ : cupicef.Term (cupicef.type s6) A => B)))) + (respectful s6 s7 s12 s13 s14 s15 cstr_10 cstr_11 cstr_13 cstr_25 + cstr_27 A B R) + (respectful s6 s7 s12 s13 s14 s15 cstr_10 cstr_11 cstr_13 cstr_25 + cstr_27 A B R_27_)) + cupicef.I + (S : + cupicef.Term + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (Coq__Classes__CRelationClasses.crelation s12 s13 B) => + S_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s12) + (cupicef.axiom (cupicef.type s13))) + (Coq__Classes__CRelationClasses.crelation s12 s13 B) => + HSS_27_ : + cupicef.Term (cupicef.type s9) + (Coq__Classes__CRelationClasses.relation__equivalence s12 s8 s9 + s13 cstr_1 cstr_23 cstr_24 cstr_26 cstr_28 cstr_29 B S S_27_) => + cupicef.cast + (cupicef.rule (cupicef.rule (cupicef.type s6) (cupicef.type s12)) + (cupicef.rule + (cupicef.rule (cupicef.type s6) (cupicef.type s12)) + (cupicef.type s15))) + (cupicef.type s11) + (cupicef.prod (cupicef.rule (cupicef.type s6) (cupicef.type s12)) + (cupicef.rule + (cupicef.rule (cupicef.type s6) (cupicef.type s12)) + (cupicef.type s15)) + (cupicef.rule + (cupicef.rule (cupicef.type s6) (cupicef.type s12)) + (cupicef.rule + (cupicef.rule (cupicef.type s6) (cupicef.type s12)) + (cupicef.type s15))) + cupicef.I + (cupicef.prod (cupicef.type s6) (cupicef.type s12) + (cupicef.rule (cupicef.type s6) (cupicef.type s12)) + cupicef.I A (__ : cupicef.Term (cupicef.type s6) A => B)) + (f : + (__ : cupicef.Term (cupicef.type s6) A -> + cupicef.Term (cupicef.type s12) B) => + cupicef.prod + (cupicef.rule (cupicef.type s6) (cupicef.type s12)) + (cupicef.type s15) + (cupicef.rule + (cupicef.rule (cupicef.type s6) (cupicef.type s12)) + (cupicef.type s15)) + cupicef.I + (cupicef.prod (cupicef.type s6) (cupicef.type s12) + (cupicef.rule (cupicef.type s6) (cupicef.type s12)) + cupicef.I A (__ : cupicef.Term (cupicef.type s6) A => B)) + (g : + (__ : cupicef.Term (cupicef.type s6) A -> + cupicef.Term (cupicef.type s12) B) => + Coq__Classes__CRelationClasses.iffT s15 s15 cstr_35 cstr_36 + cstr_35 cstr_36 + (respectful s6 s7 s12 s13 s14 s15 cstr_10 cstr_11 cstr_13 + cstr_25 cstr_27 A B R S f g) + (respectful s6 s7 s12 s13 s14 s15 cstr_10 cstr_11 cstr_13 + cstr_25 cstr_27 A B R_27_ S_27_ f g)))) + (Coq__Classes__CRelationClasses.relation__equivalence s14 s10 s11 + s15 cstr_2 cstr_32 cstr_33 cstr_34 cstr_35 cstr_36 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) (cupicef.type s12))) + (cupicef.axiom (cupicef.type s14)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) (cupicef.type s12)) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) (cupicef.type s12))) + cupicef.I) + (cupicef.univ (cupicef.type s14) + (cupicef.axiom (cupicef.type s14)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s12) (cupicef.type s14))) + cstr_25 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) (cupicef.type s14))) + cstr_10 cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) (cupicef.type s12) + (cupicef.rule (cupicef.type s6) (cupicef.type s12)) + cupicef.I A (__ : cupicef.Term (cupicef.type s6) A => B))) + (respectful s6 s7 s12 s13 s14 s15 cstr_10 cstr_11 cstr_13 + cstr_25 cstr_27 A B R S) + (respectful s6 s7 s12 s13 s14 s15 cstr_10 cstr_11 cstr_13 + cstr_25 cstr_27 A B R_27_ S_27_)) + cupicef.I + (f : + (__ : cupicef.Term (cupicef.type s6) A -> + cupicef.Term (cupicef.type s12) B) => + g : + (__ : cupicef.Term (cupicef.type s6) A -> + cupicef.Term (cupicef.type s12) B) => + cupicef.cast (cupicef.type s15) (cupicef.type s15) + (Coq__Classes__CRelationClasses.iffT s15 s15 cstr_35 cstr_36 + cstr_35 cstr_36 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.type s7) (cupicef.type s13)))) + (cupicef.axiom (cupicef.type s15)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.type s7) (cupicef.type s13))) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.type s7) + (cupicef.type s13)))) + cupicef.I) + (cupicef.univ (cupicef.type s15) + (cupicef.axiom (cupicef.type s15)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s13) + (cupicef.type s15))) + cstr_27 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) (cupicef.type s15))) + cstr_13 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) (cupicef.type s15))) + cstr_11 cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) (cupicef.type s13))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s7) (cupicef.type s13)) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13))) + cupicef.I A + (y : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s7) (cupicef.type s13) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)) + cupicef.I (R x y) + (__ : cupicef.Term (cupicef.type s7) (R x y) => + S (f x) (g y)))))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.type s7) (cupicef.type s13)))) + (cupicef.axiom (cupicef.type s15)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.type s7) (cupicef.type s13))) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.type s7) + (cupicef.type s13)))) + cupicef.I) + (cupicef.univ (cupicef.type s15) + (cupicef.axiom (cupicef.type s15)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s13) + (cupicef.type s15))) + cstr_27 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) (cupicef.type s15))) + cstr_13 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) (cupicef.type s15))) + cstr_11 cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) (cupicef.type s13))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s7) (cupicef.type s13)) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13))) + cupicef.I A + (y : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s7) (cupicef.type s13) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)) + cupicef.I (R_27_ x y) + (__ : cupicef.Term (cupicef.type s7) (R_27_ x y) => + S_27_ (f x) (g y))))))) + (Coq__Classes__CRelationClasses.iffT s15 s15 cstr_35 cstr_36 + cstr_35 cstr_36 + (respectful s6 s7 s12 s13 s14 s15 cstr_10 cstr_11 cstr_13 + cstr_25 cstr_27 A B R S f g) + (respectful s6 s7 s12 s13 s14 s15 cstr_10 cstr_11 cstr_13 + cstr_25 cstr_27 A B R_27_ S_27_ f g)) + cupicef.I + (cupicef.cast + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.type s7) (cupicef.type s13))) + (cupicef.type s15) + (Coq__Init__Datatypes.prod + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.type s7) (cupicef.type s13))) + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.type s7) (cupicef.type s13))) + (cupicef.prod + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)))) + (cupicef.rule + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13))))) + cupicef.I + (cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13))) + cupicef.I A + (y : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s7) + (cupicef.type s13) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)) + cupicef.I (R x y) + (__ : cupicef.Term (cupicef.type s7) (R x y) => + S (f x) (g y))))) + (__ : + (x : cupicef.Term (cupicef.type s6) A -> + y : cupicef.Term (cupicef.type s6) A -> + __ : cupicef.Term (cupicef.type s7) (R x y) -> + cupicef.Term (cupicef.type s13) (S (f x) (g y))) => + cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13))) + cupicef.I A + (y : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s7) + (cupicef.type s13) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)) + cupicef.I (R_27_ x y) + (__0 : + cupicef.Term (cupicef.type s7) (R_27_ x y) => + S_27_ (f x) (g y)))))) + (cupicef.prod + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)))) + (cupicef.rule + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13))))) + cupicef.I + (cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13))) + cupicef.I A + (y : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s7) + (cupicef.type s13) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)) + cupicef.I (R_27_ x y) + (__ : + cupicef.Term (cupicef.type s7) (R_27_ x y) => + S_27_ (f x) (g y))))) + (__ : + (x : cupicef.Term (cupicef.type s6) A -> + y : cupicef.Term (cupicef.type s6) A -> + __ : cupicef.Term (cupicef.type s7) (R_27_ x y) -> + cupicef.Term (cupicef.type s13) + (S_27_ (f x) (g y))) => + cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13))) + cupicef.I A + (y : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s7) + (cupicef.type s13) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)) + cupicef.I (R x y) + (__0 : cupicef.Term (cupicef.type s7) (R x y) => + S (f x) (g y))))))) + (Coq__Classes__CRelationClasses.iffT s15 s15 cstr_35 cstr_36 + cstr_35 cstr_36 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.type s7) + (cupicef.type s13)))) + (cupicef.axiom (cupicef.type s15)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.type s7) + (cupicef.type s13))) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.type s7) + (cupicef.type s13)))) + cupicef.I) + (cupicef.univ (cupicef.type s15) + (cupicef.axiom (cupicef.type s15)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s13) + (cupicef.type s15))) + cstr_27 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) + (cupicef.type s15))) + cstr_13 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) + (cupicef.type s15))) + cstr_11 cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13))) + cupicef.I A + (y : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s7) + (cupicef.type s13) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)) + cupicef.I (R x y) + (__ : cupicef.Term (cupicef.type s7) (R x y) => + S (f x) (g y)))))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.type s7) + (cupicef.type s13)))) + (cupicef.axiom (cupicef.type s15)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.type s7) + (cupicef.type s13))) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.type s7) + (cupicef.type s13)))) + cupicef.I) + (cupicef.univ (cupicef.type s15) + (cupicef.axiom (cupicef.type s15)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s13) + (cupicef.type s15))) + cstr_27 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) + (cupicef.type s15))) + cstr_13 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) + (cupicef.type s15))) + cstr_11 cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13))) + cupicef.I A + (y : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s7) + (cupicef.type s13) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)) + cupicef.I (R_27_ x y) + (__ : + cupicef.Term (cupicef.type s7) (R_27_ x y) => + S_27_ (f x) (g y))))))) + cupicef.I + (Coq__Init__Datatypes.pair + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.type s7) + (cupicef.type s13)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.type s7) + (cupicef.type s13))) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.type s7) + (cupicef.type s13)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s13) + (cupicef.type (cupicef.s cupicef.z)))) + cstr_28 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) + (cupicef.type (cupicef.s cupicef.z)))) + cstr_14 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.trans (cupicef.type s6) + (cupicef.SortSome (cupicef.type s15)) cstr_11 + (cupicef.SortSome + (cupicef.type (cupicef.s cupicef.z))) + cstr_35 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)))) + (cupicef.rule + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13))))) + cupicef.I + (cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13))) + cupicef.I A + (y : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s7) + (cupicef.type s13) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)) + cupicef.I (R x y) + (__ : + cupicef.Term (cupicef.type s7) (R x y) => + S (f x) (g y))))) + (__ : + (x : cupicef.Term (cupicef.type s6) A -> + y : cupicef.Term (cupicef.type s6) A -> + __ : cupicef.Term (cupicef.type s7) (R x y) -> + cupicef.Term (cupicef.type s13) (S (f x) (g y))) => + cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13))) + cupicef.I A + (y : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s7) + (cupicef.type s13) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)) + cupicef.I (R_27_ x y) + (__0 : + cupicef.Term (cupicef.type s7) + (R_27_ x y) => + S_27_ (f x) (g y))))))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.type s7) + (cupicef.type s13)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.type s7) + (cupicef.type s13))) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.type s7) + (cupicef.type s13)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s13) + (cupicef.type (cupicef.s cupicef.z)))) + cstr_29 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) + (cupicef.type (cupicef.s cupicef.z)))) + cstr_15 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.trans (cupicef.type s6) + (cupicef.SortSome (cupicef.type s15)) cstr_11 + (cupicef.SortSome + (cupicef.type (cupicef.s cupicef.z))) + cstr_36 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)))) + (cupicef.rule + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13))))) + cupicef.I + (cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13))) + cupicef.I A + (y : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s7) + (cupicef.type s13) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)) + cupicef.I (R_27_ x y) + (__ : + cupicef.Term (cupicef.type s7) + (R_27_ x y) => + S_27_ (f x) (g y))))) + (__ : + (x : cupicef.Term (cupicef.type s6) A -> + y : cupicef.Term (cupicef.type s6) A -> + __ : cupicef.Term (cupicef.type s7) (R_27_ x y) -> + cupicef.Term (cupicef.type s13) + (S_27_ (f x) (g y))) => + cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13))) + cupicef.I A + (y : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s7) + (cupicef.type s13) + (cupicef.rule (cupicef.type s7) + (cupicef.type s13)) + cupicef.I (R x y) + (__0 : + cupicef.Term (cupicef.type s7) (R x y) => + S (f x) (g y))))))) + (H : + (x : cupicef.Term (cupicef.type s6) A -> + y : cupicef.Term (cupicef.type s6) A -> + __ : cupicef.Term (cupicef.type s7) (R x y) -> + cupicef.Term (cupicef.type s13) (S (f x) (g y))) => + x : cupicef.Term (cupicef.type s6) A => + y : cupicef.Term (cupicef.type s6) A => + Hxy : cupicef.Term (cupicef.type s7) (R_27_ x y) => + Coq__Init__Datatypes.fst + (cupicef.cast (cupicef.axiom (cupicef.type s13)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s13) + (cupicef.axiom (cupicef.type s13)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_30 + (cupicef.prod (cupicef.type s13) (cupicef.type s13) + (cupicef.rule (cupicef.type s13) + (cupicef.type s13)) + cupicef.I (S (f x) (g y)) + (__ : + cupicef.Term (cupicef.type s13) + (S (f x) (g y)) => + S_27_ (f x) (g y)))) + (cupicef.cast (cupicef.axiom (cupicef.type s13)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s13) + (cupicef.axiom (cupicef.type s13)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_31 + (cupicef.prod (cupicef.type s13) (cupicef.type s13) + (cupicef.rule (cupicef.type s13) + (cupicef.type s13)) + cupicef.I (S_27_ (f x) (g y)) + (__ : + cupicef.Term (cupicef.type s13) + (S_27_ (f x) (g y)) => + S (f x) (g y)))) + (HSS_27_ (f x) (g y)) + (H x y + (Coq__Init__Datatypes.snd + (cupicef.cast (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_16 + (cupicef.prod (cupicef.type s7) + (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.type s7)) + cupicef.I (R x y) + (__ : + cupicef.Term (cupicef.type s7) (R x y) => + R_27_ x y))) + (cupicef.cast (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_17 + (cupicef.prod (cupicef.type s7) + (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.type s7)) + cupicef.I (R_27_ x y) + (__ : + cupicef.Term (cupicef.type s7) + (R_27_ x y) => + R x y))) + (HRR_27_ x y) Hxy))) + (H : + (x : cupicef.Term (cupicef.type s6) A -> + y : cupicef.Term (cupicef.type s6) A -> + __ : cupicef.Term (cupicef.type s7) (R_27_ x y) -> + cupicef.Term (cupicef.type s13) (S_27_ (f x) (g y))) => + x : cupicef.Term (cupicef.type s6) A => + y : cupicef.Term (cupicef.type s6) A => + Hxy : cupicef.Term (cupicef.type s7) (R x y) => + Coq__Init__Datatypes.snd + (cupicef.cast (cupicef.axiom (cupicef.type s13)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s13) + (cupicef.axiom (cupicef.type s13)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_30 + (cupicef.prod (cupicef.type s13) (cupicef.type s13) + (cupicef.rule (cupicef.type s13) + (cupicef.type s13)) + cupicef.I (S (f x) (g y)) + (__ : + cupicef.Term (cupicef.type s13) + (S (f x) (g y)) => + S_27_ (f x) (g y)))) + (cupicef.cast (cupicef.axiom (cupicef.type s13)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s13) + (cupicef.axiom (cupicef.type s13)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_31 + (cupicef.prod (cupicef.type s13) (cupicef.type s13) + (cupicef.rule (cupicef.type s13) + (cupicef.type s13)) + cupicef.I (S_27_ (f x) (g y)) + (__ : + cupicef.Term (cupicef.type s13) + (S_27_ (f x) (g y)) => + S (f x) (g y)))) + (HSS_27_ (f x) (g y)) + (H x y + (Coq__Init__Datatypes.fst + (cupicef.cast (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_16 + (cupicef.prod (cupicef.type s7) + (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.type s7)) + cupicef.I (R x y) + (__ : + cupicef.Term (cupicef.type s7) (R x y) => + R_27_ x y))) + (cupicef.cast (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_17 + (cupicef.prod (cupicef.type s7) + (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.type s7)) + cupicef.I (R_27_ x y) + (__ : + cupicef.Term (cupicef.type s7) + (R_27_ x y) => + R x y))) + (HRR_27_ x y) Hxy)))))))). + +#TYPE_CLASS_INSTANCE respectful__morphism. + +thm Reflexive__partial__app__morphism : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s3)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s3)) -> + A : cupicef.Univ (cupicef.type s0) -> + B : cupicef.Univ (cupicef.type s1) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s5))) + (Coq__Classes__CRelationClasses.crelation s1 s5 B) -> + m : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) B) -> + H : + cupicef.Term (cupicef.type s3) + (Proper s2 s3 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) (cupicef.axiom (cupicef.type s2)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s2))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s2))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => B))) + (respectful s0 s4 s1 s5 s2 s3 cstr_0 cstr_1 cstr_3 cstr_2 cstr_4 A B + R R_27_) + m) -> + x : cupicef.Term (cupicef.type s0) A -> + H0 : cupicef.Term (cupicef.type s4) (ProperProxy s0 s4 A R x) -> + cupicef.Term (cupicef.type s5) (Proper s1 s5 B R_27_ (m x)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s3)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s3)) => + A : cupicef.Univ (cupicef.type s0) => + B : cupicef.Univ (cupicef.type s1) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s0 s4 A) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s5))) + (Coq__Classes__CRelationClasses.crelation s1 s5 B) => + m : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) B) => + H : + cupicef.Term (cupicef.type s3) + (Proper s2 s3 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s2))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s2))) + cstr_0 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I + A (__ : cupicef.Term (cupicef.type s0) A => B))) + (respectful s0 s4 s1 s5 s2 s3 cstr_0 cstr_1 cstr_3 cstr_2 cstr_4 A + B R R_27_) + m) => + x : cupicef.Term (cupicef.type s0) A => + H0 : cupicef.Term (cupicef.type s4) (ProperProxy s0 s4 A R x) => + cupicef.cast (cupicef.type s5) (cupicef.type s5) + (Proper s1 s5 B R_27_ (m x)) (Proper s1 s5 B R_27_ (m x)) cupicef.I + (cupicef.cast (cupicef.type s5) (cupicef.type s5) (R_27_ (m x) (m x)) + (Proper s1 s5 B R_27_ (m x)) cupicef.I + (cupicef.cast (cupicef.type s5) (cupicef.type s5) (R_27_ (m x) (m x)) + (R_27_ (m x) (m x)) cupicef.I + (cupicef.cast (cupicef.type s5) (cupicef.type s5) + (R_27_ (m x) (m x)) (R_27_ (m x) (m x)) cupicef.I + (cupicef.cast (cupicef.type s5) (cupicef.type s5) + (R_27_ (m x) (m x)) (R_27_ (m x) (m x)) cupicef.I (H x x H0))))). + +def Params : + s0 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + of : cupicef.Term (cupicef.type s0) A -> + arity : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop. + +#TYPE_CLASS Params. + +def Build__Params : + s0 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + of : cupicef.Term (cupicef.type s0) A -> + arity : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (Params s0 A of arity). + +[ ] cupicef.code_guarded? _ (Build__Params _ _ _ _) --> cupicef.guarded. + +def match____Params : + s0 : cupicef.Nat -> + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type s0) -> + of : cupicef.Term (cupicef.type s0) A -> + arity : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + P : (cupicef.Term cupicef.prop (Params s0 A of arity) -> cupicef.Univ s) -> + case__Build__Params : cupicef.Term s (P (Build__Params s0 A of arity)) -> + x : cupicef.Term cupicef.prop (Params s0 A of arity) -> + cupicef.Term s (P x). + +[ s0, s, A, of, arity, P, case__Build__Params] + match____Params s0 s A of arity P case__Build__Params + (Build__Params s0 A of arity) + --> + case__Build__Params. + +[ s0, s, A, of, arity, P, s'] + match____Params s0 s A of arity + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____Params s0 s' A of arity (x => P x). + +thm flip__respectful : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + s7 : cupicef.Nat -> + s8 : cupicef.Nat -> + s9 : cupicef.Nat -> + s10 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s6)) (cupicef.type s9)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) (cupicef.type s10)) -> + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) (cupicef.type s3)) -> + cstr_3 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) (cupicef.type s5)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s8)) -> + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) -> + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s3)) -> + cstr_8 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s4)) -> + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s8)) -> + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s8)) -> + cstr_11 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s4)) -> + cstr_12 : + cupicef.eps + (cupicef.Cumul (cupicef.type s8) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_13 : + cupicef.eps + (cupicef.Cumul (cupicef.type s8) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + B : cupicef.Univ (cupicef.type s1) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s6))) + (Coq__Classes__CRelationClasses.crelation s0 s6 A) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s1 s7 B) -> + cupicef.Term (cupicef.type s4) + (Coq__Classes__CRelationClasses.relation__equivalence s2 s3 s4 s8 cstr_2 + cstr_7 cstr_8 cstr_11 cstr_12 cstr_13 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) (cupicef.axiom (cupicef.type s2)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s2))) + cstr_6 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s2))) + cstr_4 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => B))) + (Coq__Classes__CRelationClasses.flip s2 s2 s5 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s2))) + cstr_6 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s2))) + cstr_4 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I + A (__ : cupicef.Term (cupicef.type s0) A => B))) + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s2))) + cstr_6 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s2))) + cstr_4 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I + A (__ : cupicef.Term (cupicef.type s0) A => B))) + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.axiom (cupicef.type s8))) cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + cstr_3 + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I)) + (respectful s0 s6 s1 s7 s2 s8 cstr_4 cstr_5 cstr_9 cstr_6 cstr_10 A + B R R_27_)) + (respectful s0 s6 s1 s7 s2 s8 cstr_4 cstr_5 cstr_9 cstr_6 cstr_10 A B + (Coq__Classes__CRelationClasses.flip s0 s0 s9 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s9)) + (cupicef.univ (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s9) + (cupicef.axiom (cupicef.type s9)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I)) + R) + (Coq__Classes__CRelationClasses.flip s1 s1 s10 B B + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s10)) + (cupicef.univ (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s10) + (cupicef.axiom (cupicef.type s10)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I)) + R_27_))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + s7 : cupicef.Nat => + s8 : cupicef.Nat => + s9 : cupicef.Nat => + s10 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s6)) (cupicef.type s9)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) (cupicef.type s10)) => + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) (cupicef.type s3)) => + cstr_3 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) (cupicef.type s5)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s8)) => + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) => + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s3)) => + cstr_8 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s4)) => + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s8)) => + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s8)) => + cstr_11 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s4)) => + cstr_12 : + cupicef.eps + (cupicef.Cumul (cupicef.type s8) (cupicef.type (cupicef.s cupicef.z))) => + cstr_13 : + cupicef.eps + (cupicef.Cumul (cupicef.type s8) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + B : cupicef.Univ (cupicef.type s1) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s6))) + (Coq__Classes__CRelationClasses.crelation s0 s6 A) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s1 s7 B) => + cupicef.cast + (cupicef.rule (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s6) (cupicef.type s7))))) + (cupicef.type s4) + (cupicef.prod (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s6) (cupicef.type s7)))) + (cupicef.rule (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.rule (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s6) (cupicef.type s7))))) + cupicef.I + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => B)) + (x : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) B) => + cupicef.prod (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s6) (cupicef.type s7))) + (cupicef.rule (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s6) (cupicef.type s7)))) + cupicef.I + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => B)) + (y : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) B) => + Coq__Init__Datatypes.prod + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s6) (cupicef.type s7))) + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s6) (cupicef.type s7))) + (cupicef.prod + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7))))) + cupicef.I + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7))) + cupicef.I A + (y0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s6) (cupicef.type s7) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)) + cupicef.I (R x0 y0) + (__ : cupicef.Term (cupicef.type s6) (R x0 y0) => + R_27_ (y x0) (x y0))))) + (__ : + (x0 : cupicef.Term (cupicef.type s0) A -> + y0 : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s6) (R x0 y0) -> + cupicef.Term (cupicef.type s7) (R_27_ (y x0) (x y0))) => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7))) + cupicef.I A + (y0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s6) (cupicef.type s7) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)) + cupicef.I (R y0 x0) + (__0 : cupicef.Term (cupicef.type s6) (R y0 x0) => + R_27_ (y y0) (x x0)))))) + (cupicef.prod + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7))))) + cupicef.I + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7))) + cupicef.I A + (y0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s6) (cupicef.type s7) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)) + cupicef.I (R y0 x0) + (__ : cupicef.Term (cupicef.type s6) (R y0 x0) => + R_27_ (y y0) (x x0))))) + (__ : + (x0 : cupicef.Term (cupicef.type s0) A -> + y0 : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s6) (R y0 x0) -> + cupicef.Term (cupicef.type s7) (R_27_ (y y0) (x x0))) => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7))) + cupicef.I A + (y0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s6) (cupicef.type s7) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)) + cupicef.I (R x0 y0) + (__0 : cupicef.Term (cupicef.type s6) (R x0 y0) => + R_27_ (y x0) (x y0))))))))) + (Coq__Classes__CRelationClasses.relation__equivalence s2 s3 s4 s8 cstr_2 + cstr_7 cstr_8 cstr_11 cstr_12 cstr_13 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) (cupicef.axiom (cupicef.type s2)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s2))) + cstr_6 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s2))) + cstr_4 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => B))) + (Coq__Classes__CRelationClasses.flip s2 s2 s5 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s2))) + cstr_6 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s2))) + cstr_4 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I + A (__ : cupicef.Term (cupicef.type s0) A => B))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s2))) + cstr_6 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s2))) + cstr_4 cupicef.BoolNone) + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I + A (__ : cupicef.Term (cupicef.type s0) A => B))) + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.axiom (cupicef.type s8))) cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + cstr_3 + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I)) + (respectful s0 s6 s1 s7 s2 s8 cstr_4 cstr_5 cstr_9 cstr_6 cstr_10 + A B R R_27_)) + (respectful s0 s6 s1 s7 s2 s8 cstr_4 cstr_5 cstr_9 cstr_6 cstr_10 A B + (Coq__Classes__CRelationClasses.flip s0 s0 s9 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s9)) + (cupicef.univ (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s9) + (cupicef.axiom (cupicef.type s9)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I)) + R) + (Coq__Classes__CRelationClasses.flip s1 s1 s10 B B + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s10)) + (cupicef.univ (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s10) + (cupicef.axiom (cupicef.type s10)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I)) + R_27_))) + cupicef.I + (x : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) B) => + y : + (__ : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) B) => + Coq__Init__Datatypes.pair + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s6) (cupicef.type s7)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s6) (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s6) (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.trans (cupicef.type s7) + (cupicef.SortSome (cupicef.type s8)) cstr_10 + (cupicef.SortSome (cupicef.type (cupicef.s cupicef.z))) + cstr_12 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.trans (cupicef.type s6) + (cupicef.SortSome (cupicef.type s8)) cstr_9 + (cupicef.SortSome (cupicef.type (cupicef.s cupicef.z))) + cstr_12 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.trans (cupicef.type s0) + (cupicef.SortSome (cupicef.type s8)) cstr_5 + (cupicef.SortSome (cupicef.type (cupicef.s cupicef.z))) + cstr_12 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7))))) + cupicef.I + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7))) + cupicef.I A + (y0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s6) (cupicef.type s7) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)) + cupicef.I (R x0 y0) + (__ : cupicef.Term (cupicef.type s6) (R x0 y0) => + R_27_ (y x0) (x y0))))) + (__ : + (x0 : cupicef.Term (cupicef.type s0) A -> + y0 : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s6) (R x0 y0) -> + cupicef.Term (cupicef.type s7) (R_27_ (y x0) (x y0))) => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7))) + cupicef.I A + (y0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s6) (cupicef.type s7) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)) + cupicef.I (R y0 x0) + (__0 : cupicef.Term (cupicef.type s6) (R y0 x0) => + R_27_ (y y0) (x x0))))))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s6) (cupicef.type s7)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s6) (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s6) (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.trans (cupicef.type s7) + (cupicef.SortSome (cupicef.type s8)) cstr_10 + (cupicef.SortSome (cupicef.type (cupicef.s cupicef.z))) + cstr_13 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.trans (cupicef.type s6) + (cupicef.SortSome (cupicef.type s8)) cstr_9 + (cupicef.SortSome (cupicef.type (cupicef.s cupicef.z))) + cstr_13 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.trans (cupicef.type s0) + (cupicef.SortSome (cupicef.type s8)) cstr_5 + (cupicef.SortSome (cupicef.type (cupicef.s cupicef.z))) + cstr_13 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7))))) + cupicef.I + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7))) + cupicef.I A + (y0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s6) (cupicef.type s7) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)) + cupicef.I (R y0 x0) + (__ : cupicef.Term (cupicef.type s6) (R y0 x0) => + R_27_ (y y0) (x x0))))) + (__ : + (x0 : cupicef.Term (cupicef.type s0) A -> + y0 : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s6) (R y0 x0) -> + cupicef.Term (cupicef.type s7) (R_27_ (y y0) (x x0))) => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s6) (cupicef.type s7))) + cupicef.I A + (y0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s6) (cupicef.type s7) + (cupicef.rule (cupicef.type s6) (cupicef.type s7)) + cupicef.I (R x0 y0) + (__0 : cupicef.Term (cupicef.type s6) (R x0 y0) => + R_27_ (y x0) (x y0))))))) + (X : + (x0 : cupicef.Term (cupicef.type s0) A -> + y0 : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s6) (R x0 y0) -> + cupicef.Term (cupicef.type s7) (R_27_ (y x0) (x y0))) => + x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + X0 : cupicef.Term (cupicef.type s6) (R y0 x0) => + cupicef.cast (cupicef.type s7) (cupicef.type s7) + (R_27_ (y y0) (x x0)) (R_27_ (y y0) (x x0)) cupicef.I (X y0 x0 X0)) + (X : + (x0 : cupicef.Term (cupicef.type s0) A -> + y0 : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s6) (R y0 x0) -> + cupicef.Term (cupicef.type s7) (R_27_ (y y0) (x x0))) => + x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + X0 : cupicef.Term (cupicef.type s6) (R x0 y0) => + cupicef.cast (cupicef.type s7) (cupicef.type s7) + (R_27_ (y x0) (x y0)) (R_27_ (y x0) (x y0)) cupicef.I (X y0 x0 X0))). + +thm flip1 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s3)) -> + A : cupicef.Univ (cupicef.type s0) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (Coq__Classes__CRelationClasses.crelation s0 s2 A) -> + H : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (Coq__Classes__CRelationClasses.subrelation s0 s1 s2 A R_27_ R) -> + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (Coq__Classes__CRelationClasses.subrelation s0 s1 s2 A + (Coq__Classes__CRelationClasses.flip s0 s0 s3 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + (Coq__Classes__CRelationClasses.flip s0 s0 s3 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R_27_)) + R) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s3)) => + A : cupicef.Univ (cupicef.type s0) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (Coq__Classes__CRelationClasses.crelation s0 s2 A) => + H : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (Coq__Classes__CRelationClasses.subrelation s0 s1 s2 A R_27_ R) => + H. + +thm flip2 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s3)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (Coq__Classes__CRelationClasses.crelation s0 s2 A) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) -> + H : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (Coq__Classes__CRelationClasses.subrelation s0 s2 s1 A R R_27_) -> + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (Coq__Classes__CRelationClasses.subrelation s0 s2 s1 A R + (Coq__Classes__CRelationClasses.flip s0 s0 s3 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + (Coq__Classes__CRelationClasses.flip s0 s0 s3 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R_27_))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s3)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (Coq__Classes__CRelationClasses.crelation s0 s2 A) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) => + H : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (Coq__Classes__CRelationClasses.subrelation s0 s2 s1 A R R_27_) => + H. + +thm eq__subrelation : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.type s2) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (Coq__Classes__CRelationClasses.crelation s0 s2 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (Coq__Classes__CRelationClasses.Reflexive s0 s2 A R) -> + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (Coq__Classes__CRelationClasses.subrelation s0 s1 s2 A + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_0 A))) + R) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) (cupicef.type (cupicef.s cupicef.z))) => + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.type s2) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (Coq__Classes__CRelationClasses.crelation s0 s2 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (Coq__Classes__CRelationClasses.Reflexive s0 s2 A R) => + cupicef.cast + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (Coq__Classes__CRelationClasses.subrelation s0 s1 s2 A + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_0 A))) + R) + (Coq__Classes__CRelationClasses.subrelation s0 s1 s2 A + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_0 A))) + R) + cupicef.I + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule cupicef.prop (cupicef.type s2)))) + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule cupicef.prop (cupicef.type s2))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule cupicef.prop (cupicef.type s2)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule cupicef.prop (cupicef.type s2)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule cupicef.prop (cupicef.type s2))) + cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + cupicef.prod cupicef.prop (cupicef.type s2) + (cupicef.rule cupicef.prop (cupicef.type s2)) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_0 A) + x y) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_0 A) + x y) => + R x y)))) + (Coq__Classes__CRelationClasses.subrelation s0 s1 s2 A + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_0 A))) + R) + cupicef.I + (x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_0 A) + x y) => + cupicef.cast (cupicef.type s2) (cupicef.type s2) (R x y) (R x y) + cupicef.I + (cupicef.cast (cupicef.type s2) (cupicef.type s2) + ((x0 : cupicef.Term (cupicef.type s0) A => R x0 y) x) ( + R x y) cupicef.I + (Coq__Init__Logic.eq__rect__r + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_1 A) + y + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type s0) + (cupicef.axiom (cupicef.type s2)) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + (cupicef.prod (cupicef.type s0) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cstr_2 (x0 : cupicef.Term (cupicef.type s0) A => R x0 y)) + (cupicef.cast (cupicef.type s2) (cupicef.type s2) (R y y) + (R y y) cupicef.I (H y)) + x H0)))). + +def proper__flip__proper : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s2)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) -> + m : cupicef.Term (cupicef.type s0) A -> + mor : cupicef.Term (cupicef.type s1) (Proper s0 s1 A R m) -> + cupicef.Term (cupicef.type s1) + (Proper s0 s1 A + (Coq__Classes__CRelationClasses.flip s0 s0 s2 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R) + m) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s2)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) => + m : cupicef.Term (cupicef.type s0) A => + mor : cupicef.Term (cupicef.type s1) (Proper s0 s1 A R m) => + mor. + +thm reflexive__proper : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Coq__Classes__CRelationClasses.Reflexive s0 s1 A R) -> + x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) (Proper s0 s1 A R x) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Coq__Classes__CRelationClasses.Reflexive s0 s1 A R) => + x : cupicef.Term (cupicef.type s0) A => + (X : cupicef.Term (cupicef.type s1) (R x x) => X) (H x). + +thm proper__eq : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) -> + A : cupicef.Univ (cupicef.type s0) -> + x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) + (Proper s0 s1 A + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_0 A))) + x) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) => + A : cupicef.Univ (cupicef.type s0) => + x : cupicef.Term (cupicef.type s0) A => + reflexive__proper s0 s1 A + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s0 s1 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_0 A))) + (Coq__Classes__CRelationClasses.eq__Reflexive s0 s1 cstr_0 A) x. + +def PartialApplication : + cupicef.Univ cupicef.prop. + +#TYPE_CLASS PartialApplication. + +def Build__PartialApplication : + cupicef.Term cupicef.prop PartialApplication. + +[ ] cupicef.code_guarded? _ Build__PartialApplication --> cupicef.guarded. + +def match____PartialApplication : + s : cupicef.Sort -> + P : (cupicef.Term cupicef.prop PartialApplication -> cupicef.Univ s) -> + case__Build__PartialApplication : + cupicef.Term s (P Build__PartialApplication) -> + x : cupicef.Term cupicef.prop PartialApplication -> + cupicef.Term s (P x). + +[ s, P, case__Build__PartialApplication] + match____PartialApplication s P case__Build__PartialApplication + Build__PartialApplication + --> + case__Build__PartialApplication. + +[ s, P, s'] + match____PartialApplication s + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____PartialApplication s' (x => P x). + +def normalization__done : + cupicef.Univ cupicef.prop. + +def did__normalization : + cupicef.Term cupicef.prop normalization__done. + +thm proper__proper : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + s7 : cupicef.Nat -> + s8 : cupicef.Nat -> + s9 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) (cupicef.type s2)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) (cupicef.type s9)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s0)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s1)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s0)) -> + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s1)) -> + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s2)) -> + cstr_8 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s3)) -> + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s4)) -> + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s5)) -> + cstr_11 : + cupicef.eps + (cupicef.Cumul (cupicef.type s6) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) -> + cstr_12 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s3)) -> + cstr_13 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s5)) -> + cstr_14 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_15 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_16 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_17 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_18 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_19 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s5)) -> + cstr_20 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s4)) -> + A : cupicef.Univ (cupicef.type s6) -> + cupicef.Term (cupicef.type s1) + (Proper s0 s1 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) (cupicef.type s0))) + (cupicef.trans (cupicef.type s6) + (cupicef.SortSome (cupicef.type s2)) cstr_7 + (cupicef.SortSome (cupicef.type s0)) cstr_2 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s0))) + (cupicef.trans (cupicef.axiom (cupicef.type s7)) + (cupicef.SortSome (cupicef.type s2)) cstr_0 + (cupicef.SortSome (cupicef.type s0)) cstr_2 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I (Coq__Classes__CRelationClasses.crelation s6 s7 A) + (__ : + cupicef.Term + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s6 s7 A) => + cupicef.prod (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type s6) A => + cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I)))) + (respectful s2 s3 s4 s5 s0 s1 cstr_2 cstr_3 cstr_4 cstr_5 cstr_6 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) (cupicef.type s2))) + cstr_7 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s2))) + cstr_0 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s6 s7 A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) (cupicef.type s4))) + cstr_9 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s4))) + (cupicef.trans (cupicef.axiom (cupicef.type s7)) + (cupicef.SortSome (cupicef.type s9)) cstr_1 + (cupicef.SortSome (cupicef.type s4)) cstr_20 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => + cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I))) + (Coq__Classes__CRelationClasses.relation__equivalence s6 s2 s3 s7 + cstr_0 cstr_7 cstr_8 cstr_12 cstr_15 cstr_16 A) + (respectful s6 s8 s9 s7 s4 s5 cstr_9 cstr_10 cstr_19 cstr_20 + cstr_13 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s9)) + (cupicef.univ (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s9) + (cupicef.axiom (cupicef.type s9)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s8))) + (cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s6) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s6) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s6 s8 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_11 A))) + (Coq__Classes__CRelationClasses.iffT s7 s7 cstr_15 cstr_16 + cstr_15 cstr_16))) + (Proper s6 s7 A)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + s7 : cupicef.Nat => + s8 : cupicef.Nat => + s9 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) (cupicef.type s2)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) (cupicef.type s9)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s0)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s1)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s0)) => + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s1)) => + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s2)) => + cstr_8 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s3)) => + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s4)) => + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s5)) => + cstr_11 : + cupicef.eps + (cupicef.Cumul (cupicef.type s6) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) => + cstr_12 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s3)) => + cstr_13 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s5)) => + cstr_14 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) => + cstr_15 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) => + cstr_16 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) => + cstr_17 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) => + cstr_18 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) => + cstr_19 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s5)) => + cstr_20 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s4)) => + A : cupicef.Univ (cupicef.type s6) => + cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)))) + (cupicef.type s1) + (cupicef.prod + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.type s3) (cupicef.type s5))) + (cupicef.rule + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)))) + cupicef.I (Coq__Classes__CRelationClasses.crelation s6 s7 A) + (R : + cupicef.Term + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s6 s7 A) => + cupicef.prod + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + (cupicef.rule + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.type s3) (cupicef.type s5))) + cupicef.I (Coq__Classes__CRelationClasses.crelation s6 s7 A) + (R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s6 s7 A) => + cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) cupicef.I + (Coq__Classes__CRelationClasses.relation__equivalence s6 s2 s3 + s7 cstr_0 cstr_7 cstr_8 cstr_12 cstr_15 cstr_16 A R R_27_) + (HRR_27_ : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.relation__equivalence s6 + s2 s3 s7 cstr_0 cstr_7 cstr_8 cstr_12 cstr_15 cstr_16 A + R R_27_) => + respectful s6 s8 s9 s7 s4 s5 cstr_9 cstr_10 cstr_19 cstr_20 + cstr_13 A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s9)) + (cupicef.univ (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s9) + (cupicef.axiom (cupicef.type s9)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s8))) + (cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s6) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s6) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s6 s8 A) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_11 A))) + (Coq__Classes__CRelationClasses.iffT s7 s7 cstr_15 cstr_16 + cstr_15 cstr_16) + (Proper s6 s7 A R) (Proper s6 s7 A R_27_))))) + (Proper s0 s1 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) (cupicef.type s0))) + (cupicef.trans (cupicef.type s6) + (cupicef.SortSome (cupicef.type s2)) cstr_7 + (cupicef.SortSome (cupicef.type s0)) cstr_2 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s0))) + (cupicef.trans (cupicef.axiom (cupicef.type s7)) + (cupicef.SortSome (cupicef.type s2)) cstr_0 + (cupicef.SortSome (cupicef.type s0)) cstr_2 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I (Coq__Classes__CRelationClasses.crelation s6 s7 A) + (__ : + cupicef.Term + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s6 s7 A) => + cupicef.prod (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type s6) A => + cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I)))) + (respectful s2 s3 s4 s5 s0 s1 cstr_2 cstr_3 cstr_4 cstr_5 cstr_6 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) (cupicef.type s2))) + cstr_7 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s2))) + cstr_0 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s6 s7 A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) (cupicef.type s4))) + cstr_9 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) + (cupicef.type s4))) + (cupicef.trans (cupicef.axiom (cupicef.type s7)) + (cupicef.SortSome (cupicef.type s9)) cstr_1 + (cupicef.SortSome (cupicef.type s4)) cstr_20 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => + cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I))) + (Coq__Classes__CRelationClasses.relation__equivalence s6 s2 s3 s7 + cstr_0 cstr_7 cstr_8 cstr_12 cstr_15 cstr_16 A) + (respectful s6 s8 s9 s7 s4 s5 cstr_9 cstr_10 cstr_19 cstr_20 + cstr_13 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s9)) + (cupicef.univ (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s9) + (cupicef.axiom (cupicef.type s9)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s8))) + (cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s6) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s6) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s6 s8 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_11 A))) + (Coq__Classes__CRelationClasses.iffT s7 s7 cstr_15 cstr_16 + cstr_15 cstr_16))) + (Proper s6 s7 A)) + cupicef.I + (R : + cupicef.Term + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s6 s7 A) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s6 s7 A) => + HRR_27_ : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.relation__equivalence s6 s2 s3 s7 + cstr_0 cstr_7 cstr_8 cstr_12 cstr_15 cstr_16 A R R_27_) => + cupicef.cast + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule cupicef.prop (cupicef.type s7)))) + (cupicef.type s5) + (cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule cupicef.prop (cupicef.type s7))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule cupicef.prop (cupicef.type s7)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s6) + (cupicef.rule cupicef.prop (cupicef.type s7)) + (cupicef.rule (cupicef.type s6) + (cupicef.rule cupicef.prop (cupicef.type s7))) + cupicef.I A + (y : cupicef.Term (cupicef.type s6) A => + cupicef.prod cupicef.prop (cupicef.type s7) + (cupicef.rule cupicef.prop (cupicef.type s7)) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_11 A) + x y) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_11 A) + x y) => + (y0 : cupicef.Term (cupicef.type s6) A => + Coq__Classes__CRelationClasses.iffT s7 s7 cstr_15 cstr_16 + cstr_15 cstr_16 (Proper s6 s7 A R x) + (Proper s6 s7 A R_27_ y0)) + y)))) + (respectful s6 s8 s9 s7 s4 s5 cstr_9 cstr_10 cstr_19 cstr_20 cstr_13 + A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s9)) + (cupicef.univ (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.axiom (cupicef.type s7))) cupicef.I) + (cupicef.univ (cupicef.type s9) + (cupicef.axiom (cupicef.type s9)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s8))) + (cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s6) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s6) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s6 s8 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_11 A))) + (Coq__Classes__CRelationClasses.iffT s7 s7 cstr_15 cstr_16 cstr_15 + cstr_16) + (Proper s6 s7 A R) (Proper s6 s7 A R_27_)) + cupicef.I + (x : cupicef.Term (cupicef.type s6) A => + y : cupicef.Term (cupicef.type s6) A => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_11 A) + x y) => + Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_11 A) + x + (cupicef.cast + (cupicef.rule (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.type s6) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + cupicef.I A + (y0 : cupicef.Term (cupicef.type s6) A => + cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I)) + (cupicef.prod (cupicef.type s6) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s6) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cstr_14 + (y0 : cupicef.Term (cupicef.type s6) A => + Coq__Classes__CRelationClasses.iffT s7 s7 cstr_15 cstr_16 + cstr_15 cstr_16 (Proper s6 s7 A R x) + (Proper s6 s7 A R_27_ y0))) + (Coq__Init__Datatypes.pair + (cupicef.cast (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_15 + (cupicef.prod (cupicef.type s7) (cupicef.type s7) + (cupicef.rule (cupicef.type s7) (cupicef.type s7)) + cupicef.I (Proper s6 s7 A R x) + (__ : cupicef.Term (cupicef.type s7) (Proper s6 s7 A R x) => + Proper s6 s7 A R_27_ x))) + (cupicef.cast (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_16 + (cupicef.prod (cupicef.type s7) (cupicef.type s7) + (cupicef.rule (cupicef.type s7) (cupicef.type s7)) + cupicef.I (Proper s6 s7 A R_27_ x) + (__ : + cupicef.Term (cupicef.type s7) + (Proper s6 s7 A R_27_ x) => + Proper s6 s7 A R x))) + (cupicef.cast + (cupicef.rule (cupicef.type s7) (cupicef.type s7)) + (cupicef.rule (cupicef.type s7) (cupicef.type s7)) + (cupicef.prod (cupicef.type s7) (cupicef.type s7) + (cupicef.rule (cupicef.type s7) (cupicef.type s7)) + cupicef.I (Proper s6 s7 A R x) + (X : cupicef.Term (cupicef.type s7) (Proper s6 s7 A R x) => + R_27_ x x)) + (cupicef.prod (cupicef.type s7) (cupicef.type s7) + (cupicef.rule (cupicef.type s7) (cupicef.type s7)) + cupicef.I (Proper s6 s7 A R x) + (__ : cupicef.Term (cupicef.type s7) (Proper s6 s7 A R x) => + Proper s6 s7 A R_27_ x)) + cupicef.I + (X : cupicef.Term (cupicef.type s7) (Proper s6 s7 A R x) => + Coq__Init__Datatypes.fst + (cupicef.cast (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_17 + (cupicef.prod (cupicef.type s7) (cupicef.type s7) + (cupicef.rule (cupicef.type s7) (cupicef.type s7)) + cupicef.I (R x x) + (__ : cupicef.Term (cupicef.type s7) (R x x) => + R_27_ x x))) + (cupicef.cast (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_18 + (cupicef.prod (cupicef.type s7) (cupicef.type s7) + (cupicef.rule (cupicef.type s7) (cupicef.type s7)) + cupicef.I (R_27_ x x) + (__ : cupicef.Term (cupicef.type s7) (R_27_ x x) => + R x x))) + (HRR_27_ x x) X)) + (cupicef.cast + (cupicef.rule (cupicef.type s7) (cupicef.type s7)) + (cupicef.rule (cupicef.type s7) (cupicef.type s7)) + (cupicef.prod (cupicef.type s7) (cupicef.type s7) + (cupicef.rule (cupicef.type s7) (cupicef.type s7)) + cupicef.I (Proper s6 s7 A R_27_ x) + (X : + cupicef.Term (cupicef.type s7) + (Proper s6 s7 A R_27_ x) => + R x x)) + (cupicef.prod (cupicef.type s7) (cupicef.type s7) + (cupicef.rule (cupicef.type s7) (cupicef.type s7)) + cupicef.I (Proper s6 s7 A R_27_ x) + (__ : + cupicef.Term (cupicef.type s7) + (Proper s6 s7 A R_27_ x) => + Proper s6 s7 A R x)) + cupicef.I + (X : cupicef.Term (cupicef.type s7) (Proper s6 s7 A R_27_ x) => + Coq__Init__Datatypes.snd + (cupicef.cast (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_17 + (cupicef.prod (cupicef.type s7) (cupicef.type s7) + (cupicef.rule (cupicef.type s7) (cupicef.type s7)) + cupicef.I (R x x) + (__ : cupicef.Term (cupicef.type s7) (R x x) => + R_27_ x x))) + (cupicef.cast (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_18 + (cupicef.prod (cupicef.type s7) (cupicef.type s7) + (cupicef.rule (cupicef.type s7) (cupicef.type s7)) + cupicef.I (R_27_ x x) + (__ : cupicef.Term (cupicef.type s7) (R_27_ x x) => + R x x))) + (HRR_27_ x x) X))) + y H)). + +#TYPE_CLASS_INSTANCE proper__proper. + +def Normalizes : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s1)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) -> + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + m : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (Coq__Classes__CRelationClasses.crelation s0 s3 A) -> + m_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (Coq__Classes__CRelationClasses.crelation s0 s3 A) -> + cupicef.Univ (cupicef.type s2) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s1)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) => + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + m : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (Coq__Classes__CRelationClasses.crelation s0 s3 A) => + m_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (Coq__Classes__CRelationClasses.crelation s0 s3 A) => + Coq__Classes__CRelationClasses.relation__equivalence s0 s1 s2 s3 cstr_0 + cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 A m m_27_. + +#TYPE_CLASS Normalizes. + +def normalizes : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s1)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) -> + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + m : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (Coq__Classes__CRelationClasses.crelation s0 s3 A) -> + m_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (Coq__Classes__CRelationClasses.crelation s0 s3 A) -> + Normalizes0 : + cupicef.Term (cupicef.type s2) + (Normalizes s0 s1 s2 s3 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 A m + m_27_) -> + cupicef.Term (cupicef.type s2) + (Coq__Classes__CRelationClasses.relation__equivalence s0 s1 s2 s3 cstr_0 + cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 A m m_27_) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s1)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) => + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + m : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (Coq__Classes__CRelationClasses.crelation s0 s3 A) => + m_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (Coq__Classes__CRelationClasses.crelation s0 s3 A) => + Normalizes0 : + cupicef.Term (cupicef.type s2) + (Normalizes s0 s1 s2 s3 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 A m + m_27_) => + Normalizes0. + +thm proper__normalizes__proper : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s1)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) -> + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + R0 : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (Coq__Classes__CRelationClasses.crelation s0 s3 A) -> + R1 : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (Coq__Classes__CRelationClasses.crelation s0 s3 A) -> + H : + cupicef.Term (cupicef.type s2) + (Normalizes s0 s1 s2 s3 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 A R0 + R1) -> + m : cupicef.Term (cupicef.type s0) A -> + H0 : cupicef.Term (cupicef.type s3) (Proper s0 s3 A R1 m) -> + cupicef.Term (cupicef.type s3) (Proper s0 s3 A R0 m) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s1)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) => + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + R0 : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (Coq__Classes__CRelationClasses.crelation s0 s3 A) => + R1 : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (Coq__Classes__CRelationClasses.crelation s0 s3 A) => + H : + cupicef.Term (cupicef.type s2) + (Normalizes s0 s1 s2 s3 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 A R0 + R1) => + m : cupicef.Term (cupicef.type s0) A => + H0 : cupicef.Term (cupicef.type s3) (Proper s0 s3 A R1 m) => + Coq__Init__Datatypes.snd + (cupicef.cast (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s3) (cupicef.axiom (cupicef.type s3)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_6 + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (R0 m m) (__ : cupicef.Term (cupicef.type s3) (R0 m m) => R1 m m))) + (cupicef.cast (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s3) (cupicef.axiom (cupicef.type s3)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_7 + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (R1 m m) (__ : cupicef.Term (cupicef.type s3) (R1 m m) => R0 m m))) + (H m m) H0. + +thm flip__atom : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s1)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s4)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) -> + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (Coq__Classes__CRelationClasses.crelation s0 s3 A) -> + cupicef.Term (cupicef.type s2) + (Normalizes s0 s1 s2 s3 cstr_0 cstr_2 cstr_3 cstr_4 cstr_5 cstr_6 A R + (Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + (Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s1)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s4)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) => + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (Coq__Classes__CRelationClasses.crelation s0 s3 A) => + cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3))) + (cupicef.type s2) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) (cupicef.type s3) + (cupicef.rule (cupicef.type s0) (cupicef.type s3)) cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + Coq__Init__Datatypes.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (R x y) + (__ : cupicef.Term (cupicef.type s3) (R x y) => + Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + (Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R) + x y)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + (Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R) + x y) + (__ : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + (Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ + (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R) + x y) => + R x y))))) + (Normalizes s0 s1 s2 s3 cstr_0 cstr_2 cstr_3 cstr_4 cstr_5 cstr_6 A R + (Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + (Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R))) + cupicef.I + (x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + Coq__Init__Datatypes.pair + (cupicef.cast (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s3) (cupicef.axiom (cupicef.type s3)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_5 + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (R x y) + (__ : cupicef.Term (cupicef.type s3) (R x y) => + Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + (Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R) + x y))) + (cupicef.cast (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s3) (cupicef.axiom (cupicef.type s3)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_6 + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + (Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R) + x y) + (__ : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + (Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R) + x y) => + R x y))) + (X : cupicef.Term (cupicef.type s3) (R x y) => X) + (X : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + (Coq__Classes__CRelationClasses.flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R) + x y) => + X)). + +thm flip__arrow : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + s7 : cupicef.Nat -> + s8 : cupicef.Nat -> + s9 : cupicef.Nat -> + s10 : cupicef.Nat -> + s11 : cupicef.Nat -> + s12 : cupicef.Nat -> + s13 : cupicef.Nat -> + s14 : cupicef.Nat -> + s15 : cupicef.Nat -> + s16 : cupicef.Nat -> + s17 : cupicef.Nat -> + s18 : cupicef.Nat -> + s19 : cupicef.Nat -> + s20 : cupicef.Nat -> + s21 : cupicef.Nat -> + s22 : cupicef.Nat -> + s23 : cupicef.Nat -> + s24 : cupicef.Nat -> + s25 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s0)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s10)) -> + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s6)) (cupicef.type s1)) -> + cstr_3 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s6)) (cupicef.type s16)) -> + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) (cupicef.type s7)) -> + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) (cupicef.type s12)) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) (cupicef.type s14)) -> + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) (cupicef.type s18)) -> + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) (cupicef.type s23)) -> + cstr_9 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s21)) (cupicef.type s9)) -> + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s21)) -> + cstr_11 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s23)) -> + cstr_12 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) -> + cstr_13 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s8)) -> + cstr_14 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s10)) -> + cstr_15 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s11)) -> + cstr_16 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s8)) -> + cstr_17 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s11)) -> + cstr_18 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_19 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_20 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_21 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_22 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_23 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s2)) -> + cstr_24 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s16)) -> + cstr_25 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s17)) -> + cstr_26 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s8)) -> + cstr_27 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s17)) -> + cstr_28 : + cupicef.eps + (cupicef.Cumul (cupicef.type s6) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_29 : + cupicef.eps + (cupicef.Cumul (cupicef.type s6) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_30 : + cupicef.eps + (cupicef.Cumul (cupicef.type s6) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_31 : + cupicef.eps + (cupicef.Cumul (cupicef.type s6) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_32 : + cupicef.eps + (cupicef.Cumul (cupicef.type s6) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_33 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s21)) -> + cstr_34 : + cupicef.eps + (cupicef.Cumul (cupicef.type s8) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_35 : + cupicef.eps + (cupicef.Cumul (cupicef.type s8) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_36 : + cupicef.eps + (cupicef.Cumul (cupicef.type s8) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_37 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s18)) -> + cstr_38 : cupicef.eps (cupicef.Cumul (cupicef.type s10) (cupicef.type s14)) -> + cstr_39 : cupicef.eps (cupicef.Cumul (cupicef.type s10) (cupicef.type s15)) -> + cstr_40 : cupicef.eps (cupicef.Cumul (cupicef.type s10) (cupicef.type s25)) -> + cstr_41 : cupicef.eps (cupicef.Cumul (cupicef.type s11) (cupicef.type s15)) -> + cstr_42 : cupicef.eps (cupicef.Cumul (cupicef.type s12) (cupicef.type s14)) -> + cstr_43 : cupicef.eps (cupicef.Cumul (cupicef.type s12) (cupicef.type s25)) -> + cstr_44 : cupicef.eps (cupicef.Cumul (cupicef.type s13) (cupicef.type s15)) -> + cstr_45 : cupicef.eps (cupicef.Cumul (cupicef.type s16) (cupicef.type s12)) -> + cstr_46 : cupicef.eps (cupicef.Cumul (cupicef.type s16) (cupicef.type s13)) -> + cstr_47 : cupicef.eps (cupicef.Cumul (cupicef.type s17) (cupicef.type s13)) -> + cstr_48 : cupicef.eps (cupicef.Cumul (cupicef.type s20) (cupicef.type s19)) -> + cstr_49 : cupicef.eps (cupicef.Cumul (cupicef.type s21) (cupicef.type s13)) -> + cstr_50 : cupicef.eps (cupicef.Cumul (cupicef.type s21) (cupicef.type s19)) -> + cstr_51 : cupicef.eps (cupicef.Cumul (cupicef.type s21) (cupicef.type s22)) -> + cstr_52 : + cupicef.eps + (cupicef.Cumul (cupicef.type s21) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_53 : cupicef.eps (cupicef.Cumul (cupicef.type s23) (cupicef.type s12)) -> + cstr_54 : cupicef.eps (cupicef.Cumul (cupicef.type s23) (cupicef.type s18)) -> + cstr_55 : cupicef.eps (cupicef.Cumul (cupicef.type s23) (cupicef.type s19)) -> + cstr_56 : cupicef.eps (cupicef.Cumul (cupicef.type s23) (cupicef.type s22)) -> + cstr_57 : + cupicef.eps + (cupicef.Cumul (cupicef.type s23) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) -> + cstr_58 : + cupicef.eps + (cupicef.Cumul (cupicef.type s23) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_59 : cupicef.eps (cupicef.Cumul (cupicef.type s24) (cupicef.type s22)) -> + A : cupicef.Univ (cupicef.type s3) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s3) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s3 s4 A) -> + R_27__27__27_ : + (__ : cupicef.Term (cupicef.type s3) A -> + __0 : cupicef.Term (cupicef.type s3) A -> + cupicef.Univ (cupicef.type s4)) -> + NA : + cupicef.Term (cupicef.type s11) + (Normalizes s3 s10 s11 s4 cstr_1 cstr_14 cstr_15 cstr_17 cstr_18 + cstr_19 A R + (Coq__Classes__CRelationClasses.flip s3 s3 s0 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R_27__27__27_)) -> + B : cupicef.Univ (cupicef.type s5) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s5) (cupicef.axiom (cupicef.type s6))) + (Coq__Classes__CRelationClasses.crelation s5 s6 B) -> + R_27__27_ : + (__ : cupicef.Term (cupicef.type s5) B -> + __0 : cupicef.Term (cupicef.type s5) B -> + cupicef.Univ (cupicef.type s6)) -> + NB : + cupicef.Term (cupicef.type s17) + (Normalizes s5 s16 s17 s6 cstr_3 cstr_24 cstr_25 cstr_27 cstr_28 + cstr_29 B R_27_ + (Coq__Classes__CRelationClasses.flip s5 s5 s1 B B + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + cstr_2 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I)) + R_27__27_)) -> + cupicef.Term (cupicef.type s21) + (Normalizes s2 s23 s21 s8 cstr_8 cstr_11 cstr_10 cstr_33 cstr_34 cstr_35 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) (cupicef.axiom (cupicef.type s2)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => B))) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 cstr_16 cstr_23 cstr_26 + A B R R_27_) + (Coq__Classes__CRelationClasses.flip s2 s2 s7 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) cupicef.I + A (__ : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) cupicef.I + A (__ : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom (cupicef.type s7)) + (cupicef.univ (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.axiom (cupicef.type s8))) cupicef.I) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + cstr_4 + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I)) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 cstr_16 cstr_23 + cstr_26 A B R_27__27__27_ R_27__27_))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + s7 : cupicef.Nat => + s8 : cupicef.Nat => + s9 : cupicef.Nat => + s10 : cupicef.Nat => + s11 : cupicef.Nat => + s12 : cupicef.Nat => + s13 : cupicef.Nat => + s14 : cupicef.Nat => + s15 : cupicef.Nat => + s16 : cupicef.Nat => + s17 : cupicef.Nat => + s18 : cupicef.Nat => + s19 : cupicef.Nat => + s20 : cupicef.Nat => + s21 : cupicef.Nat => + s22 : cupicef.Nat => + s23 : cupicef.Nat => + s24 : cupicef.Nat => + s25 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s0)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s10)) => + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s6)) (cupicef.type s1)) => + cstr_3 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s6)) (cupicef.type s16)) => + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) (cupicef.type s7)) => + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) (cupicef.type s12)) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) (cupicef.type s14)) => + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) (cupicef.type s18)) => + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) (cupicef.type s23)) => + cstr_9 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s21)) (cupicef.type s9)) => + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s21)) => + cstr_11 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s23)) => + cstr_12 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) => + cstr_13 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s8)) => + cstr_14 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s10)) => + cstr_15 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s11)) => + cstr_16 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s8)) => + cstr_17 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s11)) => + cstr_18 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_19 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_20 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_21 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_22 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_23 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s2)) => + cstr_24 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s16)) => + cstr_25 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s17)) => + cstr_26 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s8)) => + cstr_27 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s17)) => + cstr_28 : + cupicef.eps + (cupicef.Cumul (cupicef.type s6) (cupicef.type (cupicef.s cupicef.z))) => + cstr_29 : + cupicef.eps + (cupicef.Cumul (cupicef.type s6) (cupicef.type (cupicef.s cupicef.z))) => + cstr_30 : + cupicef.eps + (cupicef.Cumul (cupicef.type s6) (cupicef.type (cupicef.s cupicef.z))) => + cstr_31 : + cupicef.eps + (cupicef.Cumul (cupicef.type s6) (cupicef.type (cupicef.s cupicef.z))) => + cstr_32 : + cupicef.eps + (cupicef.Cumul (cupicef.type s6) (cupicef.type (cupicef.s cupicef.z))) => + cstr_33 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s21)) => + cstr_34 : + cupicef.eps + (cupicef.Cumul (cupicef.type s8) (cupicef.type (cupicef.s cupicef.z))) => + cstr_35 : + cupicef.eps + (cupicef.Cumul (cupicef.type s8) (cupicef.type (cupicef.s cupicef.z))) => + cstr_36 : + cupicef.eps + (cupicef.Cumul (cupicef.type s8) (cupicef.type (cupicef.s cupicef.z))) => + cstr_37 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s18)) => + cstr_38 : + cupicef.eps (cupicef.Cumul (cupicef.type s10) (cupicef.type s14)) => + cstr_39 : + cupicef.eps (cupicef.Cumul (cupicef.type s10) (cupicef.type s15)) => + cstr_40 : + cupicef.eps (cupicef.Cumul (cupicef.type s10) (cupicef.type s25)) => + cstr_41 : + cupicef.eps (cupicef.Cumul (cupicef.type s11) (cupicef.type s15)) => + cstr_42 : + cupicef.eps (cupicef.Cumul (cupicef.type s12) (cupicef.type s14)) => + cstr_43 : + cupicef.eps (cupicef.Cumul (cupicef.type s12) (cupicef.type s25)) => + cstr_44 : + cupicef.eps (cupicef.Cumul (cupicef.type s13) (cupicef.type s15)) => + cstr_45 : + cupicef.eps (cupicef.Cumul (cupicef.type s16) (cupicef.type s12)) => + cstr_46 : + cupicef.eps (cupicef.Cumul (cupicef.type s16) (cupicef.type s13)) => + cstr_47 : + cupicef.eps (cupicef.Cumul (cupicef.type s17) (cupicef.type s13)) => + cstr_48 : + cupicef.eps (cupicef.Cumul (cupicef.type s20) (cupicef.type s19)) => + cstr_49 : + cupicef.eps (cupicef.Cumul (cupicef.type s21) (cupicef.type s13)) => + cstr_50 : + cupicef.eps (cupicef.Cumul (cupicef.type s21) (cupicef.type s19)) => + cstr_51 : + cupicef.eps (cupicef.Cumul (cupicef.type s21) (cupicef.type s22)) => + cstr_52 : + cupicef.eps + (cupicef.Cumul (cupicef.type s21) (cupicef.type (cupicef.s cupicef.z))) => + cstr_53 : + cupicef.eps (cupicef.Cumul (cupicef.type s23) (cupicef.type s12)) => + cstr_54 : + cupicef.eps (cupicef.Cumul (cupicef.type s23) (cupicef.type s18)) => + cstr_55 : + cupicef.eps (cupicef.Cumul (cupicef.type s23) (cupicef.type s19)) => + cstr_56 : + cupicef.eps (cupicef.Cumul (cupicef.type s23) (cupicef.type s22)) => + cstr_57 : + cupicef.eps + (cupicef.Cumul (cupicef.type s23) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) => + cstr_58 : + cupicef.eps + (cupicef.Cumul (cupicef.type s23) (cupicef.type (cupicef.s cupicef.z))) => + cstr_59 : + cupicef.eps (cupicef.Cumul (cupicef.type s24) (cupicef.type s22)) => + A : cupicef.Univ (cupicef.type s3) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s3) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s3 s4 A) => + R_27__27__27_ : + (__ : cupicef.Term (cupicef.type s3) A -> + __0 : cupicef.Term (cupicef.type s3) A -> + cupicef.Univ (cupicef.type s4)) => + NA : + cupicef.Term (cupicef.type s11) + (Normalizes s3 s10 s11 s4 cstr_1 cstr_14 cstr_15 cstr_17 cstr_18 + cstr_19 A R + (Coq__Classes__CRelationClasses.flip s3 s3 s0 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R_27__27__27_)) => + B : cupicef.Univ (cupicef.type s5) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s5) (cupicef.axiom (cupicef.type s6))) + (Coq__Classes__CRelationClasses.crelation s5 s6 B) => + R_27__27_ : + (__ : cupicef.Term (cupicef.type s5) B -> + __0 : cupicef.Term (cupicef.type s5) B -> + cupicef.Univ (cupicef.type s6)) => + NB : + cupicef.Term (cupicef.type s17) + (Normalizes s5 s16 s17 s6 cstr_3 cstr_24 cstr_25 cstr_27 cstr_28 + cstr_29 B R_27_ + (Coq__Classes__CRelationClasses.flip s5 s5 s1 B B + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + cstr_2 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I)) + R_27__27_)) => + cupicef.cast (cupicef.type s21) (cupicef.type s21) + (Coq__Classes__CRelationClasses.relation__equivalence s2 s23 s21 s8 + cstr_8 cstr_11 cstr_10 cstr_33 cstr_34 cstr_35 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) (cupicef.axiom (cupicef.type s2)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => B))) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 cstr_16 cstr_23 cstr_26 + A B R R_27_) + (Coq__Classes__CRelationClasses.flip s2 s2 s7 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) cupicef.I + A (__ : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) cupicef.I + A (__ : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom (cupicef.type s7)) + (cupicef.univ (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.axiom (cupicef.type s8))) cupicef.I) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + cstr_4 + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I)) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 cstr_16 cstr_23 + cstr_26 A B R_27__27__27_ R_27__27_))) + (Normalizes s2 s23 s21 s8 cstr_8 cstr_11 cstr_10 cstr_33 cstr_34 cstr_35 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) (cupicef.axiom (cupicef.type s2)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => B))) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 cstr_16 cstr_23 cstr_26 + A B R R_27_) + (Coq__Classes__CRelationClasses.flip s2 s2 s7 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) cupicef.I + A (__ : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) cupicef.I + A (__ : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom (cupicef.type s7)) + (cupicef.univ (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.axiom (cupicef.type s8))) cupicef.I) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + cstr_4 + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I)) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 cstr_16 cstr_23 + cstr_26 A B R_27__27__27_ R_27__27_))) + cupicef.I + ((lemma : + cupicef.Term (cupicef.type s11) + (Coq__Classes__CRelationClasses.relation__equivalence s3 s10 s11 + s4 cstr_1 cstr_14 cstr_15 cstr_17 cstr_18 cstr_19 A R + (Coq__Classes__CRelationClasses.flip s3 s3 s0 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R_27__27__27_)) => + trans__co__eq__inv__arrow__morphism s23 s18 s19 s20 s9 s9 s21 s21 + cstr_9 cstr_9 cstr_54 cstr_55 cstr_57 cstr_58 cstr_48 cstr_37 + cstr_50 cupicef.I cstr_52 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s8)))) + (cupicef.axiom (cupicef.type s23)) + (cupicef.univ + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s8)))) + cupicef.I) + (cupicef.univ (cupicef.type s23) + (cupicef.axiom (cupicef.type s23)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s23))) + cstr_11 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) + (cupicef.type s23))) + cstr_8 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s2 s8 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A (__ : cupicef.Term (cupicef.type s3) A => B))))) + (Coq__Classes__CRelationClasses.relation__equivalence s2 s23 s21 s8 + cstr_8 cstr_11 cstr_10 cstr_33 cstr_34 cstr_35 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A (__ : cupicef.Term (cupicef.type s3) A => B)))) + (Coq__Classes__CRelationClasses.Equivalence__Transitive s23 s21 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s8)))) + (cupicef.axiom (cupicef.type s23)) + (cupicef.univ + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s8)))) + cupicef.I) + (cupicef.univ (cupicef.type s23) + (cupicef.axiom (cupicef.type s23)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s23))) + cstr_11 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) + (cupicef.type s23))) + cstr_8 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s2 s8 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => B))))) + (Coq__Classes__CRelationClasses.relation__equivalence s2 s23 s21 + s8 cstr_8 cstr_11 cstr_10 cstr_33 cstr_34 cstr_35 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A (__ : cupicef.Term (cupicef.type s3) A => B)))) + (Coq__Classes__CRelationClasses.relation__equivalence__equivalence + s2 s23 s21 s8 cstr_8 cstr_11 cstr_10 cstr_33 cstr_34 cstr_35 + cstr_36 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A (__ : cupicef.Term (cupicef.type s3) A => B))))) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 cstr_16 cstr_23 + cstr_26 A B R R_27_) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 cstr_16 cstr_23 + cstr_26 A B + (Coq__Classes__CRelationClasses.flip s3 s3 s0 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R_27__27__27_) + R_27_) + (respectful__morphism s14 s15 s10 s11 s12 s13 s3 s4 s16 s17 s23 s21 + s5 s6 s2 s8 cstr_1 cstr_3 cstr_8 cstr_38 cstr_39 cstr_41 cstr_42 + cstr_44 cstr_14 cstr_15 cstr_12 cstr_13 cstr_17 cstr_16 cstr_18 + cstr_19 cstr_21 cstr_22 cstr_45 cstr_46 cstr_47 cstr_53 cstr_49 + cstr_24 cstr_25 cstr_23 cstr_27 cstr_26 cstr_28 cstr_29 cstr_31 + cstr_32 cstr_11 cstr_10 cstr_33 cstr_34 cstr_35 A B R + (Coq__Classes__CRelationClasses.flip s3 s3 s0 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R_27__27__27_) + lemma R_27_ R_27_ + (reflexive__proper__proxy s16 s17 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s5) + (cupicef.axiom (cupicef.type s6)))) + (cupicef.axiom (cupicef.type s16)) + (cupicef.univ + (cupicef.sup (cupicef.type s5) + (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom + (cupicef.sup (cupicef.type s5) + (cupicef.axiom (cupicef.type s6)))) + cupicef.I) + (cupicef.univ (cupicef.type s16) + (cupicef.axiom (cupicef.type s16)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s16))) + cstr_24 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s6)) + (cupicef.type s16))) + cstr_3 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s5 s6 B)) + (Coq__Classes__CRelationClasses.relation__equivalence s5 s16 + s17 s6 cstr_3 cstr_24 cstr_25 cstr_27 cstr_28 cstr_29 B) + (Coq__Classes__CRelationClasses.Equivalence__Reflexive s16 s17 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s5) + (cupicef.axiom (cupicef.type s6)))) + (cupicef.axiom (cupicef.type s16)) + (cupicef.univ + (cupicef.sup (cupicef.type s5) + (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom + (cupicef.sup (cupicef.type s5) + (cupicef.axiom (cupicef.type s6)))) + cupicef.I) + (cupicef.univ (cupicef.type s16) + (cupicef.axiom (cupicef.type s16)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type s16))) + cstr_24 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s6)) + (cupicef.type s16))) + cstr_3 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s5 s6 B)) + (Coq__Classes__CRelationClasses.relation__equivalence s5 + s16 s17 s6 cstr_3 cstr_24 cstr_25 cstr_27 cstr_28 + cstr_29 B) + (Coq__Classes__CRelationClasses.relation__equivalence__equivalence + s5 s16 s17 s6 cstr_3 cstr_24 cstr_25 cstr_27 cstr_28 + cstr_29 cstr_30 B)) + R_27_)) + (Coq__Classes__CRelationClasses.flip s2 s2 s7 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A (__ : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A (__ : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom (cupicef.type s7)) + (cupicef.univ (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.axiom (cupicef.type s8))) cupicef.I) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + cstr_4 + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I)) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 cstr_16 cstr_23 + cstr_26 A B R_27__27__27_ R_27__27_)) + (Coq__Classes__CRelationClasses.flip s2 s2 s7 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A (__ : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A (__ : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom (cupicef.type s7)) + (cupicef.univ (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.axiom (cupicef.type s8))) cupicef.I) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + cstr_4 + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I)) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 cstr_16 cstr_23 + cstr_26 A B R_27__27__27_ R_27__27_)) + (eq__proper__proxy s23 s20 cstr_57 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s8)))) + (cupicef.axiom (cupicef.type s23)) + (cupicef.univ + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s8)))) + cupicef.I) + (cupicef.univ (cupicef.type s23) + (cupicef.axiom (cupicef.type s23)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s23))) + cstr_11 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) + (cupicef.type s23))) + cstr_8 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s2 s8 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => B))))) + (Coq__Classes__CRelationClasses.flip s2 s2 s7 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A (__ : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A (__ : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom (cupicef.type s7)) + (cupicef.univ (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.axiom (cupicef.type s8))) + cupicef.I) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + cstr_4 + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I)) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 cstr_16 cstr_23 + cstr_26 A B R_27__27__27_ R_27__27_)))) + NA + ((lemma : + cupicef.Term (cupicef.type s17) + (Coq__Classes__CRelationClasses.relation__equivalence s5 s16 + s17 s6 cstr_3 cstr_24 cstr_25 cstr_27 cstr_28 cstr_29 B + R_27_ + (Coq__Classes__CRelationClasses.flip s5 s5 s1 B B + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + cstr_2 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I)) + R_27__27_)) => + trans__co__eq__inv__arrow__morphism s23 s18 s22 s24 s9 s9 s21 s21 + cstr_9 cstr_9 cstr_54 cstr_56 cstr_57 cstr_58 cstr_59 cstr_37 + cstr_51 cupicef.I cstr_52 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s8)))) + (cupicef.axiom (cupicef.type s23)) + (cupicef.univ + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s8)))) + cupicef.I) + (cupicef.univ (cupicef.type s23) + (cupicef.axiom (cupicef.type s23)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s23))) + cstr_11 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) + (cupicef.type s23))) + cstr_8 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s2 s8 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => B))))) + (Coq__Classes__CRelationClasses.relation__equivalence s2 s23 s21 + s8 cstr_8 cstr_11 cstr_10 cstr_33 cstr_34 cstr_35 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A (__ : cupicef.Term (cupicef.type s3) A => B)))) + (Coq__Classes__CRelationClasses.Equivalence__Transitive s23 s21 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s8)))) + (cupicef.axiom (cupicef.type s23)) + (cupicef.univ + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s8)))) + cupicef.I) + (cupicef.univ (cupicef.type s23) + (cupicef.axiom (cupicef.type s23)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s23))) + cstr_11 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) + (cupicef.type s23))) + cstr_8 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s2 s8 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) + (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => B))))) + (Coq__Classes__CRelationClasses.relation__equivalence s2 s23 + s21 s8 cstr_8 cstr_11 cstr_10 cstr_33 cstr_34 cstr_35 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => B)))) + (Coq__Classes__CRelationClasses.relation__equivalence__equivalence + s2 s23 s21 s8 cstr_8 cstr_11 cstr_10 cstr_33 cstr_34 + cstr_35 cstr_36 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => B))))) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 cstr_16 cstr_23 + cstr_26 A B + (Coq__Classes__CRelationClasses.flip s3 s3 s0 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R_27__27__27_) + R_27_) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 cstr_16 cstr_23 + cstr_26 A B + (Coq__Classes__CRelationClasses.flip s3 s3 s0 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R_27__27__27_) + (Coq__Classes__CRelationClasses.flip s5 s5 s1 B B + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + cstr_2 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I)) + R_27__27_)) + (Reflexive__partial__app__morphism s10 s12 s25 s15 s11 s13 + cstr_40 cstr_39 cstr_43 cstr_41 cstr_44 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.axiom (cupicef.type s4)))) + (cupicef.axiom (cupicef.type s10)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.axiom (cupicef.type s4)))) + cupicef.I) + (cupicef.univ (cupicef.type s10) + (cupicef.axiom (cupicef.type s10)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s10))) + cstr_14 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) + (cupicef.type s10))) + cstr_1 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s3 s4 A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s5) + (cupicef.sup (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s8)))))) + (cupicef.axiom (cupicef.type s12)) + (cupicef.univ + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s5) + (cupicef.sup (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s8))))) + (cupicef.axiom + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s5) + (cupicef.sup (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.type s8)))))) + cupicef.I) + (cupicef.univ (cupicef.type s12) + (cupicef.axiom (cupicef.type s12)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s12))) + (cupicef.trans (cupicef.type s5) + (cupicef.SortSome (cupicef.type s2)) cstr_23 + (cupicef.SortSome (cupicef.type s23)) cstr_11 + (cupicef.SortSome (cupicef.type s12)) cstr_53 + cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s12))) + (cupicef.trans (cupicef.type s2) + (cupicef.SortSome (cupicef.type s23)) cstr_11 + (cupicef.SortSome (cupicef.type s12)) cstr_53 + cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) + (cupicef.type s12))) + cstr_5 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s6)) + (cupicef.type s12))) + (cupicef.trans (cupicef.axiom (cupicef.type s6)) + (cupicef.SortSome (cupicef.type s16)) cstr_3 + (cupicef.SortSome (cupicef.type s12)) cstr_45 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod + (cupicef.sup (cupicef.type s5) + (cupicef.axiom (cupicef.type s6))) + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s8))) + (cupicef.rule + (cupicef.sup (cupicef.type s5) + (cupicef.axiom (cupicef.type s6))) + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s8)))) + cupicef.I + (Coq__Classes__CRelationClasses.crelation s5 s6 B) + (R_27_0 : + cupicef.Term + (cupicef.sup (cupicef.type s5) + (cupicef.axiom (cupicef.type s6))) + (Coq__Classes__CRelationClasses.crelation s5 s6 B) => + Coq__Classes__CRelationClasses.crelation s2 s8 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) + (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) + (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) + (cupicef.type s5)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => B)))))) + (Coq__Classes__CRelationClasses.relation__equivalence s3 s10 + s11 s4 cstr_1 cstr_14 cstr_15 cstr_17 cstr_18 cstr_19 A) + (respectful s16 s17 s23 s21 s12 s13 cstr_45 cstr_46 cstr_47 + cstr_53 cstr_49 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s5) + (cupicef.axiom (cupicef.type s6)))) + (cupicef.axiom (cupicef.type s16)) + (cupicef.univ + (cupicef.sup (cupicef.type s5) + (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom + (cupicef.sup (cupicef.type s5) + (cupicef.axiom (cupicef.type s6)))) + cupicef.I) + (cupicef.univ (cupicef.type s16) + (cupicef.axiom (cupicef.type s16)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type s16))) + cstr_24 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s6)) + (cupicef.type s16))) + cstr_3 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s5 s6 B)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s8)))) + (cupicef.axiom (cupicef.type s23)) + (cupicef.univ + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s8)))) + cupicef.I) + (cupicef.univ (cupicef.type s23) + (cupicef.axiom (cupicef.type s23)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) + (cupicef.type s23))) + cstr_11 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) + (cupicef.type s23))) + cstr_8 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s2 s8 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) + (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) + (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) + (cupicef.type s5)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => B))))) + (Coq__Classes__CRelationClasses.relation__equivalence s5 + s16 s17 s6 cstr_3 cstr_24 cstr_25 cstr_27 cstr_28 + cstr_29 B) + (Coq__Classes__CRelationClasses.relation__equivalence s2 + s23 s21 s8 cstr_8 cstr_11 cstr_10 cstr_33 cstr_34 + cstr_35 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) + (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => B))))) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 cstr_16 cstr_23 + cstr_26 A B) + (respectful__morphism s25 s15 s10 s11 s12 s13 s3 s4 s16 s17 + s23 s21 s5 s6 s2 s8 cstr_1 cstr_3 cstr_8 cstr_40 cstr_39 + cstr_41 cstr_43 cstr_44 cstr_14 cstr_15 cstr_12 cstr_13 + cstr_17 cstr_16 cstr_18 cstr_19 cstr_21 cstr_22 cstr_45 + cstr_46 cstr_47 cstr_53 cstr_49 cstr_24 cstr_25 cstr_23 + cstr_27 cstr_26 cstr_28 cstr_29 cstr_31 cstr_32 cstr_11 + cstr_10 cstr_33 cstr_34 cstr_35 A B) + (Coq__Classes__CRelationClasses.flip s3 s3 s0 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R_27__27__27_) + (reflexive__proper__proxy s10 s11 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.axiom (cupicef.type s4)))) + (cupicef.axiom (cupicef.type s10)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.axiom (cupicef.type s4)))) + cupicef.I) + (cupicef.univ (cupicef.type s10) + (cupicef.axiom (cupicef.type s10)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) + (cupicef.type s10))) + cstr_14 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) + (cupicef.type s10))) + cstr_1 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s3 s4 A)) + (Coq__Classes__CRelationClasses.relation__equivalence s3 + s10 s11 s4 cstr_1 cstr_14 cstr_15 cstr_17 cstr_18 + cstr_19 A) + (Coq__Classes__CRelationClasses.Equivalence__Reflexive s10 + s11 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.axiom (cupicef.type s4)))) + (cupicef.axiom (cupicef.type s10)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.axiom (cupicef.type s4)))) + cupicef.I) + (cupicef.univ (cupicef.type s10) + (cupicef.axiom (cupicef.type s10)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) + (cupicef.type s10))) + cstr_14 + (cupicef.BoolSome + (cupicef.Cumul + (cupicef.axiom (cupicef.type s4)) + (cupicef.type s10))) + cstr_1 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s3 s4 A)) + (Coq__Classes__CRelationClasses.relation__equivalence s3 + s10 s11 s4 cstr_1 cstr_14 cstr_15 cstr_17 cstr_18 + cstr_19 A) + (Coq__Classes__CRelationClasses.relation__equivalence__equivalence + s3 s10 s11 s4 cstr_1 cstr_14 cstr_15 cstr_17 cstr_18 + cstr_19 cstr_20 A)) + (Coq__Classes__CRelationClasses.flip s3 s3 s0 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R_27__27__27_)) + R_27_ + (Coq__Classes__CRelationClasses.flip s5 s5 s1 B B + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + cstr_2 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I)) + R_27__27_) + lemma) + (Coq__Classes__CRelationClasses.flip s2 s2 s7 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A (__ : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A (__ : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom (cupicef.type s7)) + (cupicef.univ (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.axiom (cupicef.type s8))) + cupicef.I) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + cstr_4 + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I)) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 cstr_16 cstr_23 + cstr_26 A B R_27__27__27_ R_27__27_)) + (Coq__Classes__CRelationClasses.flip s2 s2 s7 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A (__ : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A (__ : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom (cupicef.type s7)) + (cupicef.univ (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.axiom (cupicef.type s8))) + cupicef.I) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + cstr_4 + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I)) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 cstr_16 cstr_23 + cstr_26 A B R_27__27__27_ R_27__27_)) + (eq__proper__proxy s23 s24 cstr_57 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s8)))) + (cupicef.axiom (cupicef.type s23)) + (cupicef.univ + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s8)))) + cupicef.I) + (cupicef.univ (cupicef.type s23) + (cupicef.axiom (cupicef.type s23)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s23))) + cstr_11 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) + (cupicef.type s23))) + cstr_8 cupicef.BoolNone) + (Coq__Classes__CRelationClasses.crelation s2 s8 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) + (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => B))))) + (Coq__Classes__CRelationClasses.flip s2 s2 s7 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom (cupicef.type s7)) + (cupicef.univ (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.axiom (cupicef.type s8))) + cupicef.I) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + cstr_4 + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I)) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 cstr_16 + cstr_23 cstr_26 A B R_27__27__27_ R_27__27_)))) + NB + (cupicef.cast + (cupicef.rule + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + (cupicef.rule + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + (cupicef.type s8))) + (cupicef.type s21) + (cupicef.prod + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + (cupicef.rule + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + (cupicef.type s8)) + (cupicef.rule + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + (cupicef.rule + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + (cupicef.type s8))) + cupicef.I + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A (__ : cupicef.Term (cupicef.type s3) A => B)) + (x : + (__ : cupicef.Term (cupicef.type s3) A -> + cupicef.Term (cupicef.type s5) B) => + cupicef.prod + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + (cupicef.type s8) + (cupicef.rule + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + (cupicef.type s8)) + cupicef.I + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => B)) + (y : + (__ : cupicef.Term (cupicef.type s3) A -> + cupicef.Term (cupicef.type s5) B) => + Coq__Init__Datatypes.prod (cupicef.type s8) + (cupicef.type s8) + (cupicef.prod (cupicef.type s8) (cupicef.type s8) + (cupicef.rule (cupicef.type s8) (cupicef.type s8)) + cupicef.I + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 + cstr_16 cstr_23 cstr_26 A B + (Coq__Classes__CRelationClasses.flip s3 s3 s0 A + A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + cupicef.I)) + R_27__27__27_) + (Coq__Classes__CRelationClasses.flip s5 s5 s1 B + B + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) + cupicef.I) + cstr_2 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) + cupicef.I)) + R_27__27_) + x y) + (__ : + cupicef.Term (cupicef.type s8) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 + cstr_16 cstr_23 cstr_26 A B + (Coq__Classes__CRelationClasses.flip s3 s3 + s0 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.axiom + (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + cupicef.I)) + R_27__27__27_) + (Coq__Classes__CRelationClasses.flip s5 s5 + s1 B B + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom + (cupicef.axiom + (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) + cupicef.I) + cstr_2 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) + cupicef.I)) + R_27__27_) + x y) => + Coq__Classes__CRelationClasses.flip s2 s2 s7 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) + (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) + (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) + (cupicef.type s5) + (cupicef.rule (cupicef.type s3) + (cupicef.type s5)) + cupicef.I A + (__0 : cupicef.Term (cupicef.type s3) A => + B))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) + (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) + (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) + (cupicef.type s5) + (cupicef.rule (cupicef.type s3) + (cupicef.type s5)) + cupicef.I A + (__0 : cupicef.Term (cupicef.type s3) A => + B))) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom (cupicef.type s7)) + (cupicef.univ + (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s8))) + cupicef.I) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I) + cstr_4 + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) + cupicef.I)) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 + cstr_16 cstr_23 cstr_26 A B R_27__27__27_ + R_27__27_) + x y)) + (cupicef.prod (cupicef.type s8) (cupicef.type s8) + (cupicef.rule (cupicef.type s8) (cupicef.type s8)) + cupicef.I + (Coq__Classes__CRelationClasses.flip s2 s2 s7 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) + (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) + (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) + (cupicef.type s5) + (cupicef.rule (cupicef.type s3) + (cupicef.type s5)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => + B))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) + (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) + (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) + (cupicef.type s5) + (cupicef.rule (cupicef.type s3) + (cupicef.type s5)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => + B))) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom (cupicef.type s7)) + (cupicef.univ + (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s8))) + cupicef.I) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I) + cstr_4 + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) + cupicef.I)) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 + cstr_16 cstr_23 cstr_26 A B R_27__27__27_ + R_27__27_) + x y) + (__ : + cupicef.Term (cupicef.type s8) + (Coq__Classes__CRelationClasses.flip s2 s2 s7 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) + (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) + (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) + (cupicef.type s5) + (cupicef.rule (cupicef.type s3) + (cupicef.type s5)) + cupicef.I A + (__ : + cupicef.Term (cupicef.type s3) A => + B))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) + (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) + (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) + (cupicef.type s5) + (cupicef.rule (cupicef.type s3) + (cupicef.type s5)) + cupicef.I A + (__ : + cupicef.Term (cupicef.type s3) A => + B))) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom (cupicef.type s7)) + (cupicef.univ + (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s8))) + cupicef.I) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I) + cstr_4 + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) + cupicef.I)) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 + cstr_13 cstr_16 cstr_23 cstr_26 A B + R_27__27__27_ R_27__27_) + x y) => + respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 + cstr_16 cstr_23 cstr_26 A B + (Coq__Classes__CRelationClasses.flip s3 s3 s0 A + A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + cupicef.I)) + R_27__27__27_) + (Coq__Classes__CRelationClasses.flip s5 s5 s1 B + B + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) + cupicef.I) + cstr_2 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) + cupicef.I)) + R_27__27_) + x y))))) + (Coq__Classes__CRelationClasses.relation__equivalence s2 s23 + s21 s8 cstr_8 cstr_11 cstr_10 cstr_33 cstr_34 cstr_35 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => B))) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 cstr_16 + cstr_23 cstr_26 A B + (Coq__Classes__CRelationClasses.flip s3 s3 s0 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R_27__27__27_) + (Coq__Classes__CRelationClasses.flip s5 s5 s1 B B + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + cstr_2 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I)) + R_27__27_)) + (Coq__Classes__CRelationClasses.flip s2 s2 s7 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) + (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) + (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) (cupicef.type s5)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom (cupicef.type s7)) + (cupicef.univ (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.axiom (cupicef.type s8))) + cupicef.I) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + cstr_4 + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I)) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 cstr_16 + cstr_23 cstr_26 A B R_27__27__27_ R_27__27_))) + cupicef.I + (x : + (__ : cupicef.Term (cupicef.type s3) A -> + cupicef.Term (cupicef.type s5) B) => + y : + (__ : cupicef.Term (cupicef.type s3) A -> + cupicef.Term (cupicef.type s5) B) => + Coq__Init__Datatypes.pair + (cupicef.cast (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_34 + (cupicef.prod (cupicef.type s8) (cupicef.type s8) + (cupicef.rule (cupicef.type s8) (cupicef.type s8)) + cupicef.I + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 cstr_16 + cstr_23 cstr_26 A B + (Coq__Classes__CRelationClasses.flip s3 s3 s0 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + cupicef.I)) + R_27__27__27_) + (Coq__Classes__CRelationClasses.flip s5 s5 s1 B B + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) + cupicef.I) + cstr_2 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) + cupicef.I)) + R_27__27_) + x y) + (__ : + cupicef.Term (cupicef.type s8) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 + cstr_16 cstr_23 cstr_26 A B + (Coq__Classes__CRelationClasses.flip s3 s3 s0 + A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + cupicef.I)) + R_27__27__27_) + (Coq__Classes__CRelationClasses.flip s5 s5 s1 + B B + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) + cupicef.I) + cstr_2 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) + cupicef.I)) + R_27__27_) + x y) => + Coq__Classes__CRelationClasses.flip s2 s2 s7 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) + (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) + (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) + (cupicef.type s5) + (cupicef.rule (cupicef.type s3) + (cupicef.type s5)) + cupicef.I A + (__0 : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) + (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) + (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) + (cupicef.type s5) + (cupicef.rule (cupicef.type s3) + (cupicef.type s5)) + cupicef.I A + (__0 : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom (cupicef.type s7)) + (cupicef.univ (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s8))) + cupicef.I) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + cstr_4 + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I)) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 + cstr_16 cstr_23 cstr_26 A B R_27__27__27_ + R_27__27_) + x y))) + (cupicef.cast (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_35 + (cupicef.prod (cupicef.type s8) (cupicef.type s8) + (cupicef.rule (cupicef.type s8) (cupicef.type s8)) + cupicef.I + (Coq__Classes__CRelationClasses.flip s2 s2 s7 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) + (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) + (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) + (cupicef.type s5) + (cupicef.rule (cupicef.type s3) + (cupicef.type s5)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) + (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) + (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) + (cupicef.type s5) + (cupicef.rule (cupicef.type s3) + (cupicef.type s5)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom (cupicef.type s7)) + (cupicef.univ (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s8))) + cupicef.I) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + cstr_4 + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I)) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 + cstr_16 cstr_23 cstr_26 A B R_27__27__27_ + R_27__27_) + x y) + (__ : + cupicef.Term (cupicef.type s8) + (Coq__Classes__CRelationClasses.flip s2 s2 s7 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) + (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) + (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) + (cupicef.type s5) + (cupicef.rule (cupicef.type s3) + (cupicef.type s5)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => + B))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) + (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) + (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) + (cupicef.type s5) + (cupicef.rule (cupicef.type s3) + (cupicef.type s5)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => + B))) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom (cupicef.type s7)) + (cupicef.univ + (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s8))) + cupicef.I) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I) + cstr_4 + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) + cupicef.I)) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 + cstr_16 cstr_23 cstr_26 A B R_27__27__27_ + R_27__27_) + x y) => + respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 cstr_16 + cstr_23 cstr_26 A B + (Coq__Classes__CRelationClasses.flip s3 s3 s0 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + cupicef.I)) + R_27__27__27_) + (Coq__Classes__CRelationClasses.flip s5 s5 s1 B B + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) + cupicef.I) + cstr_2 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) + cupicef.I)) + R_27__27_) + x y))) + (X : + cupicef.Term (cupicef.type s8) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 cstr_16 + cstr_23 cstr_26 A B + (Coq__Classes__CRelationClasses.flip s3 s3 s0 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R_27__27__27_) + (Coq__Classes__CRelationClasses.flip s5 s5 s1 B B + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + cstr_2 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I)) + R_27__27_) + x y) => + cupicef.cast + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) (cupicef.type s6)))) + (cupicef.type s8) + (cupicef.prod (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) (cupicef.type s6))) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.type s6)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s3) A => + cupicef.prod (cupicef.type s3) + (cupicef.rule (cupicef.type s4) (cupicef.type s6)) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.type s6))) + cupicef.I A + (y0 : cupicef.Term (cupicef.type s3) A => + cupicef.prod (cupicef.type s4) (cupicef.type s6) + (cupicef.rule (cupicef.type s4) + (cupicef.type s6)) + cupicef.I (R_27__27__27_ x0 y0) + (X0 : + cupicef.Term (cupicef.type s4) + (R_27__27__27_ x0 y0) => + Coq__Classes__CRelationClasses.flip s5 s5 s1 B + B + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) + cupicef.I) + cstr_2 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) + cupicef.I)) + R_27__27_ (x y0) (y x0))))) + (Coq__Classes__CRelationClasses.flip s2 s2 s7 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) + (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) + (cupicef.type s5)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) + (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) + (cupicef.type s5)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom (cupicef.type s7)) + (cupicef.univ (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.axiom (cupicef.type s8))) + cupicef.I) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + cstr_4 + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I)) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 cstr_16 + cstr_23 cstr_26 A B R_27__27__27_ R_27__27_) + x y) + cupicef.I + (x0 : cupicef.Term (cupicef.type s3) A => + y0 : cupicef.Term (cupicef.type s3) A => + X0 : + cupicef.Term (cupicef.type s4) (R_27__27__27_ x0 y0) => + (X1 : + (y1 : cupicef.Term (cupicef.type s3) A -> + __ : + cupicef.Term (cupicef.type s4) + (Coq__Classes__CRelationClasses.flip s3 s3 s0 A + A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + cupicef.I)) + R_27__27__27_ y0 y1) -> + cupicef.Term (cupicef.type s6) + (Coq__Classes__CRelationClasses.flip s5 s5 s1 B B + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) + cupicef.I) + cstr_2 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) + cupicef.I)) + R_27__27_ (x y0) (y y1))) => + (X2 : + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Classes__CRelationClasses.flip s3 s3 s0 A + A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + cupicef.I)) + R_27__27__27_ y0 x0) -> + cupicef.Term (cupicef.type s6) + (Coq__Classes__CRelationClasses.flip s5 s5 s1 B B + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) + cupicef.I) + cstr_2 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) + cupicef.I)) + R_27__27_ (x y0) (y x0))) => + (X3 : + cupicef.Term (cupicef.type s6) + (Coq__Classes__CRelationClasses.flip s5 s5 s1 B B + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) + cupicef.I) + cstr_2 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) + cupicef.I)) + R_27__27_ (x y0) (y x0)) => + X3) (X2 X0)) (X1 x0)) + (X y0))) + (X : + cupicef.Term (cupicef.type s8) + (Coq__Classes__CRelationClasses.flip s2 s2 s7 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) + (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) + (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) + (cupicef.type s5)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type s3) + (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type s3) + (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type s2))) + cstr_23 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) + (cupicef.type s2))) + cstr_12 cupicef.BoolNone) + (cupicef.prod (cupicef.type s3) (cupicef.type s5) + (cupicef.rule (cupicef.type s3) + (cupicef.type s5)) + cupicef.I A + (__ : cupicef.Term (cupicef.type s3) A => B))) + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom (cupicef.type s7)) + (cupicef.univ (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s8))) + cupicef.I) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + cstr_4 + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I)) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 + cstr_16 cstr_23 cstr_26 A B R_27__27__27_ + R_27__27_) + x y) => + cupicef.cast + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) (cupicef.type s6)))) + (cupicef.type s8) + (cupicef.prod (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) (cupicef.type s6))) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.type s6)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s3) A => + cupicef.prod (cupicef.type s3) + (cupicef.rule (cupicef.type s4) (cupicef.type s6)) + (cupicef.rule (cupicef.type s3) + (cupicef.rule (cupicef.type s4) + (cupicef.type s6))) + cupicef.I A + (y0 : cupicef.Term (cupicef.type s3) A => + cupicef.prod (cupicef.type s4) (cupicef.type s6) + (cupicef.rule (cupicef.type s4) + (cupicef.type s6)) + cupicef.I + (Coq__Classes__CRelationClasses.flip s3 s3 s0 A + A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + cupicef.I)) + R_27__27__27_ x0 y0) + (X0 : + cupicef.Term (cupicef.type s4) + (Coq__Classes__CRelationClasses.flip s3 s3 + s0 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.axiom + (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + cupicef.I)) + R_27__27__27_ x0 y0) => + R_27__27_ (y y0) (x x0))))) + (respectful s3 s4 s5 s6 s2 s8 cstr_12 cstr_13 cstr_16 + cstr_23 cstr_26 A B + (Coq__Classes__CRelationClasses.flip s3 s3 s0 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R_27__27__27_) + (Coq__Classes__CRelationClasses.flip s5 s5 s1 B B + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s6))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.axiom (cupicef.type s6)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s6))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + cstr_2 + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I)) + R_27__27_) + x y) + cupicef.I + (x0 : cupicef.Term (cupicef.type s3) A => + y0 : cupicef.Term (cupicef.type s3) A => + X0 : + cupicef.Term (cupicef.type s4) + (Coq__Classes__CRelationClasses.flip s3 s3 s0 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R_27__27__27_ x0 y0) => + (X1 : + (y1 : cupicef.Term (cupicef.type s3) A -> + __ : + cupicef.Term (cupicef.type s4) + (R_27__27__27_ y0 y1) -> + cupicef.Term (cupicef.type s6) + (R_27__27_ (y y0) (x y1))) => + (X2 : + (__ : + cupicef.Term (cupicef.type s4) + (R_27__27__27_ y0 x0) -> + cupicef.Term (cupicef.type s6) + (R_27__27_ (y y0) (x x0))) => + (X3 : + cupicef.Term (cupicef.type s6) + (R_27__27_ (y y0) (x x0)) => + X3) (X2 X0)) (X1 x0)) + (X y0))))))). + +thm proper__sym__flip : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s6)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s0)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s0)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s1)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s1)) -> + A : cupicef.Univ (cupicef.type s4) -> + R1 : + cupicef.Term + (cupicef.sup (cupicef.type s4) (cupicef.axiom (cupicef.type s5))) + (Coq__Classes__CRelationClasses.crelation s4 s5 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s4) (cupicef.type s5)) + (Coq__Classes__CRelationClasses.Symmetric s4 s5 A R1) -> + B : cupicef.Univ (cupicef.type s2) -> + R2 : + cupicef.Term + (cupicef.sup (cupicef.type s2) (cupicef.axiom (cupicef.type s3))) + (Coq__Classes__CRelationClasses.crelation s2 s3 B) -> + f : + (__ : cupicef.Term (cupicef.type s4) A -> + cupicef.Term (cupicef.type s2) B) -> + H0 : + cupicef.Term (cupicef.type s1) + (Proper s0 s1 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s2) (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.sup (cupicef.type s2) (cupicef.type s4)) + (cupicef.axiom + (cupicef.sup (cupicef.type s2) (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s4) (cupicef.type s0))) + cstr_3 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s0))) + cstr_1 cupicef.BoolNone) + (cupicef.prod (cupicef.type s4) (cupicef.type s2) + (cupicef.rule (cupicef.type s4) (cupicef.type s2)) cupicef.I A + (__ : cupicef.Term (cupicef.type s4) A => B))) + (respectful s4 s5 s2 s3 s0 s1 cstr_3 cstr_4 cstr_5 cstr_1 cstr_2 A B + R1 R2) + f) -> + cupicef.Term (cupicef.type s1) + (Proper s0 s1 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s2) (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.sup (cupicef.type s2) (cupicef.type s4)) + (cupicef.axiom (cupicef.sup (cupicef.type s2) (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s4) (cupicef.type s0))) + cstr_3 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s0))) + cstr_1 cupicef.BoolNone) + (cupicef.prod (cupicef.type s4) (cupicef.type s2) + (cupicef.rule (cupicef.type s4) (cupicef.type s2)) cupicef.I A + (__ : cupicef.Term (cupicef.type s4) A => B))) + (respectful s4 s5 s2 s3 s0 s1 cstr_3 cstr_4 cstr_5 cstr_1 cstr_2 A B + R1 + (Coq__Classes__CRelationClasses.flip s2 s2 s6 B B + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s6)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R2)) + f) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s6)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s0)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s0)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s1)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s1)) => + A : cupicef.Univ (cupicef.type s4) => + R1 : + cupicef.Term + (cupicef.sup (cupicef.type s4) (cupicef.axiom (cupicef.type s5))) + (Coq__Classes__CRelationClasses.crelation s4 s5 A) => + Sym : + cupicef.Term (cupicef.sup (cupicef.type s4) (cupicef.type s5)) + (Coq__Classes__CRelationClasses.Symmetric s4 s5 A R1) => + B : cupicef.Univ (cupicef.type s2) => + R2 : + cupicef.Term + (cupicef.sup (cupicef.type s2) (cupicef.axiom (cupicef.type s3))) + (Coq__Classes__CRelationClasses.crelation s2 s3 B) => + f : + (__ : cupicef.Term (cupicef.type s4) A -> + cupicef.Term (cupicef.type s2) B) => + Hf : + cupicef.Term (cupicef.type s1) + (Proper s0 s1 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s2) (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.sup (cupicef.type s2) (cupicef.type s4)) + (cupicef.axiom + (cupicef.sup (cupicef.type s2) (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s4) (cupicef.type s0))) + cstr_3 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s0))) + cstr_1 cupicef.BoolNone) + (cupicef.prod (cupicef.type s4) (cupicef.type s2) + (cupicef.rule (cupicef.type s4) (cupicef.type s2)) cupicef.I + A (__ : cupicef.Term (cupicef.type s4) A => B))) + (respectful s4 s5 s2 s3 s0 s1 cstr_3 cstr_4 cstr_5 cstr_1 cstr_2 A + B R1 R2) + f) => + cupicef.cast + (cupicef.rule (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.rule (cupicef.type s5) (cupicef.type s3)))) + (cupicef.type s1) + (cupicef.prod (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.rule (cupicef.type s5) (cupicef.type s3))) + (cupicef.rule (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.rule (cupicef.type s5) (cupicef.type s3)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s4) A => + cupicef.prod (cupicef.type s4) + (cupicef.rule (cupicef.type s5) (cupicef.type s3)) + (cupicef.rule (cupicef.type s4) + (cupicef.rule (cupicef.type s5) (cupicef.type s3))) + cupicef.I A + (x_27_ : cupicef.Term (cupicef.type s4) A => + cupicef.prod (cupicef.type s5) (cupicef.type s3) + (cupicef.rule (cupicef.type s5) (cupicef.type s3)) cupicef.I + (R1 x x_27_) + (Hxx_27_ : cupicef.Term (cupicef.type s5) (R1 x x_27_) => + R2 (f x_27_) (f x))))) + (Proper s0 s1 + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s2) (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.sup (cupicef.type s2) (cupicef.type s4)) + (cupicef.axiom + (cupicef.sup (cupicef.type s2) (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s4) (cupicef.type s0))) + cstr_3 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s0))) + cstr_1 cupicef.BoolNone) + (cupicef.prod (cupicef.type s4) (cupicef.type s2) + (cupicef.rule (cupicef.type s4) (cupicef.type s2)) cupicef.I A + (__ : cupicef.Term (cupicef.type s4) A => B))) + (respectful s4 s5 s2 s3 s0 s1 cstr_3 cstr_4 cstr_5 cstr_1 cstr_2 A B + R1 + (Coq__Classes__CRelationClasses.flip s2 s2 s6 B B + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s6)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s6) + (cupicef.axiom (cupicef.type s6)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R2)) + f) + cupicef.I + (x : cupicef.Term (cupicef.type s4) A => + x_27_ : cupicef.Term (cupicef.type s4) A => + Hxx_27_ : cupicef.Term (cupicef.type s5) (R1 x x_27_) => + Hf x_27_ x (Sym x x_27_ Hxx_27_)). + +thm proper__sym__flip__2 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + s7 : cupicef.Nat -> + s8 : cupicef.Nat -> + s9 : cupicef.Nat -> + s10 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s5)) (cupicef.type s10)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s3)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s0)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s1)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s4)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s0)) -> + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s1)) -> + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s1)) -> + cstr_8 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s3)) -> + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s4)) -> + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s4)) -> + A : cupicef.Univ (cupicef.type s6) -> + R1 : + cupicef.Term + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s6 s7 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s6) (cupicef.type s7)) + (Coq__Classes__CRelationClasses.Symmetric s6 s7 A R1) -> + B : cupicef.Univ (cupicef.type s8) -> + R2 : + cupicef.Term + (cupicef.sup (cupicef.type s8) (cupicef.axiom (cupicef.type s9))) + (Coq__Classes__CRelationClasses.crelation s8 s9 B) -> + H0 : + cupicef.Term (cupicef.sup (cupicef.type s8) (cupicef.type s9)) + (Coq__Classes__CRelationClasses.Symmetric s8 s9 B R2) -> + C : cupicef.Univ (cupicef.type s2) -> + R3 : + cupicef.Term + (cupicef.sup (cupicef.type s2) (cupicef.axiom (cupicef.type s5))) + (Coq__Classes__CRelationClasses.crelation s2 s5 C) -> + f : + (__ : cupicef.Term (cupicef.type s6) A -> + __0 : cupicef.Term (cupicef.type s8) B -> + cupicef.Term (cupicef.type s2) C) -> + H1 : + cupicef.Term (cupicef.type s1) + (Proper s0 s1 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s6) (cupicef.type s8)))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s6) (cupicef.type s8))) + (cupicef.axiom + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s6) (cupicef.type s8)))) + cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s8) (cupicef.type s0))) + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s3)) cstr_8 + (cupicef.SortSome (cupicef.type s0)) cstr_2 + cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) (cupicef.type s0))) + cstr_5 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s0))) + (cupicef.trans (cupicef.type s2) + (cupicef.SortSome (cupicef.type s3)) cstr_1 + (cupicef.SortSome (cupicef.type s0)) cstr_2 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s8) (cupicef.type s2)) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s8) (cupicef.type s2))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s8) (cupicef.type s2) + (cupicef.rule (cupicef.type s8) (cupicef.type s2)) + cupicef.I B (__0 : cupicef.Term (cupicef.type s8) B => C)))) + (respectful s6 s7 s3 s4 s0 s1 cstr_5 cstr_6 cstr_7 cstr_2 cstr_3 A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s2) (cupicef.type s8))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.sup (cupicef.type s2) (cupicef.type s8)) + (cupicef.axiom + (cupicef.sup (cupicef.type s2) (cupicef.type s8))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s8) (cupicef.type s3))) + cstr_8 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s3))) + cstr_1 cupicef.BoolNone) + (cupicef.prod (cupicef.type s8) (cupicef.type s2) + (cupicef.rule (cupicef.type s8) (cupicef.type s2)) + cupicef.I B (__ : cupicef.Term (cupicef.type s8) B => C))) + R1 + (respectful s8 s9 s2 s5 s3 s4 cstr_8 cstr_9 cstr_10 cstr_1 cstr_4 + B C R2 R3)) + f) -> + cupicef.Term (cupicef.type s1) + (Proper s0 s1 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s6) (cupicef.type s8)))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s6) (cupicef.type s8))) + (cupicef.axiom + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s6) (cupicef.type s8)))) + cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s8) (cupicef.type s0))) + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s3)) cstr_8 + (cupicef.SortSome (cupicef.type s0)) cstr_2 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) (cupicef.type s0))) + cstr_5 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s0))) + (cupicef.trans (cupicef.type s2) + (cupicef.SortSome (cupicef.type s3)) cstr_1 + (cupicef.SortSome (cupicef.type s0)) cstr_2 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s8) (cupicef.type s2)) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s8) (cupicef.type s2))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s8) (cupicef.type s2) + (cupicef.rule (cupicef.type s8) (cupicef.type s2)) cupicef.I + B (__0 : cupicef.Term (cupicef.type s8) B => C)))) + (respectful s6 s7 s3 s4 s0 s1 cstr_5 cstr_6 cstr_7 cstr_2 cstr_3 A + (cupicef.cast + (cupicef.axiom (cupicef.sup (cupicef.type s2) (cupicef.type s8))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.sup (cupicef.type s2) (cupicef.type s8)) + (cupicef.axiom + (cupicef.sup (cupicef.type s2) (cupicef.type s8))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s8) (cupicef.type s3))) + cstr_8 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s3))) + cstr_1 cupicef.BoolNone) + (cupicef.prod (cupicef.type s8) (cupicef.type s2) + (cupicef.rule (cupicef.type s8) (cupicef.type s2)) cupicef.I + B (__ : cupicef.Term (cupicef.type s8) B => C))) + R1 + (respectful s8 s9 s2 s5 s3 s4 cstr_8 cstr_9 cstr_10 cstr_1 cstr_4 B + C R2 + (Coq__Classes__CRelationClasses.flip s2 s2 s10 C C + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + (cupicef.axiom (cupicef.type s10)) + (cupicef.univ (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s10) + (cupicef.axiom (cupicef.type s10)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I)) + R3))) + f) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + s7 : cupicef.Nat => + s8 : cupicef.Nat => + s9 : cupicef.Nat => + s10 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s5)) (cupicef.type s10)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s3)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s0)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s1)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s4)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s0)) => + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s1)) => + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s1)) => + cstr_8 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s3)) => + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s4)) => + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s4)) => + A : cupicef.Univ (cupicef.type s6) => + R1 : + cupicef.Term + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s6 s7 A) => + Sym1 : + cupicef.Term (cupicef.sup (cupicef.type s6) (cupicef.type s7)) + (Coq__Classes__CRelationClasses.Symmetric s6 s7 A R1) => + B : cupicef.Univ (cupicef.type s8) => + R2 : + cupicef.Term + (cupicef.sup (cupicef.type s8) (cupicef.axiom (cupicef.type s9))) + (Coq__Classes__CRelationClasses.crelation s8 s9 B) => + Sym2 : + cupicef.Term (cupicef.sup (cupicef.type s8) (cupicef.type s9)) + (Coq__Classes__CRelationClasses.Symmetric s8 s9 B R2) => + C : cupicef.Univ (cupicef.type s2) => + R3 : + cupicef.Term + (cupicef.sup (cupicef.type s2) (cupicef.axiom (cupicef.type s5))) + (Coq__Classes__CRelationClasses.crelation s2 s5 C) => + f : + (__ : cupicef.Term (cupicef.type s6) A -> + __0 : cupicef.Term (cupicef.type s8) B -> + cupicef.Term (cupicef.type s2) C) => + Hf : + cupicef.Term (cupicef.type s1) + (Proper s0 s1 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s6) (cupicef.type s8)))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s6) (cupicef.type s8))) + (cupicef.axiom + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s6) (cupicef.type s8)))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s8) (cupicef.type s0))) + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s3)) cstr_8 + (cupicef.SortSome (cupicef.type s0)) cstr_2 + cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) (cupicef.type s0))) + cstr_5 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s0))) + (cupicef.trans (cupicef.type s2) + (cupicef.SortSome (cupicef.type s3)) cstr_1 + (cupicef.SortSome (cupicef.type s0)) cstr_2 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s8) (cupicef.type s2)) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s8) (cupicef.type s2))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s8) (cupicef.type s2) + (cupicef.rule (cupicef.type s8) (cupicef.type s2)) + cupicef.I B (__0 : cupicef.Term (cupicef.type s8) B => C)))) + (respectful s6 s7 s3 s4 s0 s1 cstr_5 cstr_6 cstr_7 cstr_2 cstr_3 A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s2) (cupicef.type s8))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.sup (cupicef.type s2) (cupicef.type s8)) + (cupicef.axiom + (cupicef.sup (cupicef.type s2) (cupicef.type s8))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s8) (cupicef.type s3))) + cstr_8 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s3))) + cstr_1 cupicef.BoolNone) + (cupicef.prod (cupicef.type s8) (cupicef.type s2) + (cupicef.rule (cupicef.type s8) (cupicef.type s2)) + cupicef.I B (__ : cupicef.Term (cupicef.type s8) B => C))) + R1 + (respectful s8 s9 s2 s5 s3 s4 cstr_8 cstr_9 cstr_10 cstr_1 + cstr_4 B C R2 R3)) + f) => + cupicef.cast + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) (cupicef.type s4)))) + (cupicef.type s1) + (cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) (cupicef.type s4))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) (cupicef.type s4)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s7) (cupicef.type s4)) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) (cupicef.type s4))) + cupicef.I A + (x_27_ : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s7) (cupicef.type s4) + (cupicef.rule (cupicef.type s7) (cupicef.type s4)) cupicef.I + (R1 x x_27_) + (Hxx_27_ : cupicef.Term (cupicef.type s7) (R1 x x_27_) => + respectful s8 s9 s2 s5 s3 s4 cstr_8 cstr_9 cstr_10 cstr_1 + cstr_4 B C R2 + (Coq__Classes__CRelationClasses.flip s2 s2 s10 C C + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + (cupicef.axiom (cupicef.type s10)) + (cupicef.univ (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s10) + (cupicef.axiom (cupicef.type s10)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I)) + R3) + (f x) (f x_27_))))) + (Proper s0 s1 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s6) (cupicef.type s8)))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s6) (cupicef.type s8))) + (cupicef.axiom + (cupicef.sup (cupicef.type s2) + (cupicef.sup (cupicef.type s6) (cupicef.type s8)))) + cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s8) (cupicef.type s0))) + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s3)) cstr_8 + (cupicef.SortSome (cupicef.type s0)) cstr_2 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) (cupicef.type s0))) + cstr_5 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s0))) + (cupicef.trans (cupicef.type s2) + (cupicef.SortSome (cupicef.type s3)) cstr_1 + (cupicef.SortSome (cupicef.type s0)) cstr_2 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s8) (cupicef.type s2)) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s8) (cupicef.type s2))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s8) (cupicef.type s2) + (cupicef.rule (cupicef.type s8) (cupicef.type s2)) cupicef.I + B (__0 : cupicef.Term (cupicef.type s8) B => C)))) + (respectful s6 s7 s3 s4 s0 s1 cstr_5 cstr_6 cstr_7 cstr_2 cstr_3 A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s2) (cupicef.type s8))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.sup (cupicef.type s2) (cupicef.type s8)) + (cupicef.axiom + (cupicef.sup (cupicef.type s2) (cupicef.type s8))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s8) (cupicef.type s3))) + cstr_8 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s2) (cupicef.type s3))) + cstr_1 cupicef.BoolNone) + (cupicef.prod (cupicef.type s8) (cupicef.type s2) + (cupicef.rule (cupicef.type s8) (cupicef.type s2)) cupicef.I + B (__ : cupicef.Term (cupicef.type s8) B => C))) + R1 + (respectful s8 s9 s2 s5 s3 s4 cstr_8 cstr_9 cstr_10 cstr_1 cstr_4 + B C R2 + (Coq__Classes__CRelationClasses.flip s2 s2 s10 C C + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + (cupicef.axiom (cupicef.type s10)) + (cupicef.univ (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s10) + (cupicef.axiom (cupicef.type s10)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I)) + R3))) + f) + cupicef.I + (x : cupicef.Term (cupicef.type s6) A => + x_27_ : cupicef.Term (cupicef.type s6) A => + Hxx_27_ : cupicef.Term (cupicef.type s7) (R1 x x_27_) => + cupicef.cast + (cupicef.rule (cupicef.type s8) + (cupicef.rule (cupicef.type s8) + (cupicef.rule (cupicef.type s9) (cupicef.type s5)))) + (cupicef.type s4) + (cupicef.prod (cupicef.type s8) + (cupicef.rule (cupicef.type s8) + (cupicef.rule (cupicef.type s9) (cupicef.type s5))) + (cupicef.rule (cupicef.type s8) + (cupicef.rule (cupicef.type s8) + (cupicef.rule (cupicef.type s9) (cupicef.type s5)))) + cupicef.I B + (y : cupicef.Term (cupicef.type s8) B => + cupicef.prod (cupicef.type s8) + (cupicef.rule (cupicef.type s9) (cupicef.type s5)) + (cupicef.rule (cupicef.type s8) + (cupicef.rule (cupicef.type s9) (cupicef.type s5))) + cupicef.I B + (y_27_ : cupicef.Term (cupicef.type s8) B => + cupicef.prod (cupicef.type s9) (cupicef.type s5) + (cupicef.rule (cupicef.type s9) (cupicef.type s5)) cupicef.I + (R2 y y_27_) + (Hyy_27_ : cupicef.Term (cupicef.type s9) (R2 y y_27_) => + R3 (f x_27_ y_27_) (f x y))))) + (respectful s8 s9 s2 s5 s3 s4 cstr_8 cstr_9 cstr_10 cstr_1 cstr_4 B C + R2 + (Coq__Classes__CRelationClasses.flip s2 s2 s10 C C + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + (cupicef.axiom (cupicef.type s10)) + (cupicef.univ (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s10) + (cupicef.axiom (cupicef.type s10)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I)) + R3) + (f x) (f x_27_)) + cupicef.I + (y : cupicef.Term (cupicef.type s8) B => + y_27_ : cupicef.Term (cupicef.type s8) B => + Hyy_27_ : cupicef.Term (cupicef.type s9) (R2 y y_27_) => + Hf x_27_ x (Sym1 x x_27_ Hxx_27_) y_27_ y (Sym2 y y_27_ Hyy_27_))). + +thm proper__sym__impl__iff : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + s7 : cupicef.Nat -> + cstr_0 : + cupicef.eps (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s2)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s0)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s0)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s1)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s4)) -> + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s1)) -> + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s4)) -> + cstr_8 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s4)) -> + A : cupicef.Univ (cupicef.type s5) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s5) (cupicef.axiom (cupicef.type s6))) + (Coq__Classes__CRelationClasses.crelation s5 s6 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s5) (cupicef.type s6)) + (Coq__Classes__CRelationClasses.Symmetric s5 s6 A R) -> + f : (__ : cupicef.Term (cupicef.type s5) A -> cupicef.Univ cupicef.prop) -> + H0 : + cupicef.Term (cupicef.type s1) + (Proper s0 s1 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type s5))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s0))) + cstr_3 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom cupicef.set) + (cupicef.type s0))) + (cupicef.trans (cupicef.type (cupicef.s cupicef.z)) + (cupicef.SortSome (cupicef.type s2)) cstr_0 + (cupicef.SortSome (cupicef.type s0)) cstr_1 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s5) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (respectful s5 s6 s2 s3 s0 s1 cstr_3 cstr_4 cstr_6 cstr_1 cstr_2 A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s3))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (A0 : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s2 s3 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + cupicef.I Coq__Program__Basics.impl)) + f) -> + cupicef.Term (cupicef.type s4) + (Proper s0 s4 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type s5))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s0))) + cstr_3 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0))) + (cupicef.trans (cupicef.type (cupicef.s cupicef.z)) + (cupicef.SortSome (cupicef.type s2)) cstr_0 + (cupicef.SortSome (cupicef.type s0)) cstr_1 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s5) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (respectful s5 s6 s2 s7 s0 s4 cstr_3 cstr_5 cstr_7 cstr_1 cstr_8 A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s2) (cupicef.axiom (cupicef.type s7))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (A0 : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s2 s7 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + cupicef.I Coq__Init__Logic.iff)) + f) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + s7 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s2)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s0)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s0)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s1)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s4)) => + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s1)) => + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s4)) => + cstr_8 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s4)) => + A : cupicef.Univ (cupicef.type s5) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s5) (cupicef.axiom (cupicef.type s6))) + (Coq__Classes__CRelationClasses.crelation s5 s6 A) => + Sym : + cupicef.Term (cupicef.sup (cupicef.type s5) (cupicef.type s6)) + (Coq__Classes__CRelationClasses.Symmetric s5 s6 A R) => + f : (__ : cupicef.Term (cupicef.type s5) A -> cupicef.Univ cupicef.prop) => + Hf : + cupicef.Term (cupicef.type s1) + (Proper s0 s1 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type s5))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s0))) + cstr_3 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom cupicef.set) + (cupicef.type s0))) + (cupicef.trans (cupicef.type (cupicef.s cupicef.z)) + (cupicef.SortSome (cupicef.type s2)) cstr_0 + (cupicef.SortSome (cupicef.type s0)) cstr_1 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s5) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (respectful s5 s6 s2 s3 s0 s1 cstr_3 cstr_4 cstr_6 cstr_1 cstr_2 A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s3))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (A0 : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s2 s3 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + cupicef.I Coq__Program__Basics.impl)) + f) => + cupicef.cast + (cupicef.rule (cupicef.type s5) + (cupicef.rule (cupicef.type s5) + (cupicef.rule (cupicef.type s6) cupicef.prop))) + (cupicef.type s4) + (cupicef.prod (cupicef.type s5) + (cupicef.rule (cupicef.type s5) + (cupicef.rule (cupicef.type s6) cupicef.prop)) + (cupicef.rule (cupicef.type s5) + (cupicef.rule (cupicef.type s5) + (cupicef.rule (cupicef.type s6) cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type s5) A => + cupicef.prod (cupicef.type s5) + (cupicef.rule (cupicef.type s6) cupicef.prop) + (cupicef.rule (cupicef.type s5) + (cupicef.rule (cupicef.type s6) cupicef.prop)) + cupicef.I A + (x_27_ : cupicef.Term (cupicef.type s5) A => + cupicef.prod (cupicef.type s6) cupicef.prop + (cupicef.rule (cupicef.type s6) cupicef.prop) cupicef.I + (R x x_27_) + (Hxx_27_ : cupicef.Term (cupicef.type s6) (R x x_27_) => + Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + f x) (__ : cupicef.Term cupicef.prop (f x) => f x_27_)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (f x_27_) + (__ : cupicef.Term cupicef.prop (f x_27_) => f x)))))) + (Proper s0 s4 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type s5))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type s5)) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) (cupicef.type s0))) + cstr_3 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0))) + (cupicef.trans (cupicef.type (cupicef.s cupicef.z)) + (cupicef.SortSome (cupicef.type s2)) cstr_0 + (cupicef.SortSome (cupicef.type s0)) cstr_1 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s5) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (respectful s5 s6 s2 s7 s0 s4 cstr_3 cstr_5 cstr_7 cstr_1 cstr_8 A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s2) + (cupicef.axiom (cupicef.type s7))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (A0 : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s2 s7 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + cupicef.I Coq__Init__Logic.iff)) + f) + cupicef.I + (x : cupicef.Term (cupicef.type s5) A => + x_27_ : cupicef.Term (cupicef.type s5) A => + Hxx_27_ : cupicef.Term (cupicef.type s6) (R x x_27_) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (f x) + (__ : cupicef.Term cupicef.prop (f x) => f x_27_)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (f x_27_) + (__ : cupicef.Term cupicef.prop (f x_27_) => f x)) + (H : cupicef.Term cupicef.prop (f x) => Hf x x_27_ Hxx_27_ H) + (H : cupicef.Term cupicef.prop (f x_27_) => + Hf x_27_ x (Sym x x_27_ Hxx_27_) H)). + +thm proper__sym__arrow__iffT : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s2)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s0)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) -> + cstr_3 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s0)) -> + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s1)) -> + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s1)) -> + A : cupicef.Univ (cupicef.type s4) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s4) (cupicef.axiom (cupicef.type s5))) + (Coq__Classes__CRelationClasses.crelation s4 s5 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s4) (cupicef.type s5)) + (Coq__Classes__CRelationClasses.Symmetric s4 s5 A R) -> + f : + (__ : cupicef.Term (cupicef.type s4) A -> cupicef.Univ (cupicef.type s3)) -> + H0 : + cupicef.Term (cupicef.type s1) + (Proper s0 s1 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s3)) + (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.axiom (cupicef.type s3)) + (cupicef.type s4)) + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s3)) + (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s4) (cupicef.type s0))) + cstr_5 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) + (cupicef.type s0))) + (cupicef.trans (cupicef.axiom (cupicef.type s3)) + (cupicef.SortSome (cupicef.type s2)) cstr_0 + (cupicef.SortSome (cupicef.type s0)) cstr_1 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s4) (cupicef.axiom (cupicef.type s3)) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s3))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s4) A => + cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I))) + (respectful s4 s5 s2 s3 s0 s1 cstr_5 cstr_6 cstr_7 cstr_1 cstr_2 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R (Coq__Classes__CRelationClasses.arrow s3 s3)) + f) -> + cupicef.Term (cupicef.type s1) + (Proper s0 s1 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s3)) (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.axiom (cupicef.type s3)) (cupicef.type s4)) + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s3)) + (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s4) (cupicef.type s0))) + cstr_5 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) + (cupicef.type s0))) + (cupicef.trans (cupicef.axiom (cupicef.type s3)) + (cupicef.SortSome (cupicef.type s2)) cstr_0 + (cupicef.SortSome (cupicef.type s0)) cstr_1 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s4) (cupicef.axiom (cupicef.type s3)) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s3))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s4) A => + cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I))) + (respectful s4 s5 s2 s3 s0 s1 cstr_5 cstr_6 cstr_7 cstr_1 cstr_2 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R + (Coq__Classes__CRelationClasses.iffT s3 s3 cstr_3 cstr_4 cstr_3 + cstr_4)) + f) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s2)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s0)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) => + cstr_3 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s0)) => + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s1)) => + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s1)) => + A : cupicef.Univ (cupicef.type s4) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s4) (cupicef.axiom (cupicef.type s5))) + (Coq__Classes__CRelationClasses.crelation s4 s5 A) => + Sym : + cupicef.Term (cupicef.sup (cupicef.type s4) (cupicef.type s5)) + (Coq__Classes__CRelationClasses.Symmetric s4 s5 A R) => + f : + (__ : cupicef.Term (cupicef.type s4) A -> cupicef.Univ (cupicef.type s3)) => + Hf : + cupicef.Term (cupicef.type s1) + (Proper s0 s1 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s3)) + (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.axiom (cupicef.type s3)) + (cupicef.type s4)) + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s3)) + (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s4) (cupicef.type s0))) + cstr_5 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) + (cupicef.type s0))) + (cupicef.trans (cupicef.axiom (cupicef.type s3)) + (cupicef.SortSome (cupicef.type s2)) cstr_0 + (cupicef.SortSome (cupicef.type s0)) cstr_1 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s4) + (cupicef.axiom (cupicef.type s3)) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s3))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s4) A => + cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I))) + (respectful s4 s5 s2 s3 s0 s1 cstr_5 cstr_6 cstr_7 cstr_1 cstr_2 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R (Coq__Classes__CRelationClasses.arrow s3 s3)) + f) => + cupicef.cast + (cupicef.rule (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.rule (cupicef.type s5) (cupicef.type s3)))) + (cupicef.type s1) + (cupicef.prod (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.rule (cupicef.type s5) (cupicef.type s3))) + (cupicef.rule (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.rule (cupicef.type s5) (cupicef.type s3)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s4) A => + cupicef.prod (cupicef.type s4) + (cupicef.rule (cupicef.type s5) (cupicef.type s3)) + (cupicef.rule (cupicef.type s4) + (cupicef.rule (cupicef.type s5) (cupicef.type s3))) + cupicef.I A + (x_27_ : cupicef.Term (cupicef.type s4) A => + cupicef.prod (cupicef.type s5) (cupicef.type s3) + (cupicef.rule (cupicef.type s5) (cupicef.type s3)) cupicef.I + (R x x_27_) + (Hxx_27_ : cupicef.Term (cupicef.type s5) (R x x_27_) => + Coq__Init__Datatypes.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (f x) + (__ : cupicef.Term (cupicef.type s3) (f x) => f x_27_)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (f x_27_) + (__ : cupicef.Term (cupicef.type s3) (f x_27_) => f x)))))) + (Proper s0 s1 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s3)) + (cupicef.type s4))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.axiom (cupicef.type s3)) + (cupicef.type s4)) + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s3)) + (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s4) (cupicef.type s0))) + cstr_5 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) + (cupicef.type s0))) + (cupicef.trans (cupicef.axiom (cupicef.type s3)) + (cupicef.SortSome (cupicef.type s2)) cstr_0 + (cupicef.SortSome (cupicef.type s0)) cstr_1 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s4) (cupicef.axiom (cupicef.type s3)) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s3))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s4) A => + cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I))) + (respectful s4 s5 s2 s3 s0 s1 cstr_5 cstr_6 cstr_7 cstr_1 cstr_2 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R + (Coq__Classes__CRelationClasses.iffT s3 s3 cstr_3 cstr_4 cstr_3 + cstr_4)) + f) + cupicef.I + (x : cupicef.Term (cupicef.type s4) A => + x_27_ : cupicef.Term (cupicef.type s4) A => + Hxx_27_ : cupicef.Term (cupicef.type s5) (R x x_27_) => + Coq__Init__Datatypes.pair + (cupicef.cast (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s3) (cupicef.axiom (cupicef.type s3)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_3 + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (f x) (__ : cupicef.Term (cupicef.type s3) (f x) => f x_27_))) + (cupicef.cast (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s3) (cupicef.axiom (cupicef.type s3)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_4 + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (f x_27_) + (__ : cupicef.Term (cupicef.type s3) (f x_27_) => f x))) + (X : cupicef.Term (cupicef.type s3) (f x) => Hf x x_27_ Hxx_27_ X) + (X : cupicef.Term (cupicef.type s3) (f x_27_) => + Hf x_27_ x (Sym x x_27_ Hxx_27_) X)). + +thm proper__sym__impl__iff__2 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + s7 : cupicef.Nat -> + s8 : cupicef.Nat -> + s9 : cupicef.Nat -> + s10 : cupicef.Nat -> + s11 : cupicef.Nat -> + s12 : cupicef.Nat -> + cstr_0 : + cupicef.eps (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s4)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s0)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s3)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s0)) -> + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s1)) -> + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s6)) -> + cstr_8 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s1)) -> + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s6)) -> + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s6)) -> + cstr_11 : cupicef.eps (cupicef.Cumul (cupicef.type s10) (cupicef.type s2)) -> + cstr_12 : cupicef.eps (cupicef.Cumul (cupicef.type s10) (cupicef.type s3)) -> + cstr_13 : cupicef.eps (cupicef.Cumul (cupicef.type s10) (cupicef.type s9)) -> + cstr_14 : cupicef.eps (cupicef.Cumul (cupicef.type s11) (cupicef.type s3)) -> + cstr_15 : cupicef.eps (cupicef.Cumul (cupicef.type s11) (cupicef.type s9)) -> + cstr_16 : cupicef.eps (cupicef.Cumul (cupicef.type s12) (cupicef.type s9)) -> + A : cupicef.Univ (cupicef.type s7) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s7) (cupicef.axiom (cupicef.type s8))) + (Coq__Classes__CRelationClasses.crelation s7 s8 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s7) (cupicef.type s8)) + (Coq__Classes__CRelationClasses.Symmetric s7 s8 A R) -> + B : cupicef.Univ (cupicef.type s10) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s10) (cupicef.axiom (cupicef.type s11))) + (Coq__Classes__CRelationClasses.crelation s10 s11 B) -> + H0 : + cupicef.Term (cupicef.sup (cupicef.type s10) (cupicef.type s11)) + (Coq__Classes__CRelationClasses.Symmetric s10 s11 B R_27_) -> + f : + (__ : cupicef.Term (cupicef.type s7) A -> + __0 : cupicef.Term (cupicef.type s10) B -> + cupicef.Univ cupicef.prop) -> + H1 : + cupicef.Term (cupicef.type s1) + (Proper s0 s1 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type s7) (cupicef.type s10)))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type s7) (cupicef.type s10))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type s7) (cupicef.type s10)))) + cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s10) (cupicef.type s0))) + (cupicef.trans (cupicef.type s10) + (cupicef.SortSome (cupicef.type s2)) cstr_11 + (cupicef.SortSome (cupicef.type s0)) cstr_1 + cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) (cupicef.type s0))) + cstr_5 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom cupicef.set) + (cupicef.type s0))) + (cupicef.trans (cupicef.type (cupicef.s cupicef.z)) + (cupicef.SortSome (cupicef.type s4)) cstr_0 + (cupicef.SortSome (cupicef.type s2)) cstr_3 + (cupicef.SortSome (cupicef.type s0)) cstr_1 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s7) + (cupicef.rule (cupicef.type s10) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s10) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s7) A => + cupicef.prod (cupicef.type s10) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s10) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__0 : cupicef.Term (cupicef.type s10) B => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful s7 s8 s2 s3 s0 s1 cstr_5 cstr_6 cstr_8 cstr_1 cstr_2 A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type s10))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type s10)) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type s10))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s10) (cupicef.type s2))) + cstr_11 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom cupicef.set) + (cupicef.type s2))) + (cupicef.trans (cupicef.type (cupicef.s cupicef.z)) + (cupicef.SortSome (cupicef.type s4)) cstr_0 + (cupicef.SortSome (cupicef.type s2)) cstr_3 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s10) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s10) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__ : cupicef.Term (cupicef.type s10) B => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + R + (respectful s10 s11 s4 s5 s2 s3 cstr_11 cstr_12 cstr_14 cstr_3 + cstr_4 B + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27_ + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s4) + (cupicef.axiom (cupicef.type s5))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (A0 : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (B0 : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s4 s5 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + cupicef.I Coq__Program__Basics.impl))) + f) -> + cupicef.Term (cupicef.type s6) + (Proper s0 s6 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type s7) (cupicef.type s10)))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type s7) (cupicef.type s10))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type s7) (cupicef.type s10)))) + cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s10) (cupicef.type s0))) + (cupicef.trans (cupicef.type s10) + (cupicef.SortSome (cupicef.type s2)) cstr_11 + (cupicef.SortSome (cupicef.type s0)) cstr_1 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) (cupicef.type s0))) + cstr_5 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0))) + (cupicef.trans (cupicef.type (cupicef.s cupicef.z)) + (cupicef.SortSome (cupicef.type s4)) cstr_0 + (cupicef.SortSome (cupicef.type s2)) cstr_3 + (cupicef.SortSome (cupicef.type s0)) cstr_1 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s7) + (cupicef.rule (cupicef.type s10) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s10) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s7) A => + cupicef.prod (cupicef.type s10) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s10) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__0 : cupicef.Term (cupicef.type s10) B => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful s7 s8 s2 s9 s0 s6 cstr_5 cstr_7 cstr_9 cstr_1 cstr_10 A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type s10))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type s10)) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type s10))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s10) (cupicef.type s2))) + cstr_11 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom cupicef.set) + (cupicef.type s2))) + (cupicef.trans (cupicef.type (cupicef.s cupicef.z)) + (cupicef.SortSome (cupicef.type s4)) cstr_0 + (cupicef.SortSome (cupicef.type s2)) cstr_3 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s10) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s10) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__ : cupicef.Term (cupicef.type s10) B => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + R + (respectful s10 s11 s4 s12 s2 s9 cstr_11 cstr_13 cstr_15 cstr_3 + cstr_16 B + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27_ + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s4) + (cupicef.axiom (cupicef.type s12))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (A0 : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (B0 : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s4 s12 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + cupicef.I Coq__Init__Logic.iff))) + f) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + s7 : cupicef.Nat => + s8 : cupicef.Nat => + s9 : cupicef.Nat => + s10 : cupicef.Nat => + s11 : cupicef.Nat => + s12 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s4)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s0)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s3)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s0)) => + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s1)) => + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s6)) => + cstr_8 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s1)) => + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s6)) => + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s6)) => + cstr_11 : cupicef.eps (cupicef.Cumul (cupicef.type s10) (cupicef.type s2)) => + cstr_12 : cupicef.eps (cupicef.Cumul (cupicef.type s10) (cupicef.type s3)) => + cstr_13 : cupicef.eps (cupicef.Cumul (cupicef.type s10) (cupicef.type s9)) => + cstr_14 : cupicef.eps (cupicef.Cumul (cupicef.type s11) (cupicef.type s3)) => + cstr_15 : cupicef.eps (cupicef.Cumul (cupicef.type s11) (cupicef.type s9)) => + cstr_16 : cupicef.eps (cupicef.Cumul (cupicef.type s12) (cupicef.type s9)) => + A : cupicef.Univ (cupicef.type s7) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s7) (cupicef.axiom (cupicef.type s8))) + (Coq__Classes__CRelationClasses.crelation s7 s8 A) => + Sym : + cupicef.Term (cupicef.sup (cupicef.type s7) (cupicef.type s8)) + (Coq__Classes__CRelationClasses.Symmetric s7 s8 A R) => + B : cupicef.Univ (cupicef.type s10) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s10) (cupicef.axiom (cupicef.type s11))) + (Coq__Classes__CRelationClasses.crelation s10 s11 B) => + Sym_27_ : + cupicef.Term (cupicef.sup (cupicef.type s10) (cupicef.type s11)) + (Coq__Classes__CRelationClasses.Symmetric s10 s11 B R_27_) => + f : + (__ : cupicef.Term (cupicef.type s7) A -> + __0 : cupicef.Term (cupicef.type s10) B -> + cupicef.Univ cupicef.prop) => + Hf : + cupicef.Term (cupicef.type s1) + (Proper s0 s1 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type s7) (cupicef.type s10)))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type s7) (cupicef.type s10))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type s7) (cupicef.type s10)))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s10) (cupicef.type s0))) + (cupicef.trans (cupicef.type s10) + (cupicef.SortSome (cupicef.type s2)) cstr_11 + (cupicef.SortSome (cupicef.type s0)) cstr_1 + cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) (cupicef.type s0))) + cstr_5 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom cupicef.set) + (cupicef.type s0))) + (cupicef.trans (cupicef.type (cupicef.s cupicef.z)) + (cupicef.SortSome (cupicef.type s4)) cstr_0 + (cupicef.SortSome (cupicef.type s2)) cstr_3 + (cupicef.SortSome (cupicef.type s0)) cstr_1 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s7) + (cupicef.rule (cupicef.type s10) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s10) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s7) A => + cupicef.prod (cupicef.type s10) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s10) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__0 : cupicef.Term (cupicef.type s10) B => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful s7 s8 s2 s3 s0 s1 cstr_5 cstr_6 cstr_8 cstr_1 cstr_2 A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type s10))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type s10)) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type s10))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s10) (cupicef.type s2))) + cstr_11 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom cupicef.set) + (cupicef.type s2))) + (cupicef.trans (cupicef.type (cupicef.s cupicef.z)) + (cupicef.SortSome (cupicef.type s4)) cstr_0 + (cupicef.SortSome (cupicef.type s2)) cstr_3 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s10) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s10) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__ : cupicef.Term (cupicef.type s10) B => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + R + (respectful s10 s11 s4 s5 s2 s3 cstr_11 cstr_12 cstr_14 cstr_3 + cstr_4 B + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27_ + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s4) + (cupicef.axiom (cupicef.type s5))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (A0 : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (B0 : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s4 s5 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + cupicef.I Coq__Program__Basics.impl))) + f) => + cupicef.cast + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s8) (cupicef.type s9)))) + (cupicef.type s6) + (cupicef.prod (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s8) (cupicef.type s9))) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s8) (cupicef.type s9)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s7) A => + cupicef.prod (cupicef.type s7) + (cupicef.rule (cupicef.type s8) (cupicef.type s9)) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s8) (cupicef.type s9))) + cupicef.I A + (x_27_ : cupicef.Term (cupicef.type s7) A => + cupicef.prod (cupicef.type s8) (cupicef.type s9) + (cupicef.rule (cupicef.type s8) (cupicef.type s9)) cupicef.I + (R x x_27_) + (Hxx_27_ : cupicef.Term (cupicef.type s8) (R x x_27_) => + respectful s10 s11 s4 s12 s2 s9 cstr_11 cstr_13 cstr_15 cstr_3 + cstr_16 B + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27_ + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s4) + (cupicef.axiom (cupicef.type s12))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (A0 : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (B0 : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s4 s12 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + cupicef.I Coq__Init__Logic.iff) + (f x) (f x_27_))))) + (Proper s0 s6 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type s7) (cupicef.type s10)))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type s7) (cupicef.type s10))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type s7) (cupicef.type s10)))) + cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s10) (cupicef.type s0))) + (cupicef.trans (cupicef.type s10) + (cupicef.SortSome (cupicef.type s2)) cstr_11 + (cupicef.SortSome (cupicef.type s0)) cstr_1 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) (cupicef.type s0))) + cstr_5 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0))) + (cupicef.trans (cupicef.type (cupicef.s cupicef.z)) + (cupicef.SortSome (cupicef.type s4)) cstr_0 + (cupicef.SortSome (cupicef.type s2)) cstr_3 + (cupicef.SortSome (cupicef.type s0)) cstr_1 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s7) + (cupicef.rule (cupicef.type s10) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s10) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s7) A => + cupicef.prod (cupicef.type s10) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s10) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__0 : cupicef.Term (cupicef.type s10) B => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful s7 s8 s2 s9 s0 s6 cstr_5 cstr_7 cstr_9 cstr_1 cstr_10 A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type s10))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type s10)) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type s10))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s10) (cupicef.type s2))) + cstr_11 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom cupicef.set) + (cupicef.type s2))) + (cupicef.trans (cupicef.type (cupicef.s cupicef.z)) + (cupicef.SortSome (cupicef.type s4)) cstr_0 + (cupicef.SortSome (cupicef.type s2)) cstr_3 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s10) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s10) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__ : cupicef.Term (cupicef.type s10) B => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + R + (respectful s10 s11 s4 s12 s2 s9 cstr_11 cstr_13 cstr_15 cstr_3 + cstr_16 B + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27_ + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s4) + (cupicef.axiom (cupicef.type s12))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (A0 : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (B0 : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s4 s12 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + cupicef.I Coq__Init__Logic.iff))) + f) + cupicef.I + (x : cupicef.Term (cupicef.type s7) A => + x_27_ : cupicef.Term (cupicef.type s7) A => + Hxx_27_ : cupicef.Term (cupicef.type s8) (R x x_27_) => + cupicef.cast + (cupicef.rule (cupicef.type s10) + (cupicef.rule (cupicef.type s10) + (cupicef.rule (cupicef.type s11) cupicef.prop))) + (cupicef.type s9) + (cupicef.prod (cupicef.type s10) + (cupicef.rule (cupicef.type s10) + (cupicef.rule (cupicef.type s11) cupicef.prop)) + (cupicef.rule (cupicef.type s10) + (cupicef.rule (cupicef.type s10) + (cupicef.rule (cupicef.type s11) cupicef.prop))) + cupicef.I B + (y : cupicef.Term (cupicef.type s10) B => + cupicef.prod (cupicef.type s10) + (cupicef.rule (cupicef.type s11) cupicef.prop) + (cupicef.rule (cupicef.type s10) + (cupicef.rule (cupicef.type s11) cupicef.prop)) + cupicef.I B + (y_27_ : cupicef.Term (cupicef.type s10) B => + cupicef.prod (cupicef.type s11) cupicef.prop + (cupicef.rule (cupicef.type s11) cupicef.prop) cupicef.I + (R_27_ y y_27_) + (Hyy_27_ : cupicef.Term (cupicef.type s11) (R_27_ y y_27_) => + Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (f x y) + (__ : cupicef.Term cupicef.prop (f x y) => + f x_27_ y_27_)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (f x_27_ y_27_) + (__ : cupicef.Term cupicef.prop (f x_27_ y_27_) => + f x y)))))) + (respectful s10 s11 s4 s12 s2 s9 cstr_11 cstr_13 cstr_15 cstr_3 + cstr_16 B + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R_27_ + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s4) + (cupicef.axiom (cupicef.type s12))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (A0 : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (B0 : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__CRelationClasses.crelation s4 s12 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + cupicef.I Coq__Init__Logic.iff) + (f x) (f x_27_)) + cupicef.I + (y : cupicef.Term (cupicef.type s10) B => + y_27_ : cupicef.Term (cupicef.type s10) B => + Hyy_27_ : cupicef.Term (cupicef.type s11) (R_27_ y y_27_) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (f x y) + (__ : cupicef.Term cupicef.prop (f x y) => f x_27_ y_27_)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (f x_27_ y_27_) + (__ : cupicef.Term cupicef.prop (f x_27_ y_27_) => f x y)) + (H : cupicef.Term cupicef.prop (f x y) => + Hf x x_27_ Hxx_27_ y y_27_ Hyy_27_ H) + (H : cupicef.Term cupicef.prop (f x_27_ y_27_) => + Hf x_27_ x (Sym x x_27_ Hxx_27_) y_27_ y + (Sym_27_ y y_27_ Hyy_27_) H))). + +thm proper__sym__arrow__iffT__2 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + s7 : cupicef.Nat -> + s8 : cupicef.Nat -> + s9 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s5)) (cupicef.type s4)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s0)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s3)) -> + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s5) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s5) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s0)) -> + cstr_8 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s1)) -> + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s1)) -> + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s2)) -> + cstr_11 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s3)) -> + cstr_12 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s3)) -> + A : cupicef.Univ (cupicef.type s6) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s6 s7 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s6) (cupicef.type s7)) + (Coq__Classes__CRelationClasses.Symmetric s6 s7 A R) -> + B : cupicef.Univ (cupicef.type s8) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s8) (cupicef.axiom (cupicef.type s9))) + (Coq__Classes__CRelationClasses.crelation s8 s9 B) -> + H0 : + cupicef.Term (cupicef.sup (cupicef.type s8) (cupicef.type s9)) + (Coq__Classes__CRelationClasses.Symmetric s8 s9 B R_27_) -> + f : + (__ : cupicef.Term (cupicef.type s6) A -> + __0 : cupicef.Term (cupicef.type s8) B -> + cupicef.Univ (cupicef.type s5)) -> + H1 : + cupicef.Term (cupicef.type s1) + (Proper s0 s1 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s5)) + (cupicef.sup (cupicef.type s6) (cupicef.type s8)))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.axiom (cupicef.type s5)) + (cupicef.sup (cupicef.type s6) (cupicef.type s8))) + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s5)) + (cupicef.sup (cupicef.type s6) (cupicef.type s8)))) + cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s8) (cupicef.type s0))) + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s2)) cstr_10 + (cupicef.SortSome (cupicef.type s0)) cstr_1 + cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) (cupicef.type s0))) + cstr_7 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s5)) + (cupicef.type s0))) + (cupicef.trans (cupicef.axiom (cupicef.type s5)) + (cupicef.SortSome (cupicef.type s4)) cstr_0 + (cupicef.SortSome (cupicef.type s2)) cstr_3 + (cupicef.SortSome (cupicef.type s0)) cstr_1 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s8) + (cupicef.axiom (cupicef.type s5))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s8) + (cupicef.axiom (cupicef.type s5)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s8) + (cupicef.axiom (cupicef.type s5)) + (cupicef.rule (cupicef.type s8) + (cupicef.axiom (cupicef.type s5))) + cupicef.I B + (__0 : cupicef.Term (cupicef.type s8) B => + cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I)))) + (respectful s6 s7 s2 s3 s0 s1 cstr_7 cstr_8 cstr_9 cstr_1 cstr_2 A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s5)) + (cupicef.type s8))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.axiom (cupicef.type s5)) + (cupicef.type s8)) + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s5)) + (cupicef.type s8))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s8) (cupicef.type s2))) + cstr_10 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s5)) + (cupicef.type s2))) + (cupicef.trans (cupicef.axiom (cupicef.type s5)) + (cupicef.SortSome (cupicef.type s4)) cstr_0 + (cupicef.SortSome (cupicef.type s2)) cstr_3 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s8) + (cupicef.axiom (cupicef.type s5)) + (cupicef.rule (cupicef.type s8) + (cupicef.axiom (cupicef.type s5))) + cupicef.I B + (__ : cupicef.Term (cupicef.type s8) B => + cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I))) + R + (respectful s8 s9 s4 s5 s2 s3 cstr_10 cstr_11 cstr_12 cstr_3 + cstr_4 B + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I)) + R_27_ (Coq__Classes__CRelationClasses.arrow s5 s5))) + f) -> + cupicef.Term (cupicef.type s1) + (Proper s0 s1 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s5)) + (cupicef.sup (cupicef.type s6) (cupicef.type s8)))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.axiom (cupicef.type s5)) + (cupicef.sup (cupicef.type s6) (cupicef.type s8))) + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s5)) + (cupicef.sup (cupicef.type s6) (cupicef.type s8)))) + cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s8) (cupicef.type s0))) + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s2)) cstr_10 + (cupicef.SortSome (cupicef.type s0)) cstr_1 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) (cupicef.type s0))) + cstr_7 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s5)) + (cupicef.type s0))) + (cupicef.trans (cupicef.axiom (cupicef.type s5)) + (cupicef.SortSome (cupicef.type s4)) cstr_0 + (cupicef.SortSome (cupicef.type s2)) cstr_3 + (cupicef.SortSome (cupicef.type s0)) cstr_1 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s8) + (cupicef.axiom (cupicef.type s5))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s8) + (cupicef.axiom (cupicef.type s5)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s8) + (cupicef.axiom (cupicef.type s5)) + (cupicef.rule (cupicef.type s8) + (cupicef.axiom (cupicef.type s5))) + cupicef.I B + (__0 : cupicef.Term (cupicef.type s8) B => + cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I)))) + (respectful s6 s7 s2 s3 s0 s1 cstr_7 cstr_8 cstr_9 cstr_1 cstr_2 A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s5)) + (cupicef.type s8))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.axiom (cupicef.type s5)) + (cupicef.type s8)) + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s5)) + (cupicef.type s8))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s8) (cupicef.type s2))) + cstr_10 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s5)) + (cupicef.type s2))) + (cupicef.trans (cupicef.axiom (cupicef.type s5)) + (cupicef.SortSome (cupicef.type s4)) cstr_0 + (cupicef.SortSome (cupicef.type s2)) cstr_3 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s8) + (cupicef.axiom (cupicef.type s5)) + (cupicef.rule (cupicef.type s8) + (cupicef.axiom (cupicef.type s5))) + cupicef.I B + (__ : cupicef.Term (cupicef.type s8) B => + cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I))) + R + (respectful s8 s9 s4 s5 s2 s3 cstr_10 cstr_11 cstr_12 cstr_3 cstr_4 + B + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I)) + R_27_ + (Coq__Classes__CRelationClasses.iffT s5 s5 cstr_5 cstr_6 cstr_5 + cstr_6))) + f) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + s7 : cupicef.Nat => + s8 : cupicef.Nat => + s9 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s5)) (cupicef.type s4)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s0)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s3)) => + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s5) (cupicef.type (cupicef.s cupicef.z))) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s5) (cupicef.type (cupicef.s cupicef.z))) => + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s0)) => + cstr_8 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s1)) => + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s1)) => + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s2)) => + cstr_11 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s3)) => + cstr_12 : cupicef.eps (cupicef.Cumul (cupicef.type s9) (cupicef.type s3)) => + A : cupicef.Univ (cupicef.type s6) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s6 s7 A) => + Sym : + cupicef.Term (cupicef.sup (cupicef.type s6) (cupicef.type s7)) + (Coq__Classes__CRelationClasses.Symmetric s6 s7 A R) => + B : cupicef.Univ (cupicef.type s8) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s8) (cupicef.axiom (cupicef.type s9))) + (Coq__Classes__CRelationClasses.crelation s8 s9 B) => + Sym_27_ : + cupicef.Term (cupicef.sup (cupicef.type s8) (cupicef.type s9)) + (Coq__Classes__CRelationClasses.Symmetric s8 s9 B R_27_) => + f : + (__ : cupicef.Term (cupicef.type s6) A -> + __0 : cupicef.Term (cupicef.type s8) B -> + cupicef.Univ (cupicef.type s5)) => + Hf : + cupicef.Term (cupicef.type s1) + (Proper s0 s1 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s5)) + (cupicef.sup (cupicef.type s6) (cupicef.type s8)))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.axiom (cupicef.type s5)) + (cupicef.sup (cupicef.type s6) (cupicef.type s8))) + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s5)) + (cupicef.sup (cupicef.type s6) (cupicef.type s8)))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s8) (cupicef.type s0))) + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s2)) cstr_10 + (cupicef.SortSome (cupicef.type s0)) cstr_1 + cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) (cupicef.type s0))) + cstr_7 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s5)) + (cupicef.type s0))) + (cupicef.trans (cupicef.axiom (cupicef.type s5)) + (cupicef.SortSome (cupicef.type s4)) cstr_0 + (cupicef.SortSome (cupicef.type s2)) cstr_3 + (cupicef.SortSome (cupicef.type s0)) cstr_1 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s8) + (cupicef.axiom (cupicef.type s5))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s8) + (cupicef.axiom (cupicef.type s5)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s8) + (cupicef.axiom (cupicef.type s5)) + (cupicef.rule (cupicef.type s8) + (cupicef.axiom (cupicef.type s5))) + cupicef.I B + (__0 : cupicef.Term (cupicef.type s8) B => + cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I)))) + (respectful s6 s7 s2 s3 s0 s1 cstr_7 cstr_8 cstr_9 cstr_1 cstr_2 A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s5)) + (cupicef.type s8))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.axiom (cupicef.type s5)) + (cupicef.type s8)) + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s5)) + (cupicef.type s8))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s8) (cupicef.type s2))) + cstr_10 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s5)) + (cupicef.type s2))) + (cupicef.trans (cupicef.axiom (cupicef.type s5)) + (cupicef.SortSome (cupicef.type s4)) cstr_0 + (cupicef.SortSome (cupicef.type s2)) cstr_3 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s8) + (cupicef.axiom (cupicef.type s5)) + (cupicef.rule (cupicef.type s8) + (cupicef.axiom (cupicef.type s5))) + cupicef.I B + (__ : cupicef.Term (cupicef.type s8) B => + cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I))) + R + (respectful s8 s9 s4 s5 s2 s3 cstr_10 cstr_11 cstr_12 cstr_3 + cstr_4 B + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I)) + R_27_ (Coq__Classes__CRelationClasses.arrow s5 s5))) + f) => + cupicef.cast + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) (cupicef.type s3)))) + (cupicef.type s1) + (cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) (cupicef.type s3))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) (cupicef.type s3)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s7) (cupicef.type s3)) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) (cupicef.type s3))) + cupicef.I A + (x_27_ : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s7) (cupicef.type s3) + (cupicef.rule (cupicef.type s7) (cupicef.type s3)) cupicef.I + (R x x_27_) + (Hxx_27_ : cupicef.Term (cupicef.type s7) (R x x_27_) => + respectful s8 s9 s4 s5 s2 s3 cstr_10 cstr_11 cstr_12 cstr_3 + cstr_4 B + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I)) + R_27_ + (Coq__Classes__CRelationClasses.iffT s5 s5 cstr_5 cstr_6 + cstr_5 cstr_6) + (f x) (f x_27_))))) + (Proper s0 s1 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s5)) + (cupicef.sup (cupicef.type s6) (cupicef.type s8)))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.axiom (cupicef.type s5)) + (cupicef.sup (cupicef.type s6) (cupicef.type s8))) + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s5)) + (cupicef.sup (cupicef.type s6) (cupicef.type s8)))) + cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s8) (cupicef.type s0))) + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s2)) cstr_10 + (cupicef.SortSome (cupicef.type s0)) cstr_1 cupicef.SortNone) + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) (cupicef.type s0))) + cstr_7 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s5)) + (cupicef.type s0))) + (cupicef.trans (cupicef.axiom (cupicef.type s5)) + (cupicef.SortSome (cupicef.type s4)) cstr_0 + (cupicef.SortSome (cupicef.type s2)) cstr_3 + (cupicef.SortSome (cupicef.type s0)) cstr_1 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s8) + (cupicef.axiom (cupicef.type s5))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s8) + (cupicef.axiom (cupicef.type s5)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s8) + (cupicef.axiom (cupicef.type s5)) + (cupicef.rule (cupicef.type s8) + (cupicef.axiom (cupicef.type s5))) + cupicef.I B + (__0 : cupicef.Term (cupicef.type s8) B => + cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I)))) + (respectful s6 s7 s2 s3 s0 s1 cstr_7 cstr_8 cstr_9 cstr_1 cstr_2 A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s5)) + (cupicef.type s8))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.sup (cupicef.axiom (cupicef.type s5)) + (cupicef.type s8)) + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s5)) + (cupicef.type s8))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s8) (cupicef.type s2))) + cstr_10 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s5)) + (cupicef.type s2))) + (cupicef.trans (cupicef.axiom (cupicef.type s5)) + (cupicef.SortSome (cupicef.type s4)) cstr_0 + (cupicef.SortSome (cupicef.type s2)) cstr_3 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s8) + (cupicef.axiom (cupicef.type s5)) + (cupicef.rule (cupicef.type s8) + (cupicef.axiom (cupicef.type s5))) + cupicef.I B + (__ : cupicef.Term (cupicef.type s8) B => + cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I))) + R + (respectful s8 s9 s4 s5 s2 s3 cstr_10 cstr_11 cstr_12 cstr_3 + cstr_4 B + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I)) + R_27_ + (Coq__Classes__CRelationClasses.iffT s5 s5 cstr_5 cstr_6 cstr_5 + cstr_6))) + f) + cupicef.I + (x : cupicef.Term (cupicef.type s6) A => + x_27_ : cupicef.Term (cupicef.type s6) A => + Hxx_27_ : cupicef.Term (cupicef.type s7) (R x x_27_) => + cupicef.cast + (cupicef.rule (cupicef.type s8) + (cupicef.rule (cupicef.type s8) + (cupicef.rule (cupicef.type s9) (cupicef.type s5)))) + (cupicef.type s3) + (cupicef.prod (cupicef.type s8) + (cupicef.rule (cupicef.type s8) + (cupicef.rule (cupicef.type s9) (cupicef.type s5))) + (cupicef.rule (cupicef.type s8) + (cupicef.rule (cupicef.type s8) + (cupicef.rule (cupicef.type s9) (cupicef.type s5)))) + cupicef.I B + (y : cupicef.Term (cupicef.type s8) B => + cupicef.prod (cupicef.type s8) + (cupicef.rule (cupicef.type s9) (cupicef.type s5)) + (cupicef.rule (cupicef.type s8) + (cupicef.rule (cupicef.type s9) (cupicef.type s5))) + cupicef.I B + (y_27_ : cupicef.Term (cupicef.type s8) B => + cupicef.prod (cupicef.type s9) (cupicef.type s5) + (cupicef.rule (cupicef.type s9) (cupicef.type s5)) cupicef.I + (R_27_ y y_27_) + (Hyy_27_ : cupicef.Term (cupicef.type s9) (R_27_ y y_27_) => + Coq__Init__Datatypes.prod (cupicef.type s5) + (cupicef.type s5) + (cupicef.prod (cupicef.type s5) (cupicef.type s5) + (cupicef.rule (cupicef.type s5) (cupicef.type s5)) + cupicef.I (f x y) + (__ : cupicef.Term (cupicef.type s5) (f x y) => + f x_27_ y_27_)) + (cupicef.prod (cupicef.type s5) (cupicef.type s5) + (cupicef.rule (cupicef.type s5) (cupicef.type s5)) + cupicef.I (f x_27_ y_27_) + (__ : cupicef.Term (cupicef.type s5) (f x_27_ y_27_) => + f x y)))))) + (respectful s8 s9 s4 s5 s2 s3 cstr_10 cstr_11 cstr_12 cstr_3 cstr_4 B + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom (cupicef.axiom (cupicef.type s5))) cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I)) + R_27_ + (Coq__Classes__CRelationClasses.iffT s5 s5 cstr_5 cstr_6 cstr_5 + cstr_6) + (f x) (f x_27_)) + cupicef.I + (y : cupicef.Term (cupicef.type s8) B => + y_27_ : cupicef.Term (cupicef.type s8) B => + Hyy_27_ : cupicef.Term (cupicef.type s9) (R_27_ y y_27_) => + Coq__Init__Datatypes.pair + (cupicef.cast (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_5 + (cupicef.prod (cupicef.type s5) (cupicef.type s5) + (cupicef.rule (cupicef.type s5) (cupicef.type s5)) cupicef.I + (f x y) + (__ : cupicef.Term (cupicef.type s5) (f x y) => + f x_27_ y_27_))) + (cupicef.cast (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_6 + (cupicef.prod (cupicef.type s5) (cupicef.type s5) + (cupicef.rule (cupicef.type s5) (cupicef.type s5)) cupicef.I + (f x_27_ y_27_) + (__ : cupicef.Term (cupicef.type s5) (f x_27_ y_27_) => + f x y))) + (X : cupicef.Term (cupicef.type s5) (f x y) => + Hf x x_27_ Hxx_27_ y y_27_ Hyy_27_ X) + (X : cupicef.Term (cupicef.type s5) (f x_27_ y_27_) => + Hf x_27_ x (Sym x x_27_ Hxx_27_) y_27_ y + (Sym_27_ y y_27_ Hyy_27_) X))). + +def let__X : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s2)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s5)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s0)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s0)) -> + A : cupicef.Univ (cupicef.type s1) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) -> + equ : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Equivalence s1 s4 A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) -> + preo : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PreOrder s1 s4 A R) -> + H : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.PartialOrder s1 s2 s3 s4 s5 cstr_0 + cstr_1 cstr_3 cstr_4 cstr_5 cstr_6 cstr_7 A eqA equ R preo) -> + x : cupicef.Term (cupicef.type s1) A -> + x_27_ : cupicef.Term (cupicef.type s1) A -> + Hx : cupicef.Term (cupicef.type s4) (eqA x x_27_) -> + y : cupicef.Term (cupicef.type s1) A -> + y_27_ : cupicef.Term (cupicef.type s1) A -> + Hy : cupicef.Term (cupicef.type s4) (eqA y y_27_) -> + Hr : cupicef.Term (cupicef.type s4) (R x y) -> + X : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA x x_27_) + (__ : cupicef.Term (cupicef.type s4) (eqA x x_27_) => + Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x))) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R x x_27_) (R x_27_ x)) => + eqA x x_27_))) -> + a : + (__ : cupicef.Term (cupicef.type s4) (eqA x x_27_) -> + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x))) -> + b : + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) -> + cupicef.Term (cupicef.type s4) (eqA x x_27_)) -> + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s2)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s5)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s0)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s0)) => + A : cupicef.Univ (cupicef.type s1) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) => + equ : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Equivalence s1 s4 A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) => + preo : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PreOrder s1 s4 A R) => + H : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.PartialOrder s1 s2 s3 s4 s5 cstr_0 + cstr_1 cstr_3 cstr_4 cstr_5 cstr_6 cstr_7 A eqA equ R preo) => + x : cupicef.Term (cupicef.type s1) A => + x_27_ : cupicef.Term (cupicef.type s1) A => + Hx : cupicef.Term (cupicef.type s4) (eqA x x_27_) => + y : cupicef.Term (cupicef.type s1) A => + y_27_ : cupicef.Term (cupicef.type s1) A => + Hy : cupicef.Term (cupicef.type s4) (eqA y y_27_) => + Hr : cupicef.Term (cupicef.type s4) (R x y) => + X : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA x x_27_) + (__ : cupicef.Term (cupicef.type s4) (eqA x x_27_) => + Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x))) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R x x_27_) (R x_27_ x)) => + eqA x x_27_))) => + a : + (__ : cupicef.Term (cupicef.type s4) (eqA x x_27_) -> + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x))) => + b : + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) -> + cupicef.Term (cupicef.type s4) (eqA x x_27_)) => + a Hx. + +def let__X0 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s2)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s5)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s0)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s0)) -> + A : cupicef.Univ (cupicef.type s1) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) -> + equ : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Equivalence s1 s4 A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) -> + preo : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PreOrder s1 s4 A R) -> + H : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.PartialOrder s1 s2 s3 s4 s5 cstr_0 + cstr_1 cstr_3 cstr_4 cstr_5 cstr_6 cstr_7 A eqA equ R preo) -> + x : cupicef.Term (cupicef.type s1) A -> + x_27_ : cupicef.Term (cupicef.type s1) A -> + Hx : cupicef.Term (cupicef.type s4) (eqA x x_27_) -> + y : cupicef.Term (cupicef.type s1) A -> + y_27_ : cupicef.Term (cupicef.type s1) A -> + Hy : cupicef.Term (cupicef.type s4) (eqA y y_27_) -> + Hr : cupicef.Term (cupicef.type s4) (R x y) -> + X : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA x x_27_) + (__ : cupicef.Term (cupicef.type s4) (eqA x x_27_) => + Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x))) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R x x_27_) (R x_27_ x)) => + eqA x x_27_))) -> + a : + (__ : cupicef.Term (cupicef.type s4) (eqA x x_27_) -> + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x))) -> + b : + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) -> + cupicef.Term (cupicef.type s4) (eqA x x_27_)) -> + a0 : cupicef.Term (cupicef.type s4) (R x x_27_) -> + b0 : cupicef.Term (cupicef.type s4) (R x_27_ x) -> + __ : cupicef.Term (cupicef.type s4) (R x x_27_) -> + __0 : cupicef.Term (cupicef.type s4) (R x_27_ x) -> + cupicef.Term (cupicef.type s4) (eqA x x_27_) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s2)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s5)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s0)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s0)) => + A : cupicef.Univ (cupicef.type s1) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) => + equ : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Equivalence s1 s4 A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) => + preo : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PreOrder s1 s4 A R) => + H : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.PartialOrder s1 s2 s3 s4 s5 cstr_0 + cstr_1 cstr_3 cstr_4 cstr_5 cstr_6 cstr_7 A eqA equ R preo) => + x : cupicef.Term (cupicef.type s1) A => + x_27_ : cupicef.Term (cupicef.type s1) A => + Hx : cupicef.Term (cupicef.type s4) (eqA x x_27_) => + y : cupicef.Term (cupicef.type s1) A => + y_27_ : cupicef.Term (cupicef.type s1) A => + Hy : cupicef.Term (cupicef.type s4) (eqA y y_27_) => + Hr : cupicef.Term (cupicef.type s4) (R x y) => + X : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA x x_27_) + (__ : cupicef.Term (cupicef.type s4) (eqA x x_27_) => + Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x))) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R x x_27_) (R x_27_ x)) => + eqA x x_27_))) => + a : + (__ : cupicef.Term (cupicef.type s4) (eqA x x_27_) -> + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x))) => + b : + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) -> + cupicef.Term (cupicef.type s4) (eqA x x_27_)) => + a0 : cupicef.Term (cupicef.type s4) (R x x_27_) => + b0 : cupicef.Term (cupicef.type s4) (R x_27_ x) => + X0 : cupicef.Term (cupicef.type s4) (R x x_27_) => + X1 : cupicef.Term (cupicef.type s4) (R x_27_ x) => + b + (Coq__Init__Datatypes.pair + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) (cupicef.axiom (cupicef.type s4)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_6 (R x x_27_)) + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) (cupicef.axiom (cupicef.type s4)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_7 (R x_27_ x)) + X0 X1). + +def let__X1 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s2)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s5)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s0)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s0)) -> + A : cupicef.Univ (cupicef.type s1) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) -> + equ : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Equivalence s1 s4 A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) -> + preo : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PreOrder s1 s4 A R) -> + H : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.PartialOrder s1 s2 s3 s4 s5 cstr_0 + cstr_1 cstr_3 cstr_4 cstr_5 cstr_6 cstr_7 A eqA equ R preo) -> + x : cupicef.Term (cupicef.type s1) A -> + x_27_ : cupicef.Term (cupicef.type s1) A -> + Hx : cupicef.Term (cupicef.type s4) (eqA x x_27_) -> + y : cupicef.Term (cupicef.type s1) A -> + y_27_ : cupicef.Term (cupicef.type s1) A -> + Hy : cupicef.Term (cupicef.type s4) (eqA y y_27_) -> + Hr : cupicef.Term (cupicef.type s4) (R x y) -> + X : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA x x_27_) + (__ : cupicef.Term (cupicef.type s4) (eqA x x_27_) => + Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x))) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R x x_27_) (R x_27_ x)) => + eqA x x_27_))) -> + a : + (__ : cupicef.Term (cupicef.type s4) (eqA x x_27_) -> + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x))) -> + b : + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) -> + cupicef.Term (cupicef.type s4) (eqA x x_27_)) -> + a0 : cupicef.Term (cupicef.type s4) (R x x_27_) -> + b0 : cupicef.Term (cupicef.type s4) (R x_27_ x) -> + __ : cupicef.Term (cupicef.type s4) (R x_27_ x) -> + cupicef.Term (cupicef.type s4) (eqA x x_27_) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s2)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s5)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s0)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s0)) => + A : cupicef.Univ (cupicef.type s1) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) => + equ : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Equivalence s1 s4 A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) => + preo : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PreOrder s1 s4 A R) => + H : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.PartialOrder s1 s2 s3 s4 s5 cstr_0 + cstr_1 cstr_3 cstr_4 cstr_5 cstr_6 cstr_7 A eqA equ R preo) => + x : cupicef.Term (cupicef.type s1) A => + x_27_ : cupicef.Term (cupicef.type s1) A => + Hx : cupicef.Term (cupicef.type s4) (eqA x x_27_) => + y : cupicef.Term (cupicef.type s1) A => + y_27_ : cupicef.Term (cupicef.type s1) A => + Hy : cupicef.Term (cupicef.type s4) (eqA y y_27_) => + Hr : cupicef.Term (cupicef.type s4) (R x y) => + X : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA x x_27_) + (__ : cupicef.Term (cupicef.type s4) (eqA x x_27_) => + Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x))) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R x x_27_) (R x_27_ x)) => + eqA x x_27_))) => + a : + (__ : cupicef.Term (cupicef.type s4) (eqA x x_27_) -> + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x))) => + b : + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) -> + cupicef.Term (cupicef.type s4) (eqA x x_27_)) => + a0 : cupicef.Term (cupicef.type s4) (R x x_27_) => + b0 : cupicef.Term (cupicef.type s4) (R x_27_ x) => + let__X0 s0 s1 s2 s3 s4 s5 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 cstr_6 + cstr_7 cstr_8 cstr_9 A eqA equ R preo H x x_27_ Hx y y_27_ Hy Hr X a b + a0 b0 a0. + +def let__X3 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s2)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s5)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s0)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s0)) -> + A : cupicef.Univ (cupicef.type s1) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) -> + equ : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Equivalence s1 s4 A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) -> + preo : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PreOrder s1 s4 A R) -> + H : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.PartialOrder s1 s2 s3 s4 s5 cstr_0 + cstr_1 cstr_3 cstr_4 cstr_5 cstr_6 cstr_7 A eqA equ R preo) -> + x : cupicef.Term (cupicef.type s1) A -> + x_27_ : cupicef.Term (cupicef.type s1) A -> + Hx : cupicef.Term (cupicef.type s4) (eqA x x_27_) -> + y : cupicef.Term (cupicef.type s1) A -> + y_27_ : cupicef.Term (cupicef.type s1) A -> + Hy : cupicef.Term (cupicef.type s4) (eqA y y_27_) -> + Hr : cupicef.Term (cupicef.type s4) (R x y) -> + X : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA x x_27_) + (__ : cupicef.Term (cupicef.type s4) (eqA x x_27_) => + Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x))) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R x x_27_) (R x_27_ x)) => + eqA x x_27_))) -> + a : + (__ : cupicef.Term (cupicef.type s4) (eqA x x_27_) -> + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x))) -> + b : + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) -> + cupicef.Term (cupicef.type s4) (eqA x x_27_)) -> + a0 : cupicef.Term (cupicef.type s4) (R x x_27_) -> + b0 : cupicef.Term (cupicef.type s4) (R x_27_ x) -> + cupicef.Term (cupicef.type s4) (eqA x x_27_) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s2)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s5)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s0)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s0)) => + A : cupicef.Univ (cupicef.type s1) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) => + equ : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Equivalence s1 s4 A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) => + preo : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PreOrder s1 s4 A R) => + H : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.PartialOrder s1 s2 s3 s4 s5 cstr_0 + cstr_1 cstr_3 cstr_4 cstr_5 cstr_6 cstr_7 A eqA equ R preo) => + x : cupicef.Term (cupicef.type s1) A => + x_27_ : cupicef.Term (cupicef.type s1) A => + Hx : cupicef.Term (cupicef.type s4) (eqA x x_27_) => + y : cupicef.Term (cupicef.type s1) A => + y_27_ : cupicef.Term (cupicef.type s1) A => + Hy : cupicef.Term (cupicef.type s4) (eqA y y_27_) => + Hr : cupicef.Term (cupicef.type s4) (R x y) => + X : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA x x_27_) + (__ : cupicef.Term (cupicef.type s4) (eqA x x_27_) => + Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x))) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R x x_27_) (R x_27_ x)) => + eqA x x_27_))) => + a : + (__ : cupicef.Term (cupicef.type s4) (eqA x x_27_) -> + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x))) => + b : + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) -> + cupicef.Term (cupicef.type s4) (eqA x x_27_)) => + a0 : cupicef.Term (cupicef.type s4) (R x x_27_) => + b0 : cupicef.Term (cupicef.type s4) (R x_27_ x) => + let__X1 s0 s1 s2 s3 s4 s5 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 cstr_6 + cstr_7 cstr_8 cstr_9 A eqA equ R preo H x x_27_ Hx y y_27_ Hy Hr X a b + a0 b0 b0. + +def let__X4 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s2)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s5)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s0)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s0)) -> + A : cupicef.Univ (cupicef.type s1) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) -> + equ : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Equivalence s1 s4 A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) -> + preo : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PreOrder s1 s4 A R) -> + H : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.PartialOrder s1 s2 s3 s4 s5 cstr_0 + cstr_1 cstr_3 cstr_4 cstr_5 cstr_6 cstr_7 A eqA equ R preo) -> + x : cupicef.Term (cupicef.type s1) A -> + x_27_ : cupicef.Term (cupicef.type s1) A -> + Hx : cupicef.Term (cupicef.type s4) (eqA x x_27_) -> + y : cupicef.Term (cupicef.type s1) A -> + y_27_ : cupicef.Term (cupicef.type s1) A -> + Hy : cupicef.Term (cupicef.type s4) (eqA y y_27_) -> + Hr : cupicef.Term (cupicef.type s4) (R x y) -> + X : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA y y_27_) + (__ : cupicef.Term (cupicef.type s4) (eqA y y_27_) => + Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y))) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R y y_27_) (R y_27_ y)) => + eqA y y_27_))) -> + a : + (__ : cupicef.Term (cupicef.type s4) (eqA y y_27_) -> + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y))) -> + b : + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) -> + cupicef.Term (cupicef.type s4) (eqA y y_27_)) -> + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s2)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s5)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s0)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s0)) => + A : cupicef.Univ (cupicef.type s1) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) => + equ : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Equivalence s1 s4 A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) => + preo : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PreOrder s1 s4 A R) => + H : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.PartialOrder s1 s2 s3 s4 s5 cstr_0 + cstr_1 cstr_3 cstr_4 cstr_5 cstr_6 cstr_7 A eqA equ R preo) => + x : cupicef.Term (cupicef.type s1) A => + x_27_ : cupicef.Term (cupicef.type s1) A => + Hx : cupicef.Term (cupicef.type s4) (eqA x x_27_) => + y : cupicef.Term (cupicef.type s1) A => + y_27_ : cupicef.Term (cupicef.type s1) A => + Hy : cupicef.Term (cupicef.type s4) (eqA y y_27_) => + Hr : cupicef.Term (cupicef.type s4) (R x y) => + X : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA y y_27_) + (__ : cupicef.Term (cupicef.type s4) (eqA y y_27_) => + Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y))) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R y y_27_) (R y_27_ y)) => + eqA y y_27_))) => + a : + (__ : cupicef.Term (cupicef.type s4) (eqA y y_27_) -> + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y))) => + b : + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) -> + cupicef.Term (cupicef.type s4) (eqA y y_27_)) => + a Hy. + +def let__X5 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s2)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s5)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s0)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s0)) -> + A : cupicef.Univ (cupicef.type s1) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) -> + equ : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Equivalence s1 s4 A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) -> + preo : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PreOrder s1 s4 A R) -> + H : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.PartialOrder s1 s2 s3 s4 s5 cstr_0 + cstr_1 cstr_3 cstr_4 cstr_5 cstr_6 cstr_7 A eqA equ R preo) -> + x : cupicef.Term (cupicef.type s1) A -> + x_27_ : cupicef.Term (cupicef.type s1) A -> + Hx : cupicef.Term (cupicef.type s4) (eqA x x_27_) -> + y : cupicef.Term (cupicef.type s1) A -> + y_27_ : cupicef.Term (cupicef.type s1) A -> + Hy : cupicef.Term (cupicef.type s4) (eqA y y_27_) -> + Hr : cupicef.Term (cupicef.type s4) (R x y) -> + X : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA y y_27_) + (__ : cupicef.Term (cupicef.type s4) (eqA y y_27_) => + Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y))) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R y y_27_) (R y_27_ y)) => + eqA y y_27_))) -> + a : + (__ : cupicef.Term (cupicef.type s4) (eqA y y_27_) -> + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y))) -> + b : + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) -> + cupicef.Term (cupicef.type s4) (eqA y y_27_)) -> + a0 : cupicef.Term (cupicef.type s4) (R y y_27_) -> + b0 : cupicef.Term (cupicef.type s4) (R y_27_ y) -> + __ : cupicef.Term (cupicef.type s4) (R y y_27_) -> + __0 : cupicef.Term (cupicef.type s4) (R y_27_ y) -> + cupicef.Term (cupicef.type s4) (eqA y y_27_) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s2)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s5)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s0)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s0)) => + A : cupicef.Univ (cupicef.type s1) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) => + equ : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Equivalence s1 s4 A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) => + preo : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PreOrder s1 s4 A R) => + H : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.PartialOrder s1 s2 s3 s4 s5 cstr_0 + cstr_1 cstr_3 cstr_4 cstr_5 cstr_6 cstr_7 A eqA equ R preo) => + x : cupicef.Term (cupicef.type s1) A => + x_27_ : cupicef.Term (cupicef.type s1) A => + Hx : cupicef.Term (cupicef.type s4) (eqA x x_27_) => + y : cupicef.Term (cupicef.type s1) A => + y_27_ : cupicef.Term (cupicef.type s1) A => + Hy : cupicef.Term (cupicef.type s4) (eqA y y_27_) => + Hr : cupicef.Term (cupicef.type s4) (R x y) => + X : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA y y_27_) + (__ : cupicef.Term (cupicef.type s4) (eqA y y_27_) => + Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y))) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R y y_27_) (R y_27_ y)) => + eqA y y_27_))) => + a : + (__ : cupicef.Term (cupicef.type s4) (eqA y y_27_) -> + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y))) => + b : + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) -> + cupicef.Term (cupicef.type s4) (eqA y y_27_)) => + a0 : cupicef.Term (cupicef.type s4) (R y y_27_) => + b0 : cupicef.Term (cupicef.type s4) (R y_27_ y) => + X0 : cupicef.Term (cupicef.type s4) (R y y_27_) => + X1 : cupicef.Term (cupicef.type s4) (R y_27_ y) => + b + (Coq__Init__Datatypes.pair + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) (cupicef.axiom (cupicef.type s4)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_6 (R y y_27_)) + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) (cupicef.axiom (cupicef.type s4)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_7 (R y_27_ y)) + X0 X1). + +def let__X6 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s2)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s5)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s0)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s0)) -> + A : cupicef.Univ (cupicef.type s1) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) -> + equ : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Equivalence s1 s4 A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) -> + preo : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PreOrder s1 s4 A R) -> + H : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.PartialOrder s1 s2 s3 s4 s5 cstr_0 + cstr_1 cstr_3 cstr_4 cstr_5 cstr_6 cstr_7 A eqA equ R preo) -> + x : cupicef.Term (cupicef.type s1) A -> + x_27_ : cupicef.Term (cupicef.type s1) A -> + Hx : cupicef.Term (cupicef.type s4) (eqA x x_27_) -> + y : cupicef.Term (cupicef.type s1) A -> + y_27_ : cupicef.Term (cupicef.type s1) A -> + Hy : cupicef.Term (cupicef.type s4) (eqA y y_27_) -> + Hr : cupicef.Term (cupicef.type s4) (R x y) -> + X : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA y y_27_) + (__ : cupicef.Term (cupicef.type s4) (eqA y y_27_) => + Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y))) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R y y_27_) (R y_27_ y)) => + eqA y y_27_))) -> + a : + (__ : cupicef.Term (cupicef.type s4) (eqA y y_27_) -> + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y))) -> + b : + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) -> + cupicef.Term (cupicef.type s4) (eqA y y_27_)) -> + a0 : cupicef.Term (cupicef.type s4) (R y y_27_) -> + b0 : cupicef.Term (cupicef.type s4) (R y_27_ y) -> + __ : cupicef.Term (cupicef.type s4) (R y_27_ y) -> + cupicef.Term (cupicef.type s4) (eqA y y_27_) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s2)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s5)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s0)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s0)) => + A : cupicef.Univ (cupicef.type s1) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) => + equ : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Equivalence s1 s4 A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) => + preo : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PreOrder s1 s4 A R) => + H : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.PartialOrder s1 s2 s3 s4 s5 cstr_0 + cstr_1 cstr_3 cstr_4 cstr_5 cstr_6 cstr_7 A eqA equ R preo) => + x : cupicef.Term (cupicef.type s1) A => + x_27_ : cupicef.Term (cupicef.type s1) A => + Hx : cupicef.Term (cupicef.type s4) (eqA x x_27_) => + y : cupicef.Term (cupicef.type s1) A => + y_27_ : cupicef.Term (cupicef.type s1) A => + Hy : cupicef.Term (cupicef.type s4) (eqA y y_27_) => + Hr : cupicef.Term (cupicef.type s4) (R x y) => + X : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA y y_27_) + (__ : cupicef.Term (cupicef.type s4) (eqA y y_27_) => + Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y))) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R y y_27_) (R y_27_ y)) => + eqA y y_27_))) => + a : + (__ : cupicef.Term (cupicef.type s4) (eqA y y_27_) -> + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y))) => + b : + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) -> + cupicef.Term (cupicef.type s4) (eqA y y_27_)) => + a0 : cupicef.Term (cupicef.type s4) (R y y_27_) => + b0 : cupicef.Term (cupicef.type s4) (R y_27_ y) => + let__X5 s0 s1 s2 s3 s4 s5 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 cstr_6 + cstr_7 cstr_8 cstr_9 A eqA equ R preo H x x_27_ Hx y y_27_ Hy Hr X a b + a0 b0 a0. + +def let__X7 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s2)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s5)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s0)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s0)) -> + A : cupicef.Univ (cupicef.type s1) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) -> + equ : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Equivalence s1 s4 A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) -> + preo : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PreOrder s1 s4 A R) -> + H : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.PartialOrder s1 s2 s3 s4 s5 cstr_0 + cstr_1 cstr_3 cstr_4 cstr_5 cstr_6 cstr_7 A eqA equ R preo) -> + x : cupicef.Term (cupicef.type s1) A -> + x_27_ : cupicef.Term (cupicef.type s1) A -> + Hx : cupicef.Term (cupicef.type s4) (eqA x x_27_) -> + y : cupicef.Term (cupicef.type s1) A -> + y_27_ : cupicef.Term (cupicef.type s1) A -> + Hy : cupicef.Term (cupicef.type s4) (eqA y y_27_) -> + Hr : cupicef.Term (cupicef.type s4) (R x y) -> + X : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA y y_27_) + (__ : cupicef.Term (cupicef.type s4) (eqA y y_27_) => + Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y))) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R y y_27_) (R y_27_ y)) => + eqA y y_27_))) -> + a : + (__ : cupicef.Term (cupicef.type s4) (eqA y y_27_) -> + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y))) -> + b : + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) -> + cupicef.Term (cupicef.type s4) (eqA y y_27_)) -> + a0 : cupicef.Term (cupicef.type s4) (R y y_27_) -> + b0 : cupicef.Term (cupicef.type s4) (R y_27_ y) -> + cupicef.Term (cupicef.type s4) (eqA y y_27_) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s2)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s5)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s0)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s0)) => + A : cupicef.Univ (cupicef.type s1) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) => + equ : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Equivalence s1 s4 A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) => + preo : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PreOrder s1 s4 A R) => + H : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.PartialOrder s1 s2 s3 s4 s5 cstr_0 + cstr_1 cstr_3 cstr_4 cstr_5 cstr_6 cstr_7 A eqA equ R preo) => + x : cupicef.Term (cupicef.type s1) A => + x_27_ : cupicef.Term (cupicef.type s1) A => + Hx : cupicef.Term (cupicef.type s4) (eqA x x_27_) => + y : cupicef.Term (cupicef.type s1) A => + y_27_ : cupicef.Term (cupicef.type s1) A => + Hy : cupicef.Term (cupicef.type s4) (eqA y y_27_) => + Hr : cupicef.Term (cupicef.type s4) (R x y) => + X : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA y y_27_) + (__ : cupicef.Term (cupicef.type s4) (eqA y y_27_) => + Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y))) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R y y_27_) (R y_27_ y)) => + eqA y y_27_))) => + a : + (__ : cupicef.Term (cupicef.type s4) (eqA y y_27_) -> + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y))) => + b : + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) -> + cupicef.Term (cupicef.type s4) (eqA y y_27_)) => + a0 : cupicef.Term (cupicef.type s4) (R y y_27_) => + b0 : cupicef.Term (cupicef.type s4) (R y_27_ y) => + let__X6 s0 s1 s2 s3 s4 s5 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 cstr_6 + cstr_7 cstr_8 cstr_9 A eqA equ R preo H x x_27_ Hx y y_27_ Hy Hr X a b + a0 b0 b0. + +thm PartialOrder__proper__type : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s2)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s5)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s0)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s0)) -> + A : cupicef.Univ (cupicef.type s1) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) -> + equ : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Equivalence s1 s4 A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) -> + preo : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PreOrder s1 s4 A R) -> + H : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.PartialOrder s1 s2 s3 s4 s5 cstr_0 + cstr_1 cstr_3 cstr_4 cstr_5 cstr_6 cstr_7 A eqA equ R preo) -> + cupicef.Term (cupicef.type s3) + (Proper s0 s3 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4)))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s4)))) + cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s0))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) + (cupicef.type s0))) + (cupicef.trans (cupicef.axiom (cupicef.type s4)) + (cupicef.SortSome (cupicef.type s5)) cstr_1 + (cupicef.SortSome (cupicef.type s0)) cstr_9 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s1) + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s4)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s1) A => + cupicef.prod (cupicef.type s1) + (cupicef.axiom (cupicef.type s4)) + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s4))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type s1) A => + cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)))) + (respectful s1 s4 s0 s3 s0 s3 cstr_2 cstr_4 cstr_5 cupicef.I cupicef.I + A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s4)))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s4)))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s0))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) + (cupicef.type s0))) + (cupicef.trans (cupicef.axiom (cupicef.type s4)) + (cupicef.SortSome (cupicef.type s5)) cstr_1 + (cupicef.SortSome (cupicef.type s0)) cstr_9 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s1) + (cupicef.axiom (cupicef.type s4)) + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s4))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s1) A => + cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I))) + eqA + (respectful s1 s4 s5 s4 s0 s3 cstr_2 cstr_4 cstr_5 cstr_9 cstr_5 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + eqA + (Coq__Classes__CRelationClasses.iffT s4 s4 cstr_6 cstr_7 cstr_6 + cstr_7))) + R) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s2)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s5)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s0)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s3)) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s0)) => + A : cupicef.Univ (cupicef.type s1) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) => + equ : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.Equivalence s1 s4 A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s4))) + (Coq__Classes__CRelationClasses.crelation s1 s4 A) => + preo : + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (Coq__Classes__CRelationClasses.PreOrder s1 s4 A R) => + H : + cupicef.Term (cupicef.type s3) + (Coq__Classes__CRelationClasses.PartialOrder s1 s2 s3 s4 s5 cstr_0 + cstr_1 cstr_3 cstr_4 cstr_5 cstr_6 cstr_7 A eqA equ R preo) => + proper__sym__arrow__iffT__2 s0 s3 s0 s3 s5 s4 s1 s4 s1 s4 cstr_1 cupicef.I + cupicef.I cstr_9 cstr_5 cstr_6 cstr_7 cstr_2 cstr_4 cstr_5 cstr_2 cstr_4 + cstr_5 A eqA + (cupicef.cast + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)))) + (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (cupicef.prod (cupicef.type s1) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s4) (cupicef.type s4))) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s1) A => + cupicef.prod (cupicef.type s1) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s4) (cupicef.type s4))) + cupicef.I A + (y : cupicef.Term (cupicef.type s1) A => + cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA x y) + (X : cupicef.Term (cupicef.type s4) (eqA x y) => eqA y x)))) + (Coq__Classes__CRelationClasses.Symmetric s1 s4 A eqA) cupicef.I + (x : cupicef.Term (cupicef.type s1) A => + y : cupicef.Term (cupicef.type s1) A => + X : cupicef.Term (cupicef.type s4) (eqA x y) => + Coq__Classes__CRelationClasses.symmetry s1 s4 A eqA + (Coq__Classes__CRelationClasses.Equivalence__Symmetric s1 s4 A eqA + equ) + x y X)) + A eqA + (cupicef.cast + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)))) + (cupicef.sup (cupicef.type s1) (cupicef.type s4)) + (cupicef.prod (cupicef.type s1) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s4) (cupicef.type s4))) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s1) A => + cupicef.prod (cupicef.type s1) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s4) (cupicef.type s4))) + cupicef.I A + (y : cupicef.Term (cupicef.type s1) A => + cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA x y) + (X : cupicef.Term (cupicef.type s4) (eqA x y) => eqA y x)))) + (Coq__Classes__CRelationClasses.Symmetric s1 s4 A eqA) cupicef.I + (x : cupicef.Term (cupicef.type s1) A => + y : cupicef.Term (cupicef.type s1) A => + X : cupicef.Term (cupicef.type s4) (eqA x y) => + Coq__Classes__CRelationClasses.symmetry s1 s4 A eqA + (Coq__Classes__CRelationClasses.Equivalence__Symmetric s1 s4 A eqA + equ) + x y X)) + R + (cupicef.cast + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s4) (cupicef.type s3)))) + (cupicef.type s3) + (cupicef.prod (cupicef.type s1) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s4) (cupicef.type s3))) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s4) (cupicef.type s3)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s1) A => + cupicef.prod (cupicef.type s1) + (cupicef.rule (cupicef.type s4) (cupicef.type s3)) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s4) (cupicef.type s3))) + cupicef.I A + (x_27_ : cupicef.Term (cupicef.type s1) A => + cupicef.prod (cupicef.type s4) (cupicef.type s3) + (cupicef.rule (cupicef.type s4) (cupicef.type s3)) cupicef.I + (eqA x x_27_) + (Hx : cupicef.Term (cupicef.type s4) (eqA x x_27_) => + respectful s1 s4 s5 s4 s0 s3 cstr_2 cstr_4 cstr_5 cstr_9 + cstr_5 A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + eqA (Coq__Classes__CRelationClasses.arrow s4 s4) ( + R x) (R x_27_))))) + (Proper s0 s3 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s4)))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s4)))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s0))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) + (cupicef.type s0))) + (cupicef.trans (cupicef.axiom (cupicef.type s4)) + (cupicef.SortSome (cupicef.type s5)) cstr_1 + (cupicef.SortSome (cupicef.type s0)) cstr_9 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s1) + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s4)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s1) A => + cupicef.prod (cupicef.type s1) + (cupicef.axiom (cupicef.type s4)) + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s4))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type s1) A => + cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)))) + (respectful s1 s4 s0 s3 s0 s3 cstr_2 cstr_4 cstr_5 cupicef.I + cupicef.I A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s4)))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s4)))) + cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s1) (cupicef.type s0))) + cstr_2 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) + (cupicef.type s0))) + (cupicef.trans (cupicef.axiom (cupicef.type s4)) + (cupicef.SortSome (cupicef.type s5)) cstr_1 + (cupicef.SortSome (cupicef.type s0)) cstr_9 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s1) + (cupicef.axiom (cupicef.type s4)) + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s4))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s1) A => + cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I))) + eqA + (respectful s1 s4 s5 s4 s0 s3 cstr_2 cstr_4 cstr_5 cstr_9 + cstr_5 A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + eqA (Coq__Classes__CRelationClasses.arrow s4 s4))) + R) + cupicef.I + (x : cupicef.Term (cupicef.type s1) A => + x_27_ : cupicef.Term (cupicef.type s1) A => + Hx : cupicef.Term (cupicef.type s4) (eqA x x_27_) => + cupicef.cast + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)))) + (cupicef.type s3) + (cupicef.prod (cupicef.type s1) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s4) (cupicef.type s4))) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)))) + cupicef.I A + (y : cupicef.Term (cupicef.type s1) A => + cupicef.prod (cupicef.type s1) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s4) (cupicef.type s4))) + cupicef.I A + (y_27_ : cupicef.Term (cupicef.type s1) A => + cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) + cupicef.I (eqA y y_27_) + (Hy : cupicef.Term (cupicef.type s4) (eqA y y_27_) => + Coq__Classes__CRelationClasses.arrow s4 s4 (R x y) + (R x_27_ y_27_))))) + (respectful s1 s4 s5 s4 s0 s3 cstr_2 cstr_4 cstr_5 cstr_9 cstr_5 A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + eqA (Coq__Classes__CRelationClasses.arrow s4 s4) (R x) + (R x_27_)) + cupicef.I + (y : cupicef.Term (cupicef.type s1) A => + y_27_ : cupicef.Term (cupicef.type s1) A => + Hy : cupicef.Term (cupicef.type s4) (eqA y y_27_) => + cupicef.cast (cupicef.rule (cupicef.type s4) (cupicef.type s4)) + (cupicef.type s4) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (R x y) + (Hr : cupicef.Term (cupicef.type s4) (R x y) => + R x_27_ y_27_)) + (Coq__Classes__CRelationClasses.arrow s4 s4 (R x y) + (R x_27_ y_27_)) + cupicef.I + (Hr : cupicef.Term (cupicef.type s4) (R x y) => + Coq__Classes__CRelationClasses.transitivity s1 s4 A R + (Coq__Classes__CRelationClasses.PreOrder__Transitive s1 s4 A + R preo) + x_27_ x y_27_ + (cupicef.cast + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I (eqA x x_27_) + (__ : + cupicef.Term (cupicef.type s4) (eqA x x_27_) => + Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R x x_27_) (R x_27_ x))) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) => + eqA x x_27_))) + (X : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I (eqA x x_27_) + (__ : + cupicef.Term (cupicef.type s4) + (eqA x x_27_) => + Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x))) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) => + eqA x x_27_))) => + (X0 : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I (eqA x x_27_) + (__ : + cupicef.Term (cupicef.type s4) + (eqA x x_27_) => + Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x))) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) + (cupicef.type s4) (R x x_27_) + (R x_27_ x)) => + eqA x x_27_))) => + R x_27_ x) X)) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) + cupicef.I + (Coq__Classes__CRelationClasses.iffT s4 s4 cstr_6 + cstr_7 cstr_6 cstr_7 (eqA x x_27_) + (Coq__Classes__CRelationClasses.relation__conjunction + s1 s4 s4 s4 cupicef.I cstr_6 cupicef.I cstr_7 A + R + (Coq__Classes__CRelationClasses.flip s1 s1 s5 A + A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ + (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + cupicef.I)) + R) + x x_27_)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Classes__CRelationClasses.iffT s4 s4 cstr_6 + cstr_7 cstr_6 cstr_7 (eqA x x_27_) + (Coq__Classes__CRelationClasses.relation__conjunction + s1 s4 s4 s4 cupicef.I cstr_6 cupicef.I + cstr_7 A R + (Coq__Classes__CRelationClasses.flip s1 s1 + s5 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ + (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.axiom + (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + cupicef.I)) + R) + x x_27_)) => + R x_27_ x)) + cupicef.I + (X : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I (eqA x x_27_) + (__ : + cupicef.Term (cupicef.type s4) + (eqA x x_27_) => + Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R x x_27_) (R x_27_ x))) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) => + eqA x x_27_))) => + Coq__Init__Datatypes.prod__rect + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_6 + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I (eqA x x_27_) + (__ : + cupicef.Term (cupicef.type s4) (eqA x x_27_) => + Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R x x_27_) (R x_27_ x)))) + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_7 + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) => + eqA x x_27_))) + (cupicef.cast + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s4))) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I (eqA x x_27_) + (__ : + cupicef.Term (cupicef.type s4) + (eqA x x_27_) => + Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x))) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) + (cupicef.type s4) (R x x_27_) + (R x_27_ x)) => + eqA x x_27_))) + (X0 : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I (eqA x x_27_) + (__ : + cupicef.Term (cupicef.type s4) + (eqA x x_27_) => + Coq__Init__Datatypes.prod + (cupicef.type s4) + (cupicef.type s4) (R x x_27_) + (R x_27_ x))) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I + (Coq__Init__Datatypes.prod + (cupicef.type s4) + (cupicef.type s4) (R x x_27_) + (R x_27_ x)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) + (cupicef.type s4) (R x x_27_) + (R x_27_ x)) => + eqA x x_27_))) => + cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + (cupicef.prod (cupicef.type s4) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I (eqA x x_27_) + (__ : + cupicef.Term (cupicef.type s4) + (eqA x x_27_) => + Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x))) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) + (cupicef.type s4) (R x x_27_) + (R x_27_ x)) => + eqA x x_27_))) + (p : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I (eqA x x_27_) + (__ : + cupicef.Term (cupicef.type s4) + (eqA x x_27_) => + Coq__Init__Datatypes.prod + (cupicef.type s4) + (cupicef.type s4) (R x x_27_) + (R x_27_ x))) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I + (Coq__Init__Datatypes.prod + (cupicef.type s4) + (cupicef.type s4) (R x x_27_) + (R x_27_ x)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) + (cupicef.type s4) (R x x_27_) + (R x_27_ x)) => + eqA x x_27_))) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cstr_8 + (X0 : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I (eqA x x_27_) + (__ : + cupicef.Term (cupicef.type s4) + (eqA x x_27_) => + Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x))) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) + (cupicef.type s4) (R x x_27_) + (R x_27_ x)) => + eqA x x_27_))) => + R x_27_ x)) + (a : + (__ : cupicef.Term (cupicef.type s4) (eqA x x_27_) -> + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R x x_27_) (R x_27_ x))) => + b : + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R x x_27_) (R x_27_ x)) -> + cupicef.Term (cupicef.type s4) (eqA x x_27_)) => + Coq__Init__Datatypes.prod__rect + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_6 (R x x_27_)) + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_7 (R x_27_ x)) + (cupicef.cast + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s4))) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R x x_27_) (R x_27_ x)) + (X0 : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) => + cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + cupicef.I)) + (cupicef.prod (cupicef.type s4) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R x x_27_) (R x_27_ x)) + (p : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cstr_8 + (X0 : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R x x_27_) (R x_27_ x)) => + R x_27_ x)) + (a0 : cupicef.Term (cupicef.type s4) (R x x_27_) => + b0 : cupicef.Term (cupicef.type s4) (R x_27_ x) => + b0) + (let__X s0 s1 s2 s3 s4 s5 cstr_0 cstr_1 cstr_2 + cstr_3 cstr_4 cstr_5 cstr_6 cstr_7 cstr_8 cstr_9 + A eqA equ R preo H x x_27_ Hx y y_27_ Hy Hr X a + b)) + X) + (Coq__Classes__CRelationClasses.partial__order__equivalence + s1 s2 s3 s4 s5 cstr_0 cstr_1 cstr_3 cstr_4 cstr_5 + cstr_6 cstr_7 A eqA equ R preo H x x_27_)) + (Coq__Classes__CRelationClasses.transitivity s1 s4 A R + (Coq__Classes__CRelationClasses.PreOrder__Transitive s1 + s4 A R preo) + x y y_27_ Hr + (cupicef.cast + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I (eqA y y_27_) + (__ : + cupicef.Term (cupicef.type s4) + (eqA y y_27_) => + Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R y y_27_) (R y_27_ y))) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R y y_27_) (R y_27_ y)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) => + eqA y y_27_))) + (X : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I (eqA y y_27_) + (__ : + cupicef.Term (cupicef.type s4) + (eqA y y_27_) => + Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y))) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) + (cupicef.type s4) (R y y_27_) + (R y_27_ y)) => + eqA y y_27_))) => + (X0 : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I (eqA y y_27_) + (__ : + cupicef.Term (cupicef.type s4) + (eqA y y_27_) => + Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y))) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) + (cupicef.type s4) (R y y_27_) + (R y_27_ y)) => + eqA y y_27_))) => + R y y_27_) X)) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) + cupicef.I + (Coq__Classes__CRelationClasses.iffT s4 s4 cstr_6 + cstr_7 cstr_6 cstr_7 (eqA y y_27_) + (Coq__Classes__CRelationClasses.relation__conjunction + s1 s4 s4 s4 cupicef.I cstr_6 cupicef.I cstr_7 + A R + (Coq__Classes__CRelationClasses.flip s1 s1 s5 + A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ + (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + cupicef.I)) + R) + y y_27_)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Classes__CRelationClasses.iffT s4 s4 + cstr_6 cstr_7 cstr_6 cstr_7 (eqA y y_27_) + (Coq__Classes__CRelationClasses.relation__conjunction + s1 s4 s4 s4 cupicef.I cstr_6 cupicef.I + cstr_7 A R + (Coq__Classes__CRelationClasses.flip s1 + s1 s5 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom + (cupicef.type s4))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ + (cupicef.axiom + (cupicef.type s4)) + (cupicef.axiom + (cupicef.axiom + (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom + (cupicef.type s5)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom + (cupicef.type s4)) + cupicef.I)) + R) + y y_27_)) => + R y y_27_)) + cupicef.I + (X : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I (eqA y y_27_) + (__ : + cupicef.Term (cupicef.type s4) + (eqA y y_27_) => + Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y))) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) => + eqA y y_27_))) => + Coq__Init__Datatypes.prod__rect + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_6 + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I (eqA y y_27_) + (__ : + cupicef.Term (cupicef.type s4) + (eqA y y_27_) => + Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R y y_27_) (R y_27_ y)))) + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_7 + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R y y_27_) (R y_27_ y)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) => + eqA y y_27_))) + (cupicef.cast + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s4))) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I (eqA y y_27_) + (__ : + cupicef.Term (cupicef.type s4) + (eqA y y_27_) => + Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y))) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) + (cupicef.type s4) (R y y_27_) + (R y_27_ y)) => + eqA y y_27_))) + (X0 : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I (eqA y y_27_) + (__ : + cupicef.Term (cupicef.type s4) + (eqA y y_27_) => + Coq__Init__Datatypes.prod + (cupicef.type s4) + (cupicef.type s4) (R y y_27_) + (R y_27_ y))) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I + (Coq__Init__Datatypes.prod + (cupicef.type s4) + (cupicef.type s4) (R y y_27_) + (R y_27_ y)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) + (cupicef.type s4) + (R y y_27_) (R y_27_ y)) => + eqA y y_27_))) => + cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + cupicef.I)) + (cupicef.prod (cupicef.type s4) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I (eqA y y_27_) + (__ : + cupicef.Term (cupicef.type s4) + (eqA y y_27_) => + Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y))) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) + (cupicef.type s4) (R y y_27_) + (R y_27_ y)) => + eqA y y_27_))) + (p : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I (eqA y y_27_) + (__ : + cupicef.Term (cupicef.type s4) + (eqA y y_27_) => + Coq__Init__Datatypes.prod + (cupicef.type s4) + (cupicef.type s4) (R y y_27_) + (R y_27_ y))) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I + (Coq__Init__Datatypes.prod + (cupicef.type s4) + (cupicef.type s4) (R y y_27_) + (R y_27_ y)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) + (cupicef.type s4) + (R y y_27_) (R y_27_ y)) => + eqA y y_27_))) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cstr_8 + (X0 : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I (eqA y y_27_) + (__ : + cupicef.Term (cupicef.type s4) + (eqA y y_27_) => + Coq__Init__Datatypes.prod + (cupicef.type s4) + (cupicef.type s4) (R y y_27_) + (R y_27_ y))) + (cupicef.prod (cupicef.type s4) + (cupicef.type s4) + (cupicef.rule (cupicef.type s4) + (cupicef.type s4)) + cupicef.I + (Coq__Init__Datatypes.prod + (cupicef.type s4) + (cupicef.type s4) (R y y_27_) + (R y_27_ y)) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) + (cupicef.type s4) (R y y_27_) + (R y_27_ y)) => + eqA y y_27_))) => + R y y_27_)) + (a : + (__ : + cupicef.Term (cupicef.type s4) (eqA y y_27_) -> + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R y y_27_) (R y_27_ y))) => + b : + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) -> + cupicef.Term (cupicef.type s4) (eqA y y_27_)) => + Coq__Init__Datatypes.prod__rect + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_6 (R y y_27_)) + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_7 (R y_27_ y)) + (cupicef.cast + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s4))) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom (cupicef.type s4))) + cupicef.I + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) + (X0 : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) + (cupicef.type s4) (R y y_27_) + (R y_27_ y)) => + cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + cupicef.I)) + (cupicef.prod (cupicef.type s4) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s4) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) + (p : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) + (cupicef.type s4) (R y y_27_) + (R y_27_ y)) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cstr_8 + (X0 : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod + (cupicef.type s4) (cupicef.type s4) + (R y y_27_) (R y_27_ y)) => + R y y_27_)) + (a0 : cupicef.Term (cupicef.type s4) (R y y_27_) => + b0 : cupicef.Term (cupicef.type s4) (R y_27_ y) => + a0) + (let__X4 s0 s1 s2 s3 s4 s5 cstr_0 cstr_1 cstr_2 + cstr_3 cstr_4 cstr_5 cstr_6 cstr_7 cstr_8 + cstr_9 A eqA equ R preo H x x_27_ Hx y y_27_ + Hy Hr X a b)) + X) + (Coq__Classes__CRelationClasses.partial__order__equivalence + s1 s2 s3 s4 s5 cstr_0 cstr_1 cstr_3 cstr_4 cstr_5 + cstr_6 cstr_7 A eqA equ R preo H y y_27_))))))). + +#TYPE_CLASS_INSTANCE PartialOrder__proper__type. + +thm PartialOrder__StrictOrder : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + s7 : cupicef.Nat -> + s8 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s5)) + (cupicef.type (cupicef.s cupicef.z))) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) (cupicef.type s3)) -> + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) (cupicef.type s8)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s1)) -> + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s2) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s4)) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s4)) -> + cstr_8 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s5)) -> + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s8)) -> + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s1)) -> + cstr_11 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s5)) -> + cstr_12 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_13 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_14 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s6) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s6 s7 A) -> + equ : + cupicef.Term (cupicef.sup (cupicef.type s6) (cupicef.type s7)) + (Coq__Classes__CRelationClasses.Equivalence s6 s7 A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s6 s7 A) -> + preo : + cupicef.Term (cupicef.sup (cupicef.type s6) (cupicef.type s7)) + (Coq__Classes__CRelationClasses.PreOrder s6 s7 A R) -> + H : + cupicef.Term (cupicef.type s5) + (Coq__Classes__CRelationClasses.PartialOrder s6 s8 s5 s7 s3 cstr_2 + cstr_1 cstr_9 cstr_8 cstr_11 cstr_12 cstr_13 A eqA equ R preo) -> + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s6))) + (Coq__Classes__CRelationClasses.StrictOrder s6 s0 s1 A + (Coq__Classes__CRelationClasses.relation__conjunction s6 s7 s2 s1 + cstr_10 cstr_12 cstr_3 cstr_4 A R + (Coq__Classes__CRelationClasses.complement s6 s7 s2 A eqA))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + s7 : cupicef.Nat => + s8 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s5)) + (cupicef.type (cupicef.s cupicef.z))) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) (cupicef.type s3)) => + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s7)) (cupicef.type s8)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s1)) => + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s2) (cupicef.type (cupicef.s cupicef.z))) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s4)) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s4)) => + cstr_8 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s5)) => + cstr_9 : cupicef.eps (cupicef.Cumul (cupicef.type s6) (cupicef.type s8)) => + cstr_10 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s1)) => + cstr_11 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s5)) => + cstr_12 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) => + cstr_13 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) => + cstr_14 : + cupicef.eps + (cupicef.Cumul (cupicef.type s7) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s6) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s6 s7 A) => + equ : + cupicef.Term (cupicef.sup (cupicef.type s6) (cupicef.type s7)) + (Coq__Classes__CRelationClasses.Equivalence s6 s7 A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s6) (cupicef.axiom (cupicef.type s7))) + (Coq__Classes__CRelationClasses.crelation s6 s7 A) => + preo : + cupicef.Term (cupicef.sup (cupicef.type s6) (cupicef.type s7)) + (Coq__Classes__CRelationClasses.PreOrder s6 s7 A R) => + H : + cupicef.Term (cupicef.type s5) + (Coq__Classes__CRelationClasses.PartialOrder s6 s8 s5 s7 s3 cstr_2 + cstr_1 cstr_9 cstr_8 cstr_11 cstr_12 cstr_13 A eqA equ R preo) => + Coq__Classes__CRelationClasses.Build__StrictOrder s6 s0 s1 A + (Coq__Classes__CRelationClasses.relation__conjunction s6 s7 s2 s1 + cstr_10 cstr_12 cstr_3 cstr_4 A R + (Coq__Classes__CRelationClasses.complement s6 s7 s2 A eqA)) + (cupicef.cast + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) cupicef.prop)) + (cupicef.sup (cupicef.type s0) (cupicef.type s6)) + (cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s7) cupicef.prop) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) cupicef.prop)) + cupicef.I A + (x : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s7) cupicef.prop + (cupicef.rule (cupicef.type s7) cupicef.prop) cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s7) cupicef.prop + (R x x) + (cupicef.prod (cupicef.type s7) cupicef.prop + (cupicef.rule (cupicef.type s7) cupicef.prop) cupicef.I + (eqA x x) + (__ : cupicef.Term (cupicef.type s7) (eqA x x) => + Coq__Init__Logic.False))) + (X : + cupicef.Term (cupicef.type s7) + (Coq__Init__Datatypes.prod (cupicef.type s7) cupicef.prop + (R x x) + (cupicef.prod (cupicef.type s7) cupicef.prop + (cupicef.rule (cupicef.type s7) cupicef.prop) + cupicef.I (eqA x x) + (__ : cupicef.Term (cupicef.type s7) (eqA x x) => + Coq__Init__Logic.False))) => + Coq__Init__Logic.False))) + (Coq__Classes__CRelationClasses.Irreflexive s6 s0 s1 A + (Coq__Classes__CRelationClasses.relation__conjunction s6 s7 s2 s1 + cstr_10 cstr_12 cstr_3 cstr_4 A R + (Coq__Classes__CRelationClasses.complement s6 s7 s2 A eqA))) + cupicef.I + (x : cupicef.Term (cupicef.type s6) A => + X : + cupicef.Term (cupicef.type s7) + (Coq__Init__Datatypes.prod (cupicef.type s7) cupicef.prop + (R x x) + (cupicef.prod (cupicef.type s7) cupicef.prop + (cupicef.rule (cupicef.type s7) cupicef.prop) cupicef.I + (eqA x x) + (__ : cupicef.Term (cupicef.type s7) (eqA x x) => + Coq__Init__Logic.False))) => + Coq__Init__Datatypes.match____prod cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_12 (R x x)) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type s7) cupicef.prop + (cupicef.rule (cupicef.type s7) cupicef.prop) cupicef.I + (eqA x x) + (__ : cupicef.Term (cupicef.type s7) (eqA x x) => + Coq__Init__Logic.False))) + (p : + cupicef.Term (cupicef.type s7) + (Coq__Init__Datatypes.prod (cupicef.type s7) cupicef.prop + (R x x) + (cupicef.prod (cupicef.type s7) cupicef.prop + (cupicef.rule (cupicef.type s7) cupicef.prop) cupicef.I + (eqA x x) + (__ : cupicef.Term (cupicef.type s7) (eqA x x) => + Coq__Init__Logic.False))) => + Coq__Init__Logic.False) + (__tmp : cupicef.Term (cupicef.type s7) (R x x) => + Hx : + (__ : cupicef.Term (cupicef.type s7) (eqA x x) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Hx + (Coq__Classes__CRelationClasses.Equivalence__Reflexive s6 s7 A + eqA equ x)) + X)) + (cupicef.cast + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s7) (cupicef.type s7)))))) + (cupicef.sup (cupicef.type s1) (cupicef.type s6)) + (cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s7) (cupicef.type s7))))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s7) (cupicef.type s7)))))) + cupicef.I A + (x : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s7) (cupicef.type s7)))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s7) (cupicef.type s7))))) + cupicef.I A + (y : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s7) (cupicef.type s7))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s7) (cupicef.type s7)))) + cupicef.I A + (z : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s7) + (cupicef.rule (cupicef.type s7) (cupicef.type s7)) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s7) (cupicef.type s7))) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s7) cupicef.prop + (R x y) + (cupicef.prod (cupicef.type s7) cupicef.prop + (cupicef.rule (cupicef.type s7) cupicef.prop) + cupicef.I (eqA x y) + (__ : cupicef.Term (cupicef.type s7) (eqA x y) => + Coq__Init__Logic.False))) + (X : + cupicef.Term (cupicef.type s7) + (Coq__Init__Datatypes.prod (cupicef.type s7) + cupicef.prop (R x y) + (cupicef.prod (cupicef.type s7) cupicef.prop + (cupicef.rule (cupicef.type s7) cupicef.prop) + cupicef.I (eqA x y) + (__ : cupicef.Term (cupicef.type s7) (eqA x y) => + Coq__Init__Logic.False))) => + cupicef.prod (cupicef.type s7) (cupicef.type s7) + (cupicef.rule (cupicef.type s7) (cupicef.type s7)) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s7) + cupicef.prop (R y z) + (cupicef.prod (cupicef.type s7) cupicef.prop + (cupicef.rule (cupicef.type s7) cupicef.prop) + cupicef.I (eqA y z) + (__ : cupicef.Term (cupicef.type s7) (eqA y z) => + Coq__Init__Logic.False))) + (__ : + cupicef.Term (cupicef.type s7) + (Coq__Init__Datatypes.prod (cupicef.type s7) + cupicef.prop (R y z) + (cupicef.prod (cupicef.type s7) cupicef.prop + (cupicef.rule (cupicef.type s7) + cupicef.prop) + cupicef.I (eqA y z) + (__ : + cupicef.Term (cupicef.type s7) (eqA y z) => + Coq__Init__Logic.False))) => + Coq__Init__Datatypes.prod (cupicef.type s7) + cupicef.prop (R x z) + (cupicef.prod (cupicef.type s7) cupicef.prop + (cupicef.rule (cupicef.type s7) cupicef.prop) + cupicef.I (eqA x z) + (__0 : cupicef.Term (cupicef.type s7) (eqA x z) => + Coq__Init__Logic.False)))))))) + (Coq__Classes__CRelationClasses.Transitive s6 s1 A + (Coq__Classes__CRelationClasses.relation__conjunction s6 s7 s2 s1 + cstr_10 cstr_12 cstr_3 cstr_4 A R + (Coq__Classes__CRelationClasses.complement s6 s7 s2 A eqA))) + cupicef.I + (x : cupicef.Term (cupicef.type s6) A => + y : cupicef.Term (cupicef.type s6) A => + z : cupicef.Term (cupicef.type s6) A => + X : + cupicef.Term (cupicef.type s7) + (Coq__Init__Datatypes.prod (cupicef.type s7) cupicef.prop + (R x y) + (cupicef.prod (cupicef.type s7) cupicef.prop + (cupicef.rule (cupicef.type s7) cupicef.prop) cupicef.I + (eqA x y) + (__ : cupicef.Term (cupicef.type s7) (eqA x y) => + Coq__Init__Logic.False))) => + Coq__Init__Datatypes.match____prod (cupicef.type s7) + (cupicef.cast (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_12 (R x y)) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type s7) cupicef.prop + (cupicef.rule (cupicef.type s7) cupicef.prop) cupicef.I + (eqA x y) + (__ : cupicef.Term (cupicef.type s7) (eqA x y) => + Coq__Init__Logic.False))) + (p : + cupicef.Term (cupicef.type s7) + (Coq__Init__Datatypes.prod (cupicef.type s7) cupicef.prop + (R x y) + (cupicef.prod (cupicef.type s7) cupicef.prop + (cupicef.rule (cupicef.type s7) cupicef.prop) cupicef.I + (eqA x y) + (__ : cupicef.Term (cupicef.type s7) (eqA x y) => + Coq__Init__Logic.False))) => + cupicef.prod (cupicef.type s7) (cupicef.type s7) + (cupicef.rule (cupicef.type s7) (cupicef.type s7)) cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s7) cupicef.prop + (R y z) + (cupicef.prod (cupicef.type s7) cupicef.prop + (cupicef.rule (cupicef.type s7) cupicef.prop) cupicef.I + (eqA y z) + (__ : cupicef.Term (cupicef.type s7) (eqA y z) => + Coq__Init__Logic.False))) + (__ : + cupicef.Term (cupicef.type s7) + (Coq__Init__Datatypes.prod (cupicef.type s7) cupicef.prop + (R y z) + (cupicef.prod (cupicef.type s7) cupicef.prop + (cupicef.rule (cupicef.type s7) cupicef.prop) + cupicef.I (eqA y z) + (__ : cupicef.Term (cupicef.type s7) (eqA y z) => + Coq__Init__Logic.False))) => + Coq__Init__Datatypes.prod (cupicef.type s7) cupicef.prop + (R x z) + (cupicef.prod (cupicef.type s7) cupicef.prop + (cupicef.rule (cupicef.type s7) cupicef.prop) cupicef.I + (eqA x z) + (__0 : cupicef.Term (cupicef.type s7) (eqA x z) => + Coq__Init__Logic.False)))) + (Hxy : cupicef.Term (cupicef.type s7) (R x y) => + Hxy_27_ : + (__ : cupicef.Term (cupicef.type s7) (eqA x y) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + X0 : + cupicef.Term (cupicef.type s7) + (Coq__Init__Datatypes.prod (cupicef.type s7) cupicef.prop + (R y z) + (cupicef.prod (cupicef.type s7) cupicef.prop + (cupicef.rule (cupicef.type s7) cupicef.prop) cupicef.I + (eqA y z) + (__ : cupicef.Term (cupicef.type s7) (eqA y z) => + Coq__Init__Logic.False))) => + Coq__Init__Datatypes.match____prod (cupicef.type s7) + (cupicef.cast (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_12 (R y z)) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type s7) cupicef.prop + (cupicef.rule (cupicef.type s7) cupicef.prop) cupicef.I + (eqA y z) + (__ : cupicef.Term (cupicef.type s7) (eqA y z) => + Coq__Init__Logic.False))) + (p : + cupicef.Term (cupicef.type s7) + (Coq__Init__Datatypes.prod (cupicef.type s7) cupicef.prop + (R y z) + (cupicef.prod (cupicef.type s7) cupicef.prop + (cupicef.rule (cupicef.type s7) cupicef.prop) + cupicef.I (eqA y z) + (__ : cupicef.Term (cupicef.type s7) (eqA y z) => + Coq__Init__Logic.False))) => + Coq__Init__Datatypes.prod (cupicef.type s7) cupicef.prop + (R x z) + (cupicef.prod (cupicef.type s7) cupicef.prop + (cupicef.rule (cupicef.type s7) cupicef.prop) cupicef.I + (eqA x z) + (__ : cupicef.Term (cupicef.type s7) (eqA x z) => + Coq__Init__Logic.False))) + (Hyz : cupicef.Term (cupicef.type s7) (R y z) => + Hyz_27_ : + (__ : cupicef.Term (cupicef.type s7) (eqA y z) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Datatypes.pair + (cupicef.cast (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_12 (R x z)) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type s7) cupicef.prop + (cupicef.rule (cupicef.type s7) cupicef.prop) + cupicef.I (eqA x z) + (__ : cupicef.Term (cupicef.type s7) (eqA x z) => + Coq__Init__Logic.False))) + (Coq__Classes__CRelationClasses.PreOrder__Transitive s6 s7 A + R preo x y z Hxy Hyz) + (Hxz : cupicef.Term (cupicef.type s7) (eqA x z) => + Hxy_27_ + (Coq__Classes__CRelationClasses.partial__order__antisym + s6 s8 s5 s3 s7 cstr_2 cstr_1 cstr_9 cstr_8 cstr_11 + cstr_12 cstr_13 A eqA equ R preo H x y Hxy + ((lemma : cupicef.Term (cupicef.type s7) (eqA x z) => + Reflexive__partial__app__morphism s6 s4 s4 s5 s7 s5 + cstr_7 cstr_8 cupicef.I cstr_11 cupicef.I A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) + (cupicef.type s4))) + cstr_7 + (cupicef.BoolSome + (cupicef.Cumul + (cupicef.axiom (cupicef.type s7)) + (cupicef.type s4))) + (cupicef.trans + (cupicef.axiom (cupicef.type s7)) + (cupicef.SortSome (cupicef.type s3)) + cstr_1 + (cupicef.SortSome (cupicef.type s4)) + cstr_5 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => + cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I))) + eqA + (respectful s6 s7 s3 s7 s4 s5 cstr_7 cstr_8 + cstr_11 cstr_5 cstr_11 A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I)) + eqA + (Coq__Classes__CRelationClasses.flip s3 s3 s3 + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I)) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I)) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I)) + (Coq__Classes__CRelationClasses.arrow s7 s7))) + R + (subrelation__proper s4 s5 cstr_0 cstr_6 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) + (cupicef.type s4))) + cstr_7 + (cupicef.BoolSome + (cupicef.Cumul + (cupicef.axiom (cupicef.type s7)) + (cupicef.type s4))) + (cupicef.trans + (cupicef.axiom (cupicef.type s7)) + (cupicef.SortSome (cupicef.type s3)) + cstr_1 + (cupicef.SortSome (cupicef.type s4)) + cstr_5 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (cupicef.rule (cupicef.type s6) + (cupicef.rule (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => + cupicef.prod (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + cupicef.I A + (__0 : + cupicef.Term (cupicef.type s6) A => + cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I)))) + (respectful s6 s7 s4 s5 s4 s5 cstr_7 cstr_8 + cstr_11 cupicef.I cupicef.I A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) + (cupicef.type s4))) + cstr_7 + (cupicef.BoolSome + (cupicef.Cumul + (cupicef.axiom (cupicef.type s7)) + (cupicef.type s4))) + (cupicef.trans + (cupicef.axiom (cupicef.type s7)) + (cupicef.SortSome (cupicef.type s3)) + cstr_1 + (cupicef.SortSome (cupicef.type s4)) + cstr_5 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => + cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I))) + eqA + (respectful s6 s7 s3 s7 s4 s5 cstr_7 cstr_8 + cstr_11 cstr_5 cstr_11 A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I)) + eqA + (Coq__Classes__CRelationClasses.iffT s7 + s7 cstr_12 cstr_13 cstr_12 cstr_13))) + R + (PartialOrder__proper__type s4 s6 s8 s5 s7 s3 + cstr_2 cstr_1 cstr_7 cstr_9 cstr_8 cstr_11 + cstr_12 cstr_13 cstr_14 cstr_5 A eqA equ R + preo H) + (respectful s6 s7 s4 s5 s4 s5 cstr_7 cstr_8 + cstr_11 cupicef.I cupicef.I A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) + (cupicef.type s4))) + cstr_7 + (cupicef.BoolSome + (cupicef.Cumul + (cupicef.axiom (cupicef.type s7)) + (cupicef.type s4))) + (cupicef.trans + (cupicef.axiom (cupicef.type s7)) + (cupicef.SortSome (cupicef.type s3)) + cstr_1 + (cupicef.SortSome (cupicef.type s4)) + cstr_5 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => + cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I))) + eqA + (respectful s6 s7 s3 s7 s4 s5 cstr_7 cstr_8 + cstr_11 cstr_5 cstr_11 A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I)) + eqA + (Coq__Classes__CRelationClasses.flip s3 + s3 s3 + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom + (cupicef.axiom + (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I)) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom + (cupicef.axiom + (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I)) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom + (cupicef.axiom + (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I)) + (Coq__Classes__CRelationClasses.arrow + s7 s7)))) + Coq__Init__Datatypes.tt + (subrelation__respectful s6 s4 s5 s5 s7 s5 s7 + s4 s5 cstr_7 cstr_8 cstr_8 cstr_11 cupicef.I + cstr_11 cupicef.I cupicef.I A eqA eqA + (subrelation__refl s6 s7 A eqA) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom + (cupicef.sup (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s6) + (cupicef.type s4))) + cstr_7 + (cupicef.BoolSome + (cupicef.Cumul + (cupicef.axiom (cupicef.type s7)) + (cupicef.type s4))) + (cupicef.trans + (cupicef.axiom (cupicef.type s7)) + (cupicef.SortSome (cupicef.type s3)) + cstr_1 + (cupicef.SortSome (cupicef.type s4)) + cstr_5 cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s6) + (cupicef.axiom (cupicef.type s7)) + (cupicef.rule (cupicef.type s6) + (cupicef.axiom (cupicef.type s7))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s6) A => + cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I))) + (respectful s6 s7 s3 s7 s4 s5 cstr_7 cstr_8 + cstr_11 cstr_5 cstr_11 A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I)) + eqA + (Coq__Classes__CRelationClasses.iffT s7 + s7 cstr_12 cstr_13 cstr_12 cstr_13)) + (respectful s6 s7 s3 s7 s4 s5 cstr_7 cstr_8 + cstr_11 cstr_5 cstr_11 A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I)) + eqA + (Coq__Classes__CRelationClasses.flip s3 + s3 s3 + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom + (cupicef.axiom + (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I)) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom + (cupicef.axiom + (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I)) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom + (cupicef.axiom + (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I)) + (Coq__Classes__CRelationClasses.arrow + s7 s7))) + (subrelation__respectful s6 s4 s5 s5 s7 s7 + s7 s3 s7 cstr_7 cstr_8 cstr_8 cstr_11 + cstr_11 cstr_11 cstr_5 cstr_11 A eqA eqA + (subrelation__refl s6 s7 A eqA) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I)) + (Coq__Classes__CRelationClasses.iffT s7 + s7 cstr_12 cstr_13 cstr_12 cstr_13) + (Coq__Classes__CRelationClasses.flip s3 + s3 s3 + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom + (cupicef.axiom + (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I)) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom + (cupicef.axiom + (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I)) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom + (cupicef.axiom + (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I)) + (Coq__Classes__CRelationClasses.arrow + s7 s7)) + (flip2 s3 s7 s7 s3 cstr_1 + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s7))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s7)) + (cupicef.axiom + (cupicef.axiom + (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s7) + (cupicef.axiom (cupicef.type s7)) + cupicef.I)) + (Coq__Classes__CRelationClasses.iffT + s7 s7 cstr_12 cstr_13 cstr_12 + cstr_13) + (y0 : cupicef.Univ (cupicef.type s7) => + x0 : cupicef.Univ (cupicef.type s7) => + cupicef.prod (cupicef.type s7) + (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.type s7)) + cupicef.I x0 + (__ : + cupicef.Term (cupicef.type s7) + x0 => + y0)) + (iffT__flip__arrow__subrelation s3 s7 + s3 s7 cstr_1 cstr_1 cupicef.I + cstr_12 cstr_13 cstr_14))))) + y + (reflexive__proper__proxy s6 s7 A eqA + (Coq__Classes__CRelationClasses.Equivalence__Reflexive + s6 s7 A eqA equ) + y) + x z lemma) + Hxz Hyz)))) + X0) + X)). + +thm StrictOrder__PreOrder : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + s7 : cupicef.Nat -> + s8 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s2)) (cupicef.type s3)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s2)) (cupicef.type s5)) -> + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s6)) + (cupicef.type (cupicef.s cupicef.z))) -> + cstr_3 : cupicef.eps (cupicef.Cumul cupicef.set (cupicef.type s0)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s0)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s4)) -> + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s6)) -> + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.type s2) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s2) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_9 : + cupicef.eps + (cupicef.Cumul (cupicef.type s2) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_10 : + cupicef.eps + (cupicef.Cumul (cupicef.type s2) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_11 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s5)) -> + cstr_12 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s6)) -> + cstr_13 : + cupicef.eps + (cupicef.Cumul (cupicef.type s5) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_14 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s4)) -> + cstr_15 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s5)) -> + cstr_16 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s0)) -> + cstr_17 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s4)) -> + cstr_18 : + cupicef.eps + (cupicef.Cumul (cupicef.type s8) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s7) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s7) (cupicef.axiom (cupicef.type s8))) + (Coq__Classes__CRelationClasses.crelation s7 s8 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s7) (cupicef.type s8)) + (Coq__Classes__CRelationClasses.Equivalence s7 s8 A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type s2)) (cupicef.type s7)) + (Coq__Classes__CRelationClasses.crelation s7 s2 A) -> + H0 : + cupicef.Term + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) (cupicef.type s7))) + (Coq__Classes__CRelationClasses.StrictOrder s7 s1 s2 A R) -> + H1 : + cupicef.Term (cupicef.type s6) + (Proper s5 s6 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ + (cupicef.sup (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7)) + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s5) (cupicef.axiom (cupicef.type s5)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) (cupicef.type s5))) + cstr_15 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s2)) + (cupicef.type s5))) + cstr_1 cupicef.BoolNone) + (cupicef.prod (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s2)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s7) A => + cupicef.prod (cupicef.type s7) + (cupicef.axiom (cupicef.type s2)) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s2))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type s7) A => + cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)))) + (respectful s7 s8 s5 s6 s5 s6 cstr_15 + (cupicef.trans (cupicef.type s7) + (cupicef.SortSome (cupicef.type s4)) cstr_14 + (cupicef.SortSome (cupicef.type s6)) cstr_12 cupicef.SortNone) + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s4)) cstr_17 + (cupicef.SortSome (cupicef.type s6)) cstr_12 cupicef.SortNone) + cupicef.I cupicef.I A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ + (cupicef.sup (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7)) + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) (cupicef.type s5))) + cstr_15 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s2)) + (cupicef.type s5))) + cstr_1 cupicef.BoolNone) + (cupicef.prod (cupicef.type s7) + (cupicef.axiom (cupicef.type s2)) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s2))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s7) A => + cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I))) + eqA + (respectful s7 s8 s3 s2 s5 s6 cstr_15 + (cupicef.trans (cupicef.type s7) + (cupicef.SortSome (cupicef.type s4)) cstr_14 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s4)) cstr_17 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + cstr_11 cstr_6 A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + eqA + (Coq__Classes__CRelationClasses.iffT s2 s2 cstr_8 cstr_9 + cstr_8 cstr_9))) + R) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s7)) + (Coq__Classes__CRelationClasses.PreOrder s7 s0 A + (Coq__Classes__CRelationClasses.relation__disjunction s7 s2 s8 s0 + cupicef.I cstr_4 cstr_7 cstr_16 cstr_18 A R eqA)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + s7 : cupicef.Nat => + s8 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s2)) (cupicef.type s3)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s2)) (cupicef.type s5)) => + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s6)) + (cupicef.type (cupicef.s cupicef.z))) => + cstr_3 : cupicef.eps (cupicef.Cumul cupicef.set (cupicef.type s0)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s0)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s4)) => + cstr_6 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s6)) => + cstr_7 : + cupicef.eps + (cupicef.Cumul (cupicef.type s2) (cupicef.type (cupicef.s cupicef.z))) => + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s2) (cupicef.type (cupicef.s cupicef.z))) => + cstr_9 : + cupicef.eps + (cupicef.Cumul (cupicef.type s2) (cupicef.type (cupicef.s cupicef.z))) => + cstr_10 : + cupicef.eps + (cupicef.Cumul (cupicef.type s2) (cupicef.type (cupicef.s cupicef.z))) => + cstr_11 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s5)) => + cstr_12 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s6)) => + cstr_13 : + cupicef.eps + (cupicef.Cumul (cupicef.type s5) (cupicef.type (cupicef.s cupicef.z))) => + cstr_14 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s4)) => + cstr_15 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s5)) => + cstr_16 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s0)) => + cstr_17 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s4)) => + cstr_18 : + cupicef.eps + (cupicef.Cumul (cupicef.type s8) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s7) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s7) (cupicef.axiom (cupicef.type s8))) + (Coq__Classes__CRelationClasses.crelation s7 s8 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s7) (cupicef.type s8)) + (Coq__Classes__CRelationClasses.Equivalence s7 s8 A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type s2)) (cupicef.type s7)) + (Coq__Classes__CRelationClasses.crelation s7 s2 A) => + H0 : + cupicef.Term + (cupicef.sup (cupicef.type s1) + (cupicef.sup (cupicef.type s2) (cupicef.type s7))) + (Coq__Classes__CRelationClasses.StrictOrder s7 s1 s2 A R) => + H1 : + cupicef.Term (cupicef.type s6) + (Proper s5 s6 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ + (cupicef.sup (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7)) + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) (cupicef.type s5))) + cstr_15 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s2)) + (cupicef.type s5))) + cstr_1 cupicef.BoolNone) + (cupicef.prod (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s2)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s7) A => + cupicef.prod (cupicef.type s7) + (cupicef.axiom (cupicef.type s2)) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s2))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type s7) A => + cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)))) + (respectful s7 s8 s5 s6 s5 s6 cstr_15 + (cupicef.trans (cupicef.type s7) + (cupicef.SortSome (cupicef.type s4)) cstr_14 + (cupicef.SortSome (cupicef.type s6)) cstr_12 cupicef.SortNone) + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s4)) cstr_17 + (cupicef.SortSome (cupicef.type s6)) cstr_12 cupicef.SortNone) + cupicef.I cupicef.I A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ + (cupicef.sup (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7)) + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) (cupicef.type s5))) + cstr_15 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s2)) + (cupicef.type s5))) + cstr_1 cupicef.BoolNone) + (cupicef.prod (cupicef.type s7) + (cupicef.axiom (cupicef.type s2)) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s2))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s7) A => + cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I))) + eqA + (respectful s7 s8 s3 s2 s5 s6 cstr_15 + (cupicef.trans (cupicef.type s7) + (cupicef.SortSome (cupicef.type s4)) cstr_14 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s4)) cstr_17 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + cstr_11 cstr_6 A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + eqA + (Coq__Classes__CRelationClasses.iffT s2 s2 cstr_8 cstr_9 + cstr_8 cstr_9))) + R) => + Coq__Classes__CRelationClasses.Build__PreOrder s7 s0 A + (Coq__Classes__CRelationClasses.relation__disjunction s7 s2 s8 s0 + cupicef.I cstr_4 cstr_7 cstr_16 cstr_18 A R eqA) + (cupicef.cast + (cupicef.rule (cupicef.type s7) + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s2) (cupicef.type s8)))) + (cupicef.sup (cupicef.type s0) (cupicef.type s7)) + (cupicef.prod (cupicef.type s7) + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s2) (cupicef.type s8))) + (cupicef.rule (cupicef.type s7) + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s2) (cupicef.type s8)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s7) A => + Coq__Init__Datatypes.sum (cupicef.type s2) (cupicef.type s8) + (R x x) (eqA x x))) + (Coq__Classes__CRelationClasses.Reflexive s7 s0 A + (Coq__Classes__CRelationClasses.relation__disjunction s7 s2 s8 s0 + cupicef.I cstr_4 cstr_7 cstr_16 cstr_18 A R eqA)) + cupicef.I + (x : cupicef.Term (cupicef.type s7) A => + Coq__Init__Datatypes.inr + (cupicef.cast (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_7 (R x x)) + (cupicef.cast (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_18 (eqA x x)) + (Coq__Classes__CRelationClasses.reflexivity s7 s8 A eqA + (Coq__Classes__CRelationClasses.Equivalence__Reflexive s7 s8 A + eqA H) + x))) + (cupicef.cast + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s0)))))) + (cupicef.sup (cupicef.type s0) (cupicef.type s7)) + (cupicef.prod (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s0))))) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s0)))))) + cupicef.I A + (x : cupicef.Term (cupicef.type s7) A => + cupicef.prod (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s0)))) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s0))))) + cupicef.I A + (y : cupicef.Term (cupicef.type s7) A => + cupicef.prod (cupicef.type s7) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s0))) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s0)))) + cupicef.I A + (z : cupicef.Term (cupicef.type s7) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s0)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s0))) + cupicef.I + (Coq__Classes__CRelationClasses.relation__disjunction s7 + s2 s8 s0 cupicef.I cstr_4 cstr_7 cstr_16 cstr_18 A R + eqA x y) + (X : + cupicef.Term (cupicef.type s0) + (Coq__Classes__CRelationClasses.relation__disjunction + s7 s2 s8 s0 cupicef.I cstr_4 cstr_7 cstr_16 + cstr_18 A R eqA x y) => + cupicef.prod (cupicef.type s0) (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s0)) + cupicef.I + (Coq__Classes__CRelationClasses.relation__disjunction + s7 s2 s8 s0 cupicef.I cstr_4 cstr_7 cstr_16 cstr_18 + A R eqA y z) + (__ : + cupicef.Term (cupicef.type s0) + (Coq__Classes__CRelationClasses.relation__disjunction + s7 s2 s8 s0 cupicef.I cstr_4 cstr_7 cstr_16 + cstr_18 A R eqA y z) => + Coq__Classes__CRelationClasses.relation__disjunction + s7 s2 s8 s0 cupicef.I cstr_4 cstr_7 cstr_16 cstr_18 + A R eqA x z)))))) + (Coq__Classes__CRelationClasses.Transitive s7 s0 A + (Coq__Classes__CRelationClasses.relation__disjunction s7 s2 s8 s0 + cupicef.I cstr_4 cstr_7 cstr_16 cstr_18 A R eqA)) + cupicef.I + (x : cupicef.Term (cupicef.type s7) A => + y : cupicef.Term (cupicef.type s7) A => + z : cupicef.Term (cupicef.type s7) A => + X : + cupicef.Term (cupicef.type s0) + (Coq__Classes__CRelationClasses.relation__disjunction s7 s2 s8 + s0 cupicef.I cstr_4 cstr_7 cstr_16 cstr_18 A R eqA x y) => + Coq__Init__Datatypes.match____sum (cupicef.type s0) + (cupicef.cast (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_7 (R x y)) + (cupicef.cast (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_18 (eqA x y)) + (s : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s2) (cupicef.type s8))) + (Coq__Init__Datatypes.sum (cupicef.type s2) (cupicef.type s8) + (R x y) (eqA x y)) => + cupicef.prod (cupicef.type s0) (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s0)) cupicef.I + (Coq__Classes__CRelationClasses.relation__disjunction s7 s2 s8 + s0 cupicef.I cstr_4 cstr_7 cstr_16 cstr_18 A R eqA y z) + (__ : + cupicef.Term (cupicef.type s0) + (Coq__Classes__CRelationClasses.relation__disjunction s7 + s2 s8 s0 cupicef.I cstr_4 cstr_7 cstr_16 cstr_18 A R + eqA y z) => + Coq__Classes__CRelationClasses.relation__disjunction s7 s2 s8 + s0 cupicef.I cstr_4 cstr_7 cstr_16 cstr_18 A R eqA x z)) + (Hxy : cupicef.Term (cupicef.type s2) (R x y) => + X0 : + cupicef.Term (cupicef.type s0) + (Coq__Classes__CRelationClasses.relation__disjunction s7 s2 + s8 s0 cupicef.I cstr_4 cstr_7 cstr_16 cstr_18 A R eqA y z) => + Coq__Init__Datatypes.match____sum (cupicef.type s0) + (cupicef.cast (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_7 (R y z)) + (cupicef.cast (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_18 (eqA y z)) + (s : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s2) (cupicef.type s8))) + (Coq__Init__Datatypes.sum (cupicef.type s2) + (cupicef.type s8) (R y z) (eqA y z)) => + Coq__Classes__CRelationClasses.relation__disjunction s7 s2 s8 + s0 cupicef.I cstr_4 cstr_7 cstr_16 cstr_18 A R eqA x z) + (Hyz : cupicef.Term (cupicef.type s2) (R y z) => + Coq__Init__Datatypes.inl + (cupicef.cast (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_7 (R x z)) + (cupicef.cast (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_18 (eqA x z)) + (Coq__Classes__CRelationClasses.transitivity s7 s2 A R + (Coq__Classes__CRelationClasses.StrictOrder__Transitive + s7 s1 s2 A R H0) + x y z Hxy Hyz)) + (Hyz : cupicef.Term (cupicef.type s8) (eqA y z) => + Coq__Init__Datatypes.inl + (cupicef.cast (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_7 (R x z)) + (cupicef.cast (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_18 (eqA x z)) + ((lemma : cupicef.Term (cupicef.type s8) (eqA y z) => + Reflexive__partial__app__morphism s7 s5 s5 s6 s8 s4 + cstr_15 + (cupicef.trans (cupicef.type s7) + (cupicef.SortSome (cupicef.type s4)) cstr_14 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + cupicef.I + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s4)) cstr_17 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + cstr_12 A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ + (cupicef.sup (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7)) + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) + (cupicef.type s5))) + cstr_15 + (cupicef.BoolSome + (cupicef.Cumul + (cupicef.axiom (cupicef.type s2)) + (cupicef.type s5))) + cstr_1 cupicef.BoolNone) + (cupicef.prod (cupicef.type s7) + (cupicef.axiom (cupicef.type s2)) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s2))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s7) A => + cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I))) + eqA + (respectful s7 s8 s3 s2 s5 s4 cstr_15 cstr_14 cstr_17 + cstr_11 cstr_5 A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + eqA + (Coq__Classes__CRelationClasses.flip s3 s3 s3 + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + (Coq__Classes__CRelationClasses.arrow s2 s2))) + R + (subrelation__proper s5 s6 cstr_2 cstr_13 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ + (cupicef.sup (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7)) + (cupicef.axiom + (cupicef.sup + (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) + (cupicef.type s5))) + cstr_15 + (cupicef.BoolSome + (cupicef.Cumul + (cupicef.axiom (cupicef.type s2)) + (cupicef.type s5))) + cstr_1 cupicef.BoolNone) + (cupicef.prod (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s2)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s7) A => + cupicef.prod (cupicef.type s7) + (cupicef.axiom (cupicef.type s2)) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s2))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type s7) A => + cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)))) + (respectful s7 s8 s5 s6 s5 s6 cstr_15 + (cupicef.trans (cupicef.type s7) + (cupicef.SortSome (cupicef.type s4)) cstr_14 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s4)) cstr_17 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + cupicef.I cupicef.I A + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ + (cupicef.sup + (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7)) + (cupicef.axiom + (cupicef.sup + (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) + (cupicef.type s5))) + cstr_15 + (cupicef.BoolSome + (cupicef.Cumul + (cupicef.axiom (cupicef.type s2)) + (cupicef.type s5))) + cstr_1 cupicef.BoolNone) + (cupicef.prod (cupicef.type s7) + (cupicef.axiom (cupicef.type s2)) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s2))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s7) A => + cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I))) + eqA + (respectful s7 s8 s3 s2 s5 s6 cstr_15 + (cupicef.trans (cupicef.type s7) + (cupicef.SortSome (cupicef.type s4)) cstr_14 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s4)) cstr_17 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + cstr_11 cstr_6 A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + eqA + (Coq__Classes__CRelationClasses.iffT s2 s2 + cstr_8 cstr_9 cstr_8 cstr_9))) + R H1 + (respectful s7 s8 s5 s4 s5 s6 cstr_15 + (cupicef.trans (cupicef.type s7) + (cupicef.SortSome (cupicef.type s4)) cstr_14 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s4)) cstr_17 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + cupicef.I cstr_12 A + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ + (cupicef.sup + (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7)) + (cupicef.axiom + (cupicef.sup + (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) + (cupicef.type s5))) + cstr_15 + (cupicef.BoolSome + (cupicef.Cumul + (cupicef.axiom (cupicef.type s2)) + (cupicef.type s5))) + cstr_1 cupicef.BoolNone) + (cupicef.prod (cupicef.type s7) + (cupicef.axiom (cupicef.type s2)) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s2))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s7) A => + cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I))) + eqA + (respectful s7 s8 s3 s2 s5 s4 cstr_15 cstr_14 + cstr_17 cstr_11 cstr_5 A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + eqA + (Coq__Classes__CRelationClasses.flip s3 s3 s3 + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + (Coq__Classes__CRelationClasses.arrow s2 s2)))) + Coq__Init__Datatypes.tt + (subrelation__respectful s7 s5 s6 s6 s8 s6 s8 s5 s4 + cstr_15 + (cupicef.trans (cupicef.type s7) + (cupicef.SortSome (cupicef.type s4)) cstr_14 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + (cupicef.trans (cupicef.type s7) + (cupicef.SortSome (cupicef.type s4)) cstr_14 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s4)) cstr_17 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + cupicef.I + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s4)) cstr_17 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + cupicef.I cstr_12 A eqA eqA + (subrelation__refl s7 s8 A eqA) + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ + (cupicef.sup + (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7)) + (cupicef.axiom + (cupicef.sup + (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) + (cupicef.type s5))) + cstr_15 + (cupicef.BoolSome + (cupicef.Cumul + (cupicef.axiom (cupicef.type s2)) + (cupicef.type s5))) + cstr_1 cupicef.BoolNone) + (cupicef.prod (cupicef.type s7) + (cupicef.axiom (cupicef.type s2)) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s2))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s7) A => + cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I))) + (respectful s7 s8 s3 s2 s5 s6 cstr_15 + (cupicef.trans (cupicef.type s7) + (cupicef.SortSome (cupicef.type s4)) cstr_14 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s4)) cstr_17 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + cstr_11 cstr_6 A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + eqA + (Coq__Classes__CRelationClasses.iffT s2 s2 + cstr_8 cstr_9 cstr_8 cstr_9)) + (respectful s7 s8 s3 s2 s5 s4 cstr_15 cstr_14 + cstr_17 cstr_11 cstr_5 A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + eqA + (Coq__Classes__CRelationClasses.flip s3 s3 s3 + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + (Coq__Classes__CRelationClasses.arrow s2 s2))) + (subrelation__respectful s7 s5 s6 s4 s8 s2 s8 s3 + s2 cstr_15 + (cupicef.trans (cupicef.type s7) + (cupicef.SortSome (cupicef.type s4)) cstr_14 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + cstr_14 + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s4)) cstr_17 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + cstr_6 cstr_17 cstr_11 cstr_5 A eqA eqA + (subrelation__refl s7 s8 A eqA) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + (Coq__Classes__CRelationClasses.iffT s2 s2 + cstr_8 cstr_9 cstr_8 cstr_9) + (Coq__Classes__CRelationClasses.flip s3 s3 s3 + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + (Coq__Classes__CRelationClasses.arrow s2 s2)) + (flip2 s3 s2 s2 s3 cstr_0 + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + (Coq__Classes__CRelationClasses.iffT s2 s2 + cstr_8 cstr_9 cstr_8 cstr_9) + (y0 : cupicef.Univ (cupicef.type s2) => + x0 : cupicef.Univ (cupicef.type s2) => + cupicef.prod (cupicef.type s2) + (cupicef.type s2) + (cupicef.rule (cupicef.type s2) + (cupicef.type s2)) + cupicef.I x0 + (__ : cupicef.Term (cupicef.type s2) x0 => + y0)) + (iffT__flip__arrow__subrelation s3 s2 s3 s2 + cstr_0 cstr_0 cupicef.I cstr_8 cstr_9 + cstr_10))))) + x + (reflexive__proper__proxy s7 s8 A eqA + (Coq__Classes__CRelationClasses.Equivalence__Reflexive + s7 s8 A eqA H) + x) + z y + (Coq__Classes__CRelationClasses.symmetry s7 s8 A eqA + (Coq__Classes__CRelationClasses.Equivalence__Symmetric + s7 s8 A eqA H) + y z lemma)) + Hyz Hxy)) + X0) + (Hxy : cupicef.Term (cupicef.type s8) (eqA x y) => + X0 : + cupicef.Term (cupicef.type s0) + (Coq__Classes__CRelationClasses.relation__disjunction s7 s2 + s8 s0 cupicef.I cstr_4 cstr_7 cstr_16 cstr_18 A R eqA y z) => + Coq__Init__Datatypes.match____sum (cupicef.type s0) + (cupicef.cast (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_7 (R y z)) + (cupicef.cast (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_18 (eqA y z)) + (s : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s2) (cupicef.type s8))) + (Coq__Init__Datatypes.sum (cupicef.type s2) + (cupicef.type s8) (R y z) (eqA y z)) => + Coq__Classes__CRelationClasses.relation__disjunction s7 s2 s8 + s0 cupicef.I cstr_4 cstr_7 cstr_16 cstr_18 A R eqA x z) + (Hyz : cupicef.Term (cupicef.type s2) (R y z) => + Coq__Init__Datatypes.inl + (cupicef.cast (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_7 (R x z)) + (cupicef.cast (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_18 (eqA x z)) + ((lemma : cupicef.Term (cupicef.type s8) (eqA x y) => + subrelation__proper s5 s6 cstr_2 cstr_13 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ + (cupicef.sup (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7)) + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) + (cupicef.type s5))) + cstr_15 + (cupicef.BoolSome + (cupicef.Cumul + (cupicef.axiom (cupicef.type s2)) + (cupicef.type s5))) + cstr_1 cupicef.BoolNone) + (cupicef.prod (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s2)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s7) A => + cupicef.prod (cupicef.type s7) + (cupicef.axiom (cupicef.type s2)) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s2))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type s7) A => + cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)))) + (respectful s7 s8 s5 s6 s5 s6 cstr_15 + (cupicef.trans (cupicef.type s7) + (cupicef.SortSome (cupicef.type s4)) cstr_14 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s4)) cstr_17 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + cupicef.I cupicef.I A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ + (cupicef.sup (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7)) + (cupicef.axiom + (cupicef.sup + (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) + (cupicef.type s5))) + cstr_15 + (cupicef.BoolSome + (cupicef.Cumul + (cupicef.axiom (cupicef.type s2)) + (cupicef.type s5))) + cstr_1 cupicef.BoolNone) + (cupicef.prod (cupicef.type s7) + (cupicef.axiom (cupicef.type s2)) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s2))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s7) A => + cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I))) + eqA + (respectful s7 s8 s3 s2 s5 s6 cstr_15 + (cupicef.trans (cupicef.type s7) + (cupicef.SortSome (cupicef.type s4)) cstr_14 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s4)) cstr_17 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + cstr_11 cstr_6 A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + eqA + (Coq__Classes__CRelationClasses.iffT s2 s2 cstr_8 + cstr_9 cstr_8 cstr_9))) + R H1 + (respectful s7 s8 s5 s4 s5 s6 cstr_15 + (cupicef.trans (cupicef.type s7) + (cupicef.SortSome (cupicef.type s4)) cstr_14 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s4)) cstr_17 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + cupicef.I cstr_12 A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ + (cupicef.sup (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7)) + (cupicef.axiom + (cupicef.sup + (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) + (cupicef.type s5))) + cstr_15 + (cupicef.BoolSome + (cupicef.Cumul + (cupicef.axiom (cupicef.type s2)) + (cupicef.type s5))) + cstr_1 cupicef.BoolNone) + (cupicef.prod (cupicef.type s7) + (cupicef.axiom (cupicef.type s2)) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s2))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s7) A => + cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I))) + eqA + (respectful s7 s8 s3 s2 s5 s4 cstr_15 cstr_14 cstr_17 + cstr_11 cstr_5 A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + eqA + (Coq__Classes__CRelationClasses.flip s3 s3 s3 + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + (Coq__Classes__CRelationClasses.arrow s2 s2)))) + Coq__Init__Datatypes.tt + (subrelation__respectful s7 s5 s6 s6 s8 s6 s8 s5 s4 + cstr_15 + (cupicef.trans (cupicef.type s7) + (cupicef.SortSome (cupicef.type s4)) cstr_14 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + (cupicef.trans (cupicef.type s7) + (cupicef.SortSome (cupicef.type s4)) cstr_14 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s4)) cstr_17 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + cupicef.I + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s4)) cstr_17 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + cupicef.I cstr_12 A eqA eqA + (subrelation__refl s7 s8 A eqA) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.univ + (cupicef.sup (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7)) + (cupicef.axiom + (cupicef.sup + (cupicef.axiom (cupicef.type s2)) + (cupicef.type s7))) + cupicef.I) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) + (cupicef.type s5))) + cstr_15 + (cupicef.BoolSome + (cupicef.Cumul + (cupicef.axiom (cupicef.type s2)) + (cupicef.type s5))) + cstr_1 cupicef.BoolNone) + (cupicef.prod (cupicef.type s7) + (cupicef.axiom (cupicef.type s2)) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s2))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s7) A => + cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I))) + (respectful s7 s8 s3 s2 s5 s6 cstr_15 + (cupicef.trans (cupicef.type s7) + (cupicef.SortSome (cupicef.type s4)) cstr_14 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s4)) cstr_17 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + cstr_11 cstr_6 A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + eqA + (Coq__Classes__CRelationClasses.iffT s2 s2 cstr_8 + cstr_9 cstr_8 cstr_9)) + (respectful s7 s8 s3 s2 s5 s4 cstr_15 cstr_14 cstr_17 + cstr_11 cstr_5 A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + eqA + (Coq__Classes__CRelationClasses.flip s3 s3 s3 + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + (Coq__Classes__CRelationClasses.arrow s2 s2))) + (subrelation__respectful s7 s5 s6 s4 s8 s2 s8 s3 s2 + cstr_15 + (cupicef.trans (cupicef.type s7) + (cupicef.SortSome (cupicef.type s4)) cstr_14 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + cstr_14 + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s4)) cstr_17 + (cupicef.SortSome (cupicef.type s6)) cstr_12 + cupicef.SortNone) + cstr_6 cstr_17 cstr_11 cstr_5 A eqA eqA + (subrelation__refl s7 s8 A eqA) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + (Coq__Classes__CRelationClasses.iffT s2 s2 cstr_8 + cstr_9 cstr_8 cstr_9) + (Coq__Classes__CRelationClasses.flip s3 s3 s3 + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + (Coq__Classes__CRelationClasses.arrow s2 s2)) + (flip2 s3 s2 s2 s3 cstr_0 + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + (Coq__Classes__CRelationClasses.iffT s2 s2 + cstr_8 cstr_9 cstr_8 cstr_9) + (y0 : cupicef.Univ (cupicef.type s2) => + x0 : cupicef.Univ (cupicef.type s2) => + cupicef.prod (cupicef.type s2) + (cupicef.type s2) + (cupicef.rule (cupicef.type s2) + (cupicef.type s2)) + cupicef.I x0 + (__ : cupicef.Term (cupicef.type s2) x0 => + y0)) + (iffT__flip__arrow__subrelation s3 s2 s3 s2 + cstr_0 cstr_0 cupicef.I cstr_8 cstr_9 + cstr_10)))) + x y lemma z z + (reflexive__proper__proxy s7 s8 A eqA + (Coq__Classes__CRelationClasses.Equivalence__Reflexive + s7 s8 A eqA H) + z)) + Hxy Hyz)) + (Hyz : cupicef.Term (cupicef.type s8) (eqA y z) => + Coq__Init__Datatypes.inr + (cupicef.cast (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_7 (R x z)) + (cupicef.cast (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_18 (eqA x z)) + (Coq__Classes__CRelationClasses.transitivity s7 s8 A eqA + (Coq__Classes__CRelationClasses.Equivalence__Transitive + s7 s8 A eqA H) + x y z Hxy Hyz)) + X0) + X)). + +thm StrictOrder__PartialOrder : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + s5 : cupicef.Nat -> + s6 : cupicef.Nat -> + s7 : cupicef.Nat -> + s8 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) + (cupicef.type (cupicef.s cupicef.z))) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s5)) (cupicef.type s0)) -> + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s5)) (cupicef.type s2)) -> + cstr_3 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) (cupicef.type s3)) -> + cstr_4 : cupicef.eps (cupicef.Cumul cupicef.set (cupicef.type s5)) -> + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s4)) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s1)) -> + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s5) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_9 : + cupicef.eps + (cupicef.Cumul (cupicef.type s5) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_10 : + cupicef.eps + (cupicef.Cumul (cupicef.type s5) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_11 : + cupicef.eps + (cupicef.Cumul (cupicef.type s5) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_12 : + cupicef.eps + (cupicef.Cumul (cupicef.type s5) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_13 : + cupicef.eps + (cupicef.Cumul (cupicef.type s5) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_14 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s0)) -> + cstr_15 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s1)) -> + cstr_16 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s4)) -> + cstr_17 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s5)) -> + cstr_18 : + cupicef.eps + (cupicef.Cumul (cupicef.type s8) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_19 : + cupicef.eps + (cupicef.Cumul (cupicef.type s8) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_20 : + cupicef.eps + (cupicef.Cumul (cupicef.type s8) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_21 : + cupicef.eps + (cupicef.Cumul (cupicef.type s8) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s7) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type s5)) (cupicef.type s7)) + (Coq__Classes__CRelationClasses.crelation s7 s5 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s5) (cupicef.type s7)) + (Coq__Classes__CRelationClasses.Equivalence s7 s5 A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s7) (cupicef.axiom (cupicef.type s8))) + (Coq__Classes__CRelationClasses.crelation s7 s8 A) -> + H0 : + cupicef.Term + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.type s7) (cupicef.type s8))) + (Coq__Classes__CRelationClasses.StrictOrder s7 s6 s8 A R) -> + H1 : + cupicef.Term (cupicef.type s1) + (Proper s4 s1 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s7) + (cupicef.axiom (cupicef.type s8)))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ + (cupicef.sup (cupicef.type s7) + (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom + (cupicef.sup (cupicef.type s7) + (cupicef.axiom (cupicef.type s8)))) + cupicef.I) + (cupicef.univ (cupicef.type s4) (cupicef.axiom (cupicef.type s4)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) (cupicef.type s4))) + cstr_16 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) + (cupicef.type s4))) + (cupicef.trans (cupicef.axiom (cupicef.type s8)) + (cupicef.SortSome (cupicef.type s3)) cstr_3 + (cupicef.SortSome (cupicef.type s4)) cstr_5 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s8))) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s8)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s7) A => + cupicef.prod (cupicef.type s7) + (cupicef.axiom (cupicef.type s8)) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s8))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type s7) A => + cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I)))) + (respectful s7 s5 s4 s1 s4 s1 cstr_16 cstr_15 cstr_7 cupicef.I + cupicef.I A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s7) + (cupicef.axiom (cupicef.type s8)))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ + (cupicef.sup (cupicef.type s7) + (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom + (cupicef.sup (cupicef.type s7) + (cupicef.axiom (cupicef.type s8)))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) (cupicef.type s4))) + cstr_16 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) + (cupicef.type s4))) + (cupicef.trans (cupicef.axiom (cupicef.type s8)) + (cupicef.SortSome (cupicef.type s3)) cstr_3 + (cupicef.SortSome (cupicef.type s4)) cstr_5 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s7) + (cupicef.axiom (cupicef.type s8)) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s8))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s7) A => + cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I))) + eqA + (respectful s7 s5 s3 s8 s4 s1 cstr_16 cstr_15 cstr_7 cstr_5 + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s5)) cstr_17 + (cupicef.SortSome (cupicef.type s1)) cstr_7 + cupicef.SortNone) + A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.axiom (cupicef.type s8))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_3 + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I)) + eqA + (Coq__Classes__CRelationClasses.iffT s8 s8 cstr_19 cstr_20 + cstr_19 cstr_20))) + R) -> + cupicef.Term (cupicef.type s1) + (Coq__Classes__CRelationClasses.PartialOrder s7 s0 s1 s5 s2 cstr_1 cstr_2 + cstr_14 cstr_15 cstr_7 cstr_11 cstr_12 A eqA H + (Coq__Classes__CRelationClasses.relation__disjunction s7 s8 s5 s5 + cupicef.I cstr_17 cstr_18 cupicef.I cstr_9 A R eqA) + (StrictOrder__PreOrder s5 s6 s8 s3 s1 s4 s1 s7 s5 cstr_3 + (cupicef.trans (cupicef.axiom (cupicef.type s8)) + (cupicef.SortSome (cupicef.type s3)) cstr_3 + (cupicef.SortSome (cupicef.type s4)) cstr_5 cupicef.SortNone) + cstr_0 cupicef.I cstr_17 + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s5)) cstr_17 + (cupicef.SortSome (cupicef.type s1)) cstr_7 cupicef.SortNone) + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s5)) cstr_17 + (cupicef.SortSome (cupicef.type s1)) cstr_7 cupicef.SortNone) + cstr_18 cstr_19 cstr_20 cstr_21 cstr_5 cupicef.I cstr_6 cstr_15 + cstr_16 cupicef.I cstr_7 cstr_9 A eqA H R H0 H1)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + s5 : cupicef.Nat => + s6 : cupicef.Nat => + s7 : cupicef.Nat => + s8 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) + (cupicef.type (cupicef.s cupicef.z))) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s5)) (cupicef.type s0)) => + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s5)) (cupicef.type s2)) => + cstr_3 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) (cupicef.type s3)) => + cstr_4 : cupicef.eps (cupicef.Cumul cupicef.set (cupicef.type s5)) => + cstr_5 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s4)) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_7 : cupicef.eps (cupicef.Cumul (cupicef.type s5) (cupicef.type s1)) => + cstr_8 : + cupicef.eps + (cupicef.Cumul (cupicef.type s5) (cupicef.type (cupicef.s cupicef.z))) => + cstr_9 : + cupicef.eps + (cupicef.Cumul (cupicef.type s5) (cupicef.type (cupicef.s cupicef.z))) => + cstr_10 : + cupicef.eps + (cupicef.Cumul (cupicef.type s5) (cupicef.type (cupicef.s cupicef.z))) => + cstr_11 : + cupicef.eps + (cupicef.Cumul (cupicef.type s5) (cupicef.type (cupicef.s cupicef.z))) => + cstr_12 : + cupicef.eps + (cupicef.Cumul (cupicef.type s5) (cupicef.type (cupicef.s cupicef.z))) => + cstr_13 : + cupicef.eps + (cupicef.Cumul (cupicef.type s5) (cupicef.type (cupicef.s cupicef.z))) => + cstr_14 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s0)) => + cstr_15 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s1)) => + cstr_16 : cupicef.eps (cupicef.Cumul (cupicef.type s7) (cupicef.type s4)) => + cstr_17 : cupicef.eps (cupicef.Cumul (cupicef.type s8) (cupicef.type s5)) => + cstr_18 : + cupicef.eps + (cupicef.Cumul (cupicef.type s8) (cupicef.type (cupicef.s cupicef.z))) => + cstr_19 : + cupicef.eps + (cupicef.Cumul (cupicef.type s8) (cupicef.type (cupicef.s cupicef.z))) => + cstr_20 : + cupicef.eps + (cupicef.Cumul (cupicef.type s8) (cupicef.type (cupicef.s cupicef.z))) => + cstr_21 : + cupicef.eps + (cupicef.Cumul (cupicef.type s8) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s7) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type s5)) (cupicef.type s7)) + (Coq__Classes__CRelationClasses.crelation s7 s5 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s5) (cupicef.type s7)) + (Coq__Classes__CRelationClasses.Equivalence s7 s5 A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s7) (cupicef.axiom (cupicef.type s8))) + (Coq__Classes__CRelationClasses.crelation s7 s8 A) => + H0 : + cupicef.Term + (cupicef.sup (cupicef.type s6) + (cupicef.sup (cupicef.type s7) (cupicef.type s8))) + (Coq__Classes__CRelationClasses.StrictOrder s7 s6 s8 A R) => + H1 : + cupicef.Term (cupicef.type s1) + (Proper s4 s1 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s7) + (cupicef.axiom (cupicef.type s8)))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ + (cupicef.sup (cupicef.type s7) + (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom + (cupicef.sup (cupicef.type s7) + (cupicef.axiom (cupicef.type s8)))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) (cupicef.type s4))) + cstr_16 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) + (cupicef.type s4))) + (cupicef.trans (cupicef.axiom (cupicef.type s8)) + (cupicef.SortSome (cupicef.type s3)) cstr_3 + (cupicef.SortSome (cupicef.type s4)) cstr_5 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s8))) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s8)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s7) A => + cupicef.prod (cupicef.type s7) + (cupicef.axiom (cupicef.type s8)) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s8))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type s7) A => + cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I)))) + (respectful s7 s5 s4 s1 s4 s1 cstr_16 cstr_15 cstr_7 cupicef.I + cupicef.I A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s7) + (cupicef.axiom (cupicef.type s8)))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ + (cupicef.sup (cupicef.type s7) + (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom + (cupicef.sup (cupicef.type s7) + (cupicef.axiom (cupicef.type s8)))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s7) (cupicef.type s4))) + cstr_16 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s8)) + (cupicef.type s4))) + (cupicef.trans (cupicef.axiom (cupicef.type s8)) + (cupicef.SortSome (cupicef.type s3)) cstr_3 + (cupicef.SortSome (cupicef.type s4)) cstr_5 + cupicef.SortNone) + cupicef.BoolNone) + (cupicef.prod (cupicef.type s7) + (cupicef.axiom (cupicef.type s8)) + (cupicef.rule (cupicef.type s7) + (cupicef.axiom (cupicef.type s8))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s7) A => + cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I))) + eqA + (respectful s7 s5 s3 s8 s4 s1 cstr_16 cstr_15 cstr_7 cstr_5 + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s5)) cstr_17 + (cupicef.SortSome (cupicef.type s1)) cstr_7 + cupicef.SortNone) + A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s8))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.axiom (cupicef.type s8))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_3 + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I)) + eqA + (Coq__Classes__CRelationClasses.iffT s8 s8 cstr_19 cstr_20 + cstr_19 cstr_20))) + R) => + cupicef.cast + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s7) (cupicef.type s5))) + (cupicef.type s1) + (cupicef.prod (cupicef.type s7) + (cupicef.rule (cupicef.type s7) (cupicef.type s5)) + (cupicef.rule (cupicef.type s7) + (cupicef.rule (cupicef.type s7) (cupicef.type s5))) + cupicef.I A + (x : cupicef.Term (cupicef.type s7) A => + cupicef.prod (cupicef.type s7) (cupicef.type s5) + (cupicef.rule (cupicef.type s7) (cupicef.type s5)) cupicef.I A + (y : cupicef.Term (cupicef.type s7) A => + Coq__Classes__CRelationClasses.iffT s5 s5 cstr_11 cstr_12 cstr_11 + cstr_12 (eqA x y) + (Coq__Classes__CRelationClasses.relation__conjunction s7 s5 s5 + s5 cupicef.I cstr_11 cupicef.I cstr_12 A + (Coq__Classes__CRelationClasses.relation__disjunction s7 s8 + s5 s5 cupicef.I cstr_17 cstr_18 cupicef.I cstr_9 A R eqA) + (Coq__Classes__CRelationClasses.flip s7 s7 s2 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_2 + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I)) + (Coq__Classes__CRelationClasses.relation__disjunction s7 + s8 s5 s5 cupicef.I cstr_17 cstr_18 cupicef.I cstr_9 A + R eqA)) + x y)))) + (Coq__Classes__CRelationClasses.PartialOrder s7 s0 s1 s5 s2 cstr_1 + cstr_2 cstr_14 cstr_15 cstr_7 cstr_11 cstr_12 A eqA H + (Coq__Classes__CRelationClasses.relation__disjunction s7 s8 s5 s5 + cupicef.I cstr_17 cstr_18 cupicef.I cstr_9 A R eqA) + (StrictOrder__PreOrder s5 s6 s8 s3 s1 s4 s1 s7 s5 cstr_3 + (cupicef.trans (cupicef.axiom (cupicef.type s8)) + (cupicef.SortSome (cupicef.type s3)) cstr_3 + (cupicef.SortSome (cupicef.type s4)) cstr_5 cupicef.SortNone) + cstr_0 cupicef.I cstr_17 + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s5)) cstr_17 + (cupicef.SortSome (cupicef.type s1)) cstr_7 cupicef.SortNone) + (cupicef.trans (cupicef.type s8) + (cupicef.SortSome (cupicef.type s5)) cstr_17 + (cupicef.SortSome (cupicef.type s1)) cstr_7 cupicef.SortNone) + cstr_18 cstr_19 cstr_20 cstr_21 cstr_5 cupicef.I cstr_6 cstr_15 + cstr_16 cupicef.I cstr_7 cstr_9 A eqA H R H0 H1)) + cupicef.I + (x : cupicef.Term (cupicef.type s7) A => + y : cupicef.Term (cupicef.type s7) A => + cupicef.cast + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.type s5) + (Coq__Init__Datatypes.prod + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.prod (cupicef.type s5) + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.rule (cupicef.type s5) + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8)))) + cupicef.I (eqA x y) + (__ : cupicef.Term (cupicef.type s5) (eqA x y) => + Coq__Init__Datatypes.prod + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R x y) (eqA x y)) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R y x) (eqA y x)))) + (cupicef.prod + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.type s5) + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.type s5)) + cupicef.I + (Coq__Init__Datatypes.prod + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R x y) (eqA x y)) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R y x) (eqA y x))) + (__ : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (Coq__Init__Datatypes.prod + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R x y) (eqA x y)) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R y x) (eqA y x))) => + eqA x y))) + (Coq__Classes__CRelationClasses.iffT s5 s5 cstr_11 cstr_12 cstr_11 + cstr_12 (eqA x y) + (Coq__Classes__CRelationClasses.relation__conjunction s7 s5 s5 s5 + cupicef.I cstr_11 cupicef.I cstr_12 A + (Coq__Classes__CRelationClasses.relation__disjunction s7 s8 s5 + s5 cupicef.I cstr_17 cstr_18 cupicef.I cstr_9 A R eqA) + (Coq__Classes__CRelationClasses.flip s7 s7 s2 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom (cupicef.axiom (cupicef.type s5))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_2 + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I)) + (Coq__Classes__CRelationClasses.relation__disjunction s7 s8 + s5 s5 cupicef.I cstr_17 cstr_18 cupicef.I cstr_9 A R eqA)) + x y)) + cupicef.I + (Coq__Init__Datatypes.pair + (cupicef.cast + (cupicef.axiom + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.axiom + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s8) + (cupicef.type (cupicef.s cupicef.z)))) + cstr_19 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)))) + cstr_11 cupicef.BoolNone) + (cupicef.prod (cupicef.type s5) + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.rule (cupicef.type s5) + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8)))) + cupicef.I (eqA x y) + (__ : cupicef.Term (cupicef.type s5) (eqA x y) => + Coq__Init__Datatypes.prod + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R x y) (eqA x y)) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R y x) (eqA y x))))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.axiom + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s8) + (cupicef.type (cupicef.s cupicef.z)))) + cstr_20 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)))) + cstr_12 cupicef.BoolNone) + (cupicef.prod + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.type s5) + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.type s5)) + cupicef.I + (Coq__Init__Datatypes.prod + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R x y) (eqA x y)) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R y x) (eqA y x))) + (__ : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (Coq__Init__Datatypes.prod + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R x y) (eqA x y)) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R y x) (eqA y x))) => + eqA x y))) + (X : cupicef.Term (cupicef.type s5) (eqA x y) => + Coq__Init__Datatypes.pair + (cupicef.cast + (cupicef.axiom + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.axiom + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s8) + (cupicef.type (cupicef.s cupicef.z)))) + cstr_19 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)))) + cstr_11 cupicef.BoolNone) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R x y) (eqA x y))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.axiom + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s8) + (cupicef.type (cupicef.s cupicef.z)))) + cstr_20 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)))) + cstr_12 cupicef.BoolNone) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R y x) (eqA y x))) + (Coq__Init__Datatypes.inr + (cupicef.cast (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_18 (R x y)) + (cupicef.cast (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_9 (eqA x y)) + X) + (Coq__Init__Datatypes.inr + (cupicef.cast (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_18 (R y x)) + (cupicef.cast (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_9 (eqA y x)) + (cupicef.cast (cupicef.type s5) (cupicef.type s5) (eqA y x) + (eqA y x) cupicef.I + (Coq__Classes__CRelationClasses.symmetry s7 s5 A eqA + (Coq__Classes__CRelationClasses.Equivalence__Symmetric + s7 s5 A eqA H) + x y X)))) + (X : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (Coq__Init__Datatypes.prod + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R x y) (eqA x y)) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R y x) (eqA y x))) => + Coq__Init__Datatypes.prod__rect + (cupicef.cast + (cupicef.axiom + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.axiom + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s8) + (cupicef.type (cupicef.s cupicef.z)))) + cstr_19 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)))) + cstr_11 cupicef.BoolNone) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R x y) (eqA x y))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.axiom + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s8) + (cupicef.type (cupicef.s cupicef.z)))) + cstr_20 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s5) + (cupicef.type (cupicef.s cupicef.z)))) + cstr_12 cupicef.BoolNone) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R y x) (eqA y x))) + (cupicef.cast + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.axiom (cupicef.type s5))) + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.axiom (cupicef.type s5))) + cupicef.I + (Coq__Init__Datatypes.prod + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R x y) (eqA x y)) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R y x) (eqA y x))) + (X0 : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (Coq__Init__Datatypes.prod + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) + (cupicef.type s8))) + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) + (cupicef.type s8))) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R x y) (eqA x y)) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R y x) (eqA y x))) => + cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I)) + (cupicef.prod + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Datatypes.prod + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R x y) (eqA x y)) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R y x) (eqA y x))) + (p : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (Coq__Init__Datatypes.prod + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) + (cupicef.type s8))) + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) + (cupicef.type s8))) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R x y) (eqA x y)) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R y x) (eqA y x))) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cstr_13 + (X0 : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (Coq__Init__Datatypes.prod + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R x y) (eqA x y)) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R y x) (eqA y x))) => + eqA x y)) + (a : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R x y) (eqA x y)) => + b : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R y x) (eqA y x)) => + Coq__Init__Datatypes.sum__rect + (cupicef.cast (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_18 (R x y)) + (cupicef.cast (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_9 (eqA x y)) + (cupicef.cast + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.axiom (cupicef.type s5))) + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.axiom (cupicef.type s5))) + cupicef.I + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R x y) (eqA x y)) + (a0 : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) + (cupicef.type s8))) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R x y) (eqA x y)) => + cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I)) + (cupicef.prod + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R x y) (eqA x y)) + (s : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) + (cupicef.type s8))) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R x y) (eqA x y)) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cstr_10 + (a0 : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R x y) (eqA x y)) => + eqA x y)) + (a0 : cupicef.Term (cupicef.type s8) (R x y) => + Coq__Init__Datatypes.sum__rect + (cupicef.cast (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_18 (R y x)) + (cupicef.cast (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_9 (eqA y x)) + (cupicef.cast + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.axiom (cupicef.type s5))) + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) + (cupicef.type s8))) + (cupicef.axiom (cupicef.type s5))) + cupicef.I + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R y x) (eqA y x)) + (b0 : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) + (cupicef.type s8))) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R y x) (eqA y x)) => + cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I)) + (cupicef.prod + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) + (cupicef.type s8))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R y x) (eqA y x)) + (s : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) + (cupicef.type s8))) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R y x) (eqA y x)) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cstr_10 + (b0 : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) + (cupicef.type s8))) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R y x) (eqA y x)) => + eqA x y)) + (a1 : cupicef.Term (cupicef.type s8) (R y x) => + Coq__Init__Logic.False__rect + (cupicef.cast (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_8 (eqA x y)) + (Coq__Classes__CRelationClasses.StrictOrder__Irreflexive + s7 s6 s8 A R H0 x + (Coq__Classes__CRelationClasses.transitivity s7 s8 + A R + (Coq__Classes__CRelationClasses.StrictOrder__Transitive + s7 s6 s8 A R H0) + x y x a0 a1))) + (b0 : cupicef.Term (cupicef.type s5) (eqA y x) => + cupicef.cast (cupicef.type s5) (cupicef.type s5) + (eqA x y) (eqA x y) cupicef.I + (Coq__Classes__CRelationClasses.symmetry s7 s5 A eqA + (Coq__Classes__CRelationClasses.Equivalence__Symmetric + s7 s5 A eqA H) + y x b0)) + b) + (b0 : cupicef.Term (cupicef.type s5) (eqA x y) => + Coq__Init__Datatypes.sum__rect + (cupicef.cast (cupicef.axiom (cupicef.type s8)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s8) + (cupicef.axiom (cupicef.type s8)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_18 (R y x)) + (cupicef.cast (cupicef.axiom (cupicef.type s5)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_9 (eqA y x)) + (cupicef.cast + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.axiom (cupicef.type s5))) + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.axiom (cupicef.type s5)) + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) + (cupicef.type s8))) + (cupicef.axiom (cupicef.type s5))) + cupicef.I + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R y x) (eqA y x)) + (b1 : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) + (cupicef.type s8))) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R y x) (eqA y x)) => + cupicef.univ (cupicef.type s5) + (cupicef.axiom (cupicef.type s5)) cupicef.I)) + (cupicef.prod + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) (cupicef.type s8))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) + (cupicef.type s8))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R y x) (eqA y x)) + (s : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) + (cupicef.type s8))) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R y x) (eqA y x)) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cstr_10 + (b1 : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s5) + (cupicef.type s8))) + (Coq__Init__Datatypes.sum (cupicef.type s8) + (cupicef.type s5) (R y x) (eqA y x)) => + eqA x y)) + (a0 : cupicef.Term (cupicef.type s8) (R y x) => b0) + (b1 : cupicef.Term (cupicef.type s5) (eqA y x) => b0) b) + a) + X))). + +(; End of translation. ;) diff --git a/out/Coq__Classes__CRelationClasses.dk b/out/Coq__Classes__CRelationClasses.dk new file mode 100644 index 000000000..18762daef --- /dev/null +++ b/out/Coq__Classes__CRelationClasses.dk @@ -0,0 +1,11122 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def crelation : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + cupicef.Univ + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.axiom (cupicef.type s1)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type s0) A => + cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I)). + +def arrow : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + B : cupicef.Univ (cupicef.type s1) -> + cupicef.Univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + B : cupicef.Univ (cupicef.type s1) => + cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => B). + +def flip : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + B : cupicef.Univ (cupicef.type s1) -> + C : cupicef.Univ (cupicef.type s2) -> + f : + (__ : cupicef.Term (cupicef.type s0) A -> + __0 : cupicef.Term (cupicef.type s1) B -> + cupicef.Term (cupicef.type s2) C) -> + x : cupicef.Term (cupicef.type s1) B -> + y : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s2) C + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + B : cupicef.Univ (cupicef.type s1) => + C : cupicef.Univ (cupicef.type s2) => + f : + (__ : cupicef.Term (cupicef.type s0) A -> + __0 : cupicef.Term (cupicef.type s1) B -> + cupicef.Term (cupicef.type s2) C) => + x : cupicef.Term (cupicef.type s1) B => + y : cupicef.Term (cupicef.type s0) A => + f y x. + +def iffT : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_3 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + B : cupicef.Univ (cupicef.type s1) -> + cupicef.Univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) (cupicef.type (cupicef.s cupicef.z))) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) (cupicef.type (cupicef.s cupicef.z))) => + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) => + cstr_3 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + B : cupicef.Univ (cupicef.type s1) => + Coq__Init__Datatypes.prod + (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => B)) + (cupicef.prod (cupicef.type s1) (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s0)) cupicef.I B + (__ : cupicef.Term (cupicef.type s1) B => A)). + +def Reflexive : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + cupicef.Univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => R x x). + +#TYPE_CLASS Reflexive. + +def reflexivity : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + Reflexive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Reflexive s0 s1 A R) -> + x : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s1) (R x x) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + Reflexive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Reflexive s0 s1 A R) => + Reflexive0. + +def complement : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))))) + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))))) + (cupicef.prod (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))))) + cupicef.I + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (A : cupicef.Univ (cupicef.type s0) => + cupicef.prod + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I (crelation s0 s1 A) + (R : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))))) + (cupicef.prod (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2)))) + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))))) + cupicef.I + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (A : cupicef.Univ (cupicef.type s0) => + cupicef.prod + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2)))) + cupicef.I (crelation s0 s1 A) + (R : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + crelation s0 s2 A))) + cupicef.I + (A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s1) cupicef.prop + (cupicef.rule (cupicef.type s1) cupicef.prop) cupicef.I (R x y) + (__ : cupicef.Term (cupicef.type s1) (R x y) => + Coq__Init__Logic.False)). + +thm complement__inverse : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s2)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) -> + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) -> + cstr_3 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + (__ : cupicef.Term (cupicef.type s0) A -> + __0 : cupicef.Term (cupicef.type s0) A -> + cupicef.Univ (cupicef.type s1)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s4)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))))) + cstr_3 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))))) + cstr_2 cupicef.BoolNone) + (crelation s0 s4 A)) + (complement s0 s1 s4 A + (flip s0 s0 s2 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R)) + (flip s0 s0 s3 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + (complement s0 s1 s4 A R))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s2)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) => + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) => + cstr_3 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) => + A : cupicef.Univ (cupicef.type s0) => + R : + (__ : cupicef.Term (cupicef.type s0) A -> + __0 : cupicef.Term (cupicef.type s0) A -> + cupicef.Univ (cupicef.type s1)) => + Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s4)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))))) + cstr_3 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))))) + cstr_2 cupicef.BoolNone) + (crelation s0 s4 A)) + (flip s0 s0 s3 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) cupicef.I) + (cupicef.univ (cupicef.type s3) (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) (cupicef.axiom (cupicef.type s4)) + cupicef.I)) + (complement s0 s1 s4 A R)). + +def Irreflexive : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) -> + cupicef.Univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) => + Reflexive s0 s1 A (complement s0 s2 s1 A R). + +#TYPE_CLASS Irreflexive. + +def irreflexivity : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) -> + Irreflexive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Irreflexive s0 s1 s2 A R) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Reflexive s0 s1 A (complement s0 s2 s1 A R)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) => + Irreflexive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Irreflexive s0 s1 s2 A R) => + Irreflexive0. + +def Symmetric : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + cupicef.Univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))) + cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) cupicef.I + (R x y) (__ : cupicef.Term (cupicef.type s1) (R x y) => R y x))). + +#TYPE_CLASS Symmetric. + +def symmetry : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + Symmetric0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Symmetric s0 s1 A R) -> + x : cupicef.Term (cupicef.type s0) A -> + y : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s1) (R x y) -> + cupicef.Term (cupicef.type s1) (R y x) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + Symmetric0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Symmetric s0 s1 A R) => + Symmetric0. + +def Asymmetric : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) -> + cupicef.Univ + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s2) (cupicef.type s1))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s2) (cupicef.type s1)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s2) (cupicef.type s1)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s2) (cupicef.type s1))) + cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s2) (cupicef.type s1) + (cupicef.rule (cupicef.type s2) (cupicef.type s1)) cupicef.I + (R x y) + (__ : cupicef.Term (cupicef.type s2) (R x y) => + cupicef.cast (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + cstr_0 (complement s0 s2 s3 A R y x)))). + +#TYPE_CLASS Asymmetric. + +def asymmetry : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) -> + Asymmetric0 : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (Asymmetric s0 s1 s2 s3 cstr_0 A R) -> + x : cupicef.Term (cupicef.type s0) A -> + y : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s2) (R x y) -> + cupicef.Term (cupicef.type s3) (complement s0 s2 s3 A R y x) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s1)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) => + Asymmetric0 : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (Asymmetric s0 s1 s2 s3 cstr_0 A R) => + Asymmetric0. + +def Transitive : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + cupicef.Univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))))) + cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))) + cupicef.I A + (z : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))) + cupicef.I (R x y) + (__ : cupicef.Term (cupicef.type s1) (R x y) => + cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) cupicef.I + (R y z) + (__0 : cupicef.Term (cupicef.type s1) (R y z) => R x z))))). + +#TYPE_CLASS Transitive. + +def transitivity : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + Transitive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Transitive s0 s1 A R) -> + x : cupicef.Term (cupicef.type s0) A -> + y : cupicef.Term (cupicef.type s0) A -> + z : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s1) (R x y) -> + __0 : cupicef.Term (cupicef.type s1) (R y z) -> + cupicef.Term (cupicef.type s1) (R x z) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + Transitive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Transitive s0 s1 A R) => + Transitive0. + +def PreOrder : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + cupicef.Univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)). + +#TYPE_CLASS PreOrder. + +def Build__PreOrder : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + PreOrder__Reflexive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Reflexive s0 s1 A R) -> + PreOrder__Transitive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Transitive s0 s1 A R) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PreOrder s0 s1 A R). + +[ ] cupicef.code_guarded? _ (Build__PreOrder _ _ _ _ _ _) --> cupicef.guarded. + +def match____PreOrder : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + P : + (cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PreOrder s0 s1 A R) -> + cupicef.Univ s) -> + case__Build__PreOrder : + (PreOrder__Reflexive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Reflexive s0 s1 A R) -> + PreOrder__Transitive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Transitive s0 s1 A R) -> + cupicef.Term s + (P + (Build__PreOrder s0 s1 A R PreOrder__Reflexive0 + PreOrder__Transitive0))) -> + x : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PreOrder s0 s1 A R) -> + cupicef.Term s (P x). + +[ s0, s1, s, A, R, P, case__Build__PreOrder, PreOrder__Reflexive0, PreOrder__Transitive0] + match____PreOrder s0 s1 s A R P case__Build__PreOrder + (Build__PreOrder s0 s1 A R PreOrder__Reflexive0 PreOrder__Transitive0) + --> + case__Build__PreOrder PreOrder__Reflexive0 PreOrder__Transitive0. + +[ s0, s1, s, A, R, P, s'] + match____PreOrder s0 s1 s A R + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____PreOrder s0 s1 s' A R (x => P x). + +def PreOrder__Reflexive : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + PreOrder0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PreOrder s0 s1 A R) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Reflexive s0 s1 A R) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + PreOrder0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PreOrder s0 s1 A R) => + match____PreOrder s0 s1 (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + A R + (PreOrder1 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PreOrder s0 s1 A R) => + Reflexive s0 s1 A R) + (PreOrder__Reflexive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Reflexive s0 s1 A R) => + PreOrder__Transitive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Transitive s0 s1 A R) => + PreOrder__Reflexive0) + PreOrder0. + +#TYPE_CLASS_INSTANCE PreOrder__Reflexive. + +def PreOrder__Transitive : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + PreOrder0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PreOrder s0 s1 A R) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Transitive s0 s1 A R) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + PreOrder0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PreOrder s0 s1 A R) => + match____PreOrder s0 s1 (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + A R + (PreOrder1 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PreOrder s0 s1 A R) => + Transitive s0 s1 A R) + (PreOrder__Reflexive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Reflexive s0 s1 A R) => + PreOrder__Transitive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Transitive s0 s1 A R) => + PreOrder__Transitive0) + PreOrder0. + +#TYPE_CLASS_INSTANCE PreOrder__Transitive. + +def StrictOrder : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) -> + cupicef.Univ + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))). + +#TYPE_CLASS StrictOrder. + +def Build__StrictOrder : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) -> + StrictOrder__Irreflexive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Irreflexive s0 s1 s2 A R) -> + StrictOrder__Transitive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (Transitive s0 s2 A R) -> + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (StrictOrder s0 s1 s2 A R). + +[ ] cupicef.code_guarded? _ (Build__StrictOrder _ _ _ _ _ _ _) --> cupicef.guarded. + +def match____StrictOrder : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) -> + P : + (cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (StrictOrder s0 s1 s2 A R) -> + cupicef.Univ s) -> + case__Build__StrictOrder : + (StrictOrder__Irreflexive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Irreflexive s0 s1 s2 A R) -> + StrictOrder__Transitive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (Transitive s0 s2 A R) -> + cupicef.Term s + (P + (Build__StrictOrder s0 s1 s2 A R StrictOrder__Irreflexive0 + StrictOrder__Transitive0))) -> + x : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (StrictOrder s0 s1 s2 A R) -> + cupicef.Term s (P x). + +[ s0, s1, s2, s, A, R, P, case__Build__StrictOrder, StrictOrder__Irreflexive0, StrictOrder__Transitive0] + match____StrictOrder s0 s1 s2 s A R P case__Build__StrictOrder + (Build__StrictOrder s0 s1 s2 A R StrictOrder__Irreflexive0 + StrictOrder__Transitive0) + --> + case__Build__StrictOrder StrictOrder__Irreflexive0 StrictOrder__Transitive0. + +[ s0, s1, s2, s, A, R, P, s'] + match____StrictOrder s0 s1 s2 s A R + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____StrictOrder s0 s1 s2 s' A R (x => P x). + +def StrictOrder__Irreflexive : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) -> + StrictOrder0 : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (StrictOrder s0 s1 s2 A R) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Irreflexive s0 s1 s2 A R) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) => + StrictOrder0 : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (StrictOrder s0 s1 s2 A R) => + match____StrictOrder s0 s1 s2 + (cupicef.sup (cupicef.type s0) (cupicef.type s1)) A R + (StrictOrder1 : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (StrictOrder s0 s1 s2 A R) => + Irreflexive s0 s1 s2 A R) + (StrictOrder__Irreflexive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Irreflexive s0 s1 s2 A R) => + StrictOrder__Transitive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (Transitive s0 s2 A R) => + StrictOrder__Irreflexive0) + StrictOrder0. + +#TYPE_CLASS_INSTANCE StrictOrder__Irreflexive. + +def StrictOrder__Transitive : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) -> + StrictOrder0 : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (StrictOrder s0 s1 s2 A R) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (Transitive s0 s2 A R) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) => + StrictOrder0 : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (StrictOrder s0 s1 s2 A R) => + match____StrictOrder s0 s1 s2 + (cupicef.sup (cupicef.type s0) (cupicef.type s2)) A R + (StrictOrder1 : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (StrictOrder s0 s1 s2 A R) => + Transitive s0 s2 A R) + (StrictOrder__Irreflexive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Irreflexive s0 s1 s2 A R) => + StrictOrder__Transitive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (Transitive s0 s2 A R) => + StrictOrder__Transitive0) + StrictOrder0. + +#TYPE_CLASS_INSTANCE StrictOrder__Transitive. + +thm StrictOrder__Asymmetric : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) -> + H : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (StrictOrder s0 s1 s2 A R) -> + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s2) (cupicef.type s3))) + (Asymmetric s0 s3 s2 s3 cupicef.I A R) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) => + H : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (StrictOrder s0 s1 s2 A R) => + cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s2) (cupicef.type s3)))) + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s2) (cupicef.type s3))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s2) (cupicef.type s3))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s2) (cupicef.type s3)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s2) (cupicef.type s3)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s2) (cupicef.type s3))) + cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s2) (cupicef.type s3) + (cupicef.rule (cupicef.type s2) (cupicef.type s3)) cupicef.I + (R x y) + (X : cupicef.Term (cupicef.type s2) (R x y) => + complement s0 s2 s3 A R y x)))) + (Asymmetric s0 s3 s2 s3 cupicef.I A R) cupicef.I + (x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + X : cupicef.Term (cupicef.type s2) (R x y) => + cupicef.cast (cupicef.rule (cupicef.type s2) cupicef.prop) + (cupicef.type s3) + (cupicef.prod (cupicef.type s2) cupicef.prop + (cupicef.rule (cupicef.type s2) cupicef.prop) cupicef.I (R y x) + (X0 : cupicef.Term (cupicef.type s2) (R y x) => + Coq__Init__Logic.False)) + (complement s0 s2 s3 A R y x) cupicef.I + (X0 : cupicef.Term (cupicef.type s2) (R y x) => + match____StrictOrder s0 s1 s2 cupicef.prop A R + (s : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (StrictOrder s0 s1 s2 A R) => + Coq__Init__Logic.False) + (StrictOrder__Irreflexive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Irreflexive s0 s1 s2 A R) => + StrictOrder__Transitive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (Transitive s0 s2 A R) => + (X1 : + (y0 : cupicef.Term (cupicef.type s0) A -> + z : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s2) (R y y0) -> + __0 : cupicef.Term (cupicef.type s2) (R y0 z) -> + cupicef.Term (cupicef.type s2) (R y z)) => + (X2 : + cupicef.Term (cupicef.type s1) (complement s0 s2 s1 A R y y) => + (X3 : + (z : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s2) (R y x) -> + __0 : cupicef.Term (cupicef.type s2) (R x z) -> + cupicef.Term (cupicef.type s2) (R y z)) => + X2 (X3 y X0 X)) (X1 x)) (StrictOrder__Irreflexive0 y)) + (StrictOrder__Transitive0 y)) + H)). + +#TYPE_CLASS_INSTANCE StrictOrder__Asymmetric. + +def PER : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + cupicef.Univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)). + +#TYPE_CLASS PER. + +def Build__PER : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + PER__Symmetric0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Symmetric s0 s1 A R) -> + PER__Transitive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Transitive s0 s1 A R) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PER s0 s1 A R). + +[ ] cupicef.code_guarded? _ (Build__PER _ _ _ _ _ _) --> cupicef.guarded. + +def match____PER : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + P : + (cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PER s0 s1 A R) -> + cupicef.Univ s) -> + case__Build__PER : + (PER__Symmetric0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Symmetric s0 s1 A R) -> + PER__Transitive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Transitive s0 s1 A R) -> + cupicef.Term s + (P (Build__PER s0 s1 A R PER__Symmetric0 PER__Transitive0))) -> + x : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PER s0 s1 A R) -> + cupicef.Term s (P x). + +[ s0, s1, s, A, R, P, case__Build__PER, PER__Symmetric0, PER__Transitive0] + match____PER s0 s1 s A R P case__Build__PER + (Build__PER s0 s1 A R PER__Symmetric0 PER__Transitive0) + --> + case__Build__PER PER__Symmetric0 PER__Transitive0. + +[ s0, s1, s, A, R, P, s'] + match____PER s0 s1 s A R + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____PER s0 s1 s' A R (x => P x). + +def PER__Symmetric : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + PER0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PER s0 s1 A R) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Symmetric s0 s1 A R) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + PER0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PER s0 s1 A R) => + match____PER s0 s1 (cupicef.sup (cupicef.type s0) (cupicef.type s1)) A R + (PER1 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PER s0 s1 A R) => + Symmetric s0 s1 A R) + (PER__Symmetric0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Symmetric s0 s1 A R) => + PER__Transitive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Transitive s0 s1 A R) => + PER__Symmetric0) + PER0. + +#TYPE_CLASS_INSTANCE PER__Symmetric. + +def PER__Transitive : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + PER0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PER s0 s1 A R) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Transitive s0 s1 A R) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + PER0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PER s0 s1 A R) => + match____PER s0 s1 (cupicef.sup (cupicef.type s0) (cupicef.type s1)) A R + (PER1 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PER s0 s1 A R) => + Transitive s0 s1 A R) + (PER__Symmetric0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Symmetric s0 s1 A R) => + PER__Transitive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Transitive s0 s1 A R) => + PER__Transitive0) + PER0. + +#TYPE_CLASS_INSTANCE PER__Transitive. + +def Equivalence : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + cupicef.Univ (cupicef.sup (cupicef.type s0) (cupicef.type s1)). + +#TYPE_CLASS Equivalence. + +def Build__Equivalence : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + Equivalence__Reflexive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Reflexive s0 s1 A R) -> + Equivalence__Symmetric0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Symmetric s0 s1 A R) -> + Equivalence__Transitive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Transitive s0 s1 A R) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Equivalence s0 s1 A R). + +[ ] cupicef.code_guarded? _ (Build__Equivalence _ _ _ _ _ _ _) --> cupicef.guarded. + +def match____Equivalence : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + P : + (cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Equivalence s0 s1 A R) -> + cupicef.Univ s) -> + case__Build__Equivalence : + (Equivalence__Reflexive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Reflexive s0 s1 A R) -> + Equivalence__Symmetric0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Symmetric s0 s1 A R) -> + Equivalence__Transitive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Transitive s0 s1 A R) -> + cupicef.Term s + (P + (Build__Equivalence s0 s1 A R Equivalence__Reflexive0 + Equivalence__Symmetric0 Equivalence__Transitive0))) -> + x : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Equivalence s0 s1 A R) -> + cupicef.Term s (P x). + +[ s0, s1, s, A, R, P, case__Build__Equivalence, Equivalence__Reflexive0, Equivalence__Symmetric0, Equivalence__Transitive0] + match____Equivalence s0 s1 s A R P case__Build__Equivalence + (Build__Equivalence s0 s1 A R Equivalence__Reflexive0 + Equivalence__Symmetric0 Equivalence__Transitive0) + --> + case__Build__Equivalence Equivalence__Reflexive0 Equivalence__Symmetric0 + Equivalence__Transitive0. + +[ s0, s1, s, A, R, P, s'] + match____Equivalence s0 s1 s A R + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____Equivalence s0 s1 s' A R (x => P x). + +def Equivalence__Reflexive : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + Equivalence0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Equivalence s0 s1 A R) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Reflexive s0 s1 A R) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + Equivalence0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Equivalence s0 s1 A R) => + match____Equivalence s0 s1 + (cupicef.sup (cupicef.type s0) (cupicef.type s1)) A R + (Equivalence1 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Equivalence s0 s1 A R) => + Reflexive s0 s1 A R) + (Equivalence__Reflexive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Reflexive s0 s1 A R) => + Equivalence__Symmetric0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Symmetric s0 s1 A R) => + Equivalence__Transitive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Transitive s0 s1 A R) => + Equivalence__Reflexive0) + Equivalence0. + +#TYPE_CLASS_INSTANCE Equivalence__Reflexive. + +def Equivalence__Symmetric : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + Equivalence0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Equivalence s0 s1 A R) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Symmetric s0 s1 A R) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + Equivalence0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Equivalence s0 s1 A R) => + match____Equivalence s0 s1 + (cupicef.sup (cupicef.type s0) (cupicef.type s1)) A R + (Equivalence1 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Equivalence s0 s1 A R) => + Symmetric s0 s1 A R) + (Equivalence__Reflexive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Reflexive s0 s1 A R) => + Equivalence__Symmetric0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Symmetric s0 s1 A R) => + Equivalence__Transitive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Transitive s0 s1 A R) => + Equivalence__Symmetric0) + Equivalence0. + +#TYPE_CLASS_INSTANCE Equivalence__Symmetric. + +def Equivalence__Transitive : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + Equivalence0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Equivalence s0 s1 A R) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Transitive s0 s1 A R) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + Equivalence0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Equivalence s0 s1 A R) => + match____Equivalence s0 s1 + (cupicef.sup (cupicef.type s0) (cupicef.type s1)) A R + (Equivalence1 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Equivalence s0 s1 A R) => + Transitive s0 s1 A R) + (Equivalence__Reflexive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Reflexive s0 s1 A R) => + Equivalence__Symmetric0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Symmetric s0 s1 A R) => + Equivalence__Transitive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Transitive s0 s1 A R) => + Equivalence__Transitive0) + Equivalence0. + +#TYPE_CLASS_INSTANCE Equivalence__Transitive. + +def Equivalence__PER : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Equivalence s0 s1 A R) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PER s0 s1 A R) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Equivalence s0 s1 A R) => + Build__PER s0 s1 A R (Equivalence__Symmetric s0 s1 A R H) + (Equivalence__Transitive s0 s1 A R H). + +#TYPE_CLASS_INSTANCE Equivalence__PER. + +def Antisymmetric : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + equ : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Equivalence s0 s1 A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) -> + cupicef.Univ + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + equ : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Equivalence s0 s1 A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s1)))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s1))))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s1))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s1)))) + cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s1)) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s1))) + cupicef.I (R x y) + (__ : cupicef.Term (cupicef.type s2) (R x y) => + cupicef.prod (cupicef.type s2) (cupicef.type s1) + (cupicef.rule (cupicef.type s2) (cupicef.type s1)) cupicef.I + (R y x) + (__0 : cupicef.Term (cupicef.type s2) (R y x) => eqA x y)))). + +#TYPE_CLASS Antisymmetric. + +def antisymmetry : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + equ : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Equivalence s0 s1 A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) -> + Antisymmetric0 : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (Antisymmetric s0 s1 s2 A eqA equ R) -> + x : cupicef.Term (cupicef.type s0) A -> + y : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s2) (R x y) -> + __0 : cupicef.Term (cupicef.type s2) (R y x) -> + cupicef.Term (cupicef.type s1) (eqA x y) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + equ : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Equivalence s0 s1 A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) => + Antisymmetric0 : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (Antisymmetric s0 s1 s2 A eqA equ R) => + Antisymmetric0. + +def subrelation : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) -> + cupicef.Univ + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s2))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s2)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s2)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s2))) + cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s1) (cupicef.type s2) + (cupicef.rule (cupicef.type s1) (cupicef.type s2)) cupicef.I + (R x y) (__ : cupicef.Term (cupicef.type s1) (R x y) => R_27_ x y))). + +#TYPE_CLASS subrelation. + +def is__subrelation : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) -> + subrelation0 : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (subrelation s0 s1 s2 A R R_27_) -> + x : cupicef.Term (cupicef.type s0) A -> + y : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s1) (R x y) -> + cupicef.Term (cupicef.type s2) (R_27_ x y) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) => + subrelation0 : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (subrelation s0 s1 s2 A R R_27_) => + subrelation0. + +thm subrelation__symmetric : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s2)) (cupicef.type s1)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (Symmetric s0 s2 A R) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (subrelation s0 s2 s2 A + (flip s0 s0 s1 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + R) + R) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s2)) (cupicef.type s1)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (Symmetric s0 s2 A R) => + cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)))) + (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s2) (cupicef.type s2))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s2) (cupicef.type s2))) + cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s2) (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)) cupicef.I + (flip s0 s0 s1 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + R x y) + (H_27_ : + cupicef.Term (cupicef.type s2) + (flip s0 s0 s1 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + R x y) => + R x y)))) + (subrelation s0 s2 s2 A + (flip s0 s0 s1 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + R) + R) + cupicef.I + (x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + H_27_ : + cupicef.Term (cupicef.type s2) + (flip s0 s0 s1 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + R x y) => + symmetry s0 s2 A R H y x H_27_). + +thm flip__Reflexive : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s2)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Reflexive s0 s1 A R) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Reflexive s0 s1 A + (flip s0 s0 s2 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s2)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Reflexive s0 s1 A R) => + H. + +def flip__Irreflexive : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s2)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (crelation s0 s4 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (Irreflexive s0 s3 s4 A R) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Irreflexive s0 s1 s4 A + (flip s0 s0 s2 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s2)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (crelation s0 s4 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (Irreflexive s0 s3 s4 A R) => + irreflexivity s0 s3 s4 A R H. + +def flip__Symmetric : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s2)) (cupicef.type s1)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (Symmetric s0 s2 A R) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (Symmetric s0 s2 A + (flip s0 s0 s1 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + R)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s2)) (cupicef.type s1)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (Symmetric s0 s2 A R) => + x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + H0 : + cupicef.Term (cupicef.type s2) + (flip s0 s0 s1 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + R x y) => + symmetry s0 s2 A R H y x H0. + +def flip__Asymmetric : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s2)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) -> + H : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s3) (cupicef.type s4))) + (Asymmetric s0 s4 s3 s4 cupicef.I A R) -> + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s3))) + (Asymmetric s0 s1 s3 s1 cupicef.I A + (flip s0 s0 s2 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s2)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + H : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s3) (cupicef.type s4))) + (Asymmetric s0 s4 s3 s4 cupicef.I A R) => + x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + H0 : + cupicef.Term (cupicef.type s3) + (flip s0 s0 s2 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R x y) => + H_27_ : + cupicef.Term (cupicef.type s3) + (flip s0 s0 s2 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R y x) => + asymmetry s0 s4 s3 s4 cupicef.I A R H y x H0 H_27_. + +def flip__Transitive : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s2)) (cupicef.type s1)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (Transitive s0 s2 A R) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (Transitive s0 s2 A + (flip s0 s0 s1 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + R)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s2)) (cupicef.type s1)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (Transitive s0 s2 A R) => + x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + z : cupicef.Term (cupicef.type s0) A => + H0 : + cupicef.Term (cupicef.type s2) + (flip s0 s0 s1 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + R x y) => + H_27_ : + cupicef.Term (cupicef.type s2) + (flip s0 s0 s1 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + R y z) => + transitivity s0 s2 A R H z y x H_27_ H0. + +thm flip__Antisymmetric : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s3)) -> + A : cupicef.Univ (cupicef.type s0) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) -> + equ : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (Equivalence s0 s2 A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + H : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (Antisymmetric s0 s2 s1 A eqA equ R) -> + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (Antisymmetric s0 s2 s1 A eqA equ + (flip s0 s0 s3 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s3)) => + A : cupicef.Univ (cupicef.type s0) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) => + equ : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (Equivalence s0 s2 A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + H : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (Antisymmetric s0 s2 s1 A eqA equ R) => + cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s2))))) + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s2)))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s2))))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s2))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s2)))) + cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s2)) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s2))) + cupicef.I + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R x y) + (X : + cupicef.Term (cupicef.type s1) + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R x y) => + cupicef.prod (cupicef.type s1) (cupicef.type s2) + (cupicef.rule (cupicef.type s1) (cupicef.type s2)) cupicef.I + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R y x) + (X0 : + cupicef.Term (cupicef.type s1) + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R y x) => + eqA x y))))) + (Antisymmetric s0 s2 s1 A eqA equ + (flip s0 s0 s3 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R)) + cupicef.I + (x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + X : + cupicef.Term (cupicef.type s1) + (flip s0 s0 s3 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R x y) => + X0 : + cupicef.Term (cupicef.type s1) + (flip s0 s0 s3 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R y x) => + (X1 : + (y0 : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s1) (R y y0) -> + __0 : cupicef.Term (cupicef.type s1) (R y0 y) -> + cupicef.Term (cupicef.type s2) (eqA y y0)) => + (X2 : + (__ : cupicef.Term (cupicef.type s1) (R y x) -> + __0 : cupicef.Term (cupicef.type s1) (R x y) -> + cupicef.Term (cupicef.type s2) (eqA y x)) => + (X3 : + (__ : cupicef.Term (cupicef.type s1) (R x y) -> + cupicef.Term (cupicef.type s2) (eqA y x)) => + (X4 : cupicef.Term (cupicef.type s2) (eqA y x) => + (X5 : + (y0 : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s1) (R x y0) -> + __0 : cupicef.Term (cupicef.type s1) (R y0 x) -> + cupicef.Term (cupicef.type s2) (eqA x y0)) => + H x y X0 X) (H x)) (X3 X0)) + (X2 X)) + (X1 x)) + (H y)). + +thm flip__PreOrder : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s2)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PreOrder s0 s1 A R) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PreOrder s0 s1 A + (flip s0 s0 s2 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s2)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PreOrder s0 s1 A R) => + match____PreOrder s0 s1 (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + A R + (p : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PreOrder s0 s1 A R) => + PreOrder s0 s1 A + (flip s0 s0 s2 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R)) + (PreOrder__Reflexive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Reflexive s0 s1 A R) => + PreOrder__Transitive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Transitive s0 s1 A R) => + Build__PreOrder s0 s1 A + (flip s0 s0 s2 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R) + PreOrder__Reflexive0 + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))))) + (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))))) + cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))) + cupicef.I A + (z : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))) + cupicef.I + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R x y) + (X : + cupicef.Term (cupicef.type s1) + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) + cupicef.I)) + R x y) => + cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) + cupicef.I)) + R y z) + (X0 : + cupicef.Term (cupicef.type s1) + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) + cupicef.I)) + R y z) => + R z x)))))) + (Transitive s0 s1 A + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R)) + cupicef.I + (x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + z : cupicef.Term (cupicef.type s0) A => + X : + cupicef.Term (cupicef.type s1) + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R x y) => + X0 : + cupicef.Term (cupicef.type s1) + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R y z) => + (X1 : + (y0 : cupicef.Term (cupicef.type s0) A -> + z0 : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s1) (R z y0) -> + __0 : cupicef.Term (cupicef.type s1) (R y0 z0) -> + cupicef.Term (cupicef.type s1) (R z z0)) => + (X2 : + (z0 : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s1) (R z y) -> + __0 : cupicef.Term (cupicef.type s1) (R y z0) -> + cupicef.Term (cupicef.type s1) (R z z0)) => + (X3 : + (__ : cupicef.Term (cupicef.type s1) (R z y) -> + __0 : cupicef.Term (cupicef.type s1) (R y x) -> + cupicef.Term (cupicef.type s1) (R z x)) => + (X4 : + (__ : cupicef.Term (cupicef.type s1) (R y x) -> + cupicef.Term (cupicef.type s1) (R z x)) => + (X5 : cupicef.Term (cupicef.type s1) (R z x) => X5) (X4 X)) + (X3 X0)) + (X2 x)) + (X1 y)) + (PreOrder__Transitive0 z)))) + H. + +thm flip__StrictOrder : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s2)) (cupicef.type s4)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) -> + H : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (StrictOrder s0 s1 s2 A R) -> + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s2) (cupicef.type s3))) + (StrictOrder s0 s3 s2 A + (flip s0 s0 s4 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + R)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s2)) (cupicef.type s4)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) => + H : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (StrictOrder s0 s1 s2 A R) => + match____StrictOrder s0 s1 s2 + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s2) (cupicef.type s3))) + A R + (s : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (StrictOrder s0 s1 s2 A R) => + StrictOrder s0 s3 s2 A + (flip s0 s0 s4 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + R)) + (StrictOrder__Irreflexive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Irreflexive s0 s1 s2 A R) => + StrictOrder__Transitive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (Transitive s0 s2 A R) => + Build__StrictOrder s0 s3 s2 A + (flip s0 s0 s4 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + R) + StrictOrder__Irreflexive0 + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)))))) + (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2))))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)))))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2))))) + cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)))) + cupicef.I A + (z : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2))) + cupicef.I + (flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + R x y) + (X : + cupicef.Term (cupicef.type s2) + (flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + R x y) => + cupicef.prod (cupicef.type s2) (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)) + cupicef.I + (flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + R y z) + (X0 : + cupicef.Term (cupicef.type s2) + (flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ + (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I)) + R y z) => + R z x)))))) + (Transitive s0 s2 A + (flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + R)) + cupicef.I + (x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + z : cupicef.Term (cupicef.type s0) A => + X : + cupicef.Term (cupicef.type s2) + (flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + R x y) => + X0 : + cupicef.Term (cupicef.type s2) + (flip s0 s0 s4 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s2)) + (cupicef.axiom (cupicef.axiom (cupicef.type s2))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I)) + R y z) => + (X1 : + (y0 : cupicef.Term (cupicef.type s0) A -> + z0 : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s2) (R z y0) -> + __0 : cupicef.Term (cupicef.type s2) (R y0 z0) -> + cupicef.Term (cupicef.type s2) (R z z0)) => + (X2 : + (z0 : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s2) (R z y) -> + __0 : cupicef.Term (cupicef.type s2) (R y z0) -> + cupicef.Term (cupicef.type s2) (R z z0)) => + (X3 : + (__ : cupicef.Term (cupicef.type s2) (R z y) -> + __0 : cupicef.Term (cupicef.type s2) (R y x) -> + cupicef.Term (cupicef.type s2) (R z x)) => + (X4 : + (__ : cupicef.Term (cupicef.type s2) (R y x) -> + cupicef.Term (cupicef.type s2) (R z x)) => + (X5 : cupicef.Term (cupicef.type s2) (R z x) => X5) (X4 X)) + (X3 X0)) + (X2 x)) + (X1 y)) + (StrictOrder__Transitive0 z)))) + H. + +thm flip__PER : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s2)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PER s0 s1 A R) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PER s0 s1 A + (flip s0 s0 s2 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s2)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PER s0 s1 A R) => + match____PER s0 s1 (cupicef.sup (cupicef.type s0) (cupicef.type s1)) A R + (p : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (PER s0 s1 A R) => + PER s0 s1 A + (flip s0 s0 s2 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R)) + (PER__Symmetric0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Symmetric s0 s1 A R) => + PER__Transitive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Transitive s0 s1 A R) => + Build__PER s0 s1 A + (flip s0 s0 s2 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R) + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))) + (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))) + cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R x y) + (X : + cupicef.Term (cupicef.type s1) + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R x y) => + R x y)))) + (Symmetric s0 s1 A + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R)) + cupicef.I + (x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + X : + cupicef.Term (cupicef.type s1) + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R x y) => + (X0 : + (y0 : cupicef.Term (cupicef.type s0) A -> + z : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s1) (R y y0) -> + __0 : cupicef.Term (cupicef.type s1) (R y0 z) -> + cupicef.Term (cupicef.type s1) (R y z)) => + (X1 : + (y0 : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s1) (R y y0) -> + cupicef.Term (cupicef.type s1) (R y0 y)) => + (X2 : + (__ : cupicef.Term (cupicef.type s1) (R y x) -> + cupicef.Term (cupicef.type s1) (R x y)) => + (X3 : cupicef.Term (cupicef.type s1) (R x y) => X3) (X2 X)) + (X1 x)) + (PER__Symmetric0 y)) + (PER__Transitive0 y))) + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))))) + (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))))) + cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))) + cupicef.I A + (z : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))) + cupicef.I + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R x y) + (X : + cupicef.Term (cupicef.type s1) + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) + cupicef.I)) + R x y) => + cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) + cupicef.I)) + R y z) + (X0 : + cupicef.Term (cupicef.type s1) + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) + cupicef.I)) + R y z) => + R z x)))))) + (Transitive s0 s1 A + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R)) + cupicef.I + (x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + z : cupicef.Term (cupicef.type s0) A => + X : + cupicef.Term (cupicef.type s1) + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R x y) => + X0 : + cupicef.Term (cupicef.type s1) + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R y z) => + (X1 : + (y0 : cupicef.Term (cupicef.type s0) A -> + z0 : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s1) (R z y0) -> + __0 : cupicef.Term (cupicef.type s1) (R y0 z0) -> + cupicef.Term (cupicef.type s1) (R z z0)) => + (X2 : + (z0 : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s1) (R z y) -> + __0 : cupicef.Term (cupicef.type s1) (R y z0) -> + cupicef.Term (cupicef.type s1) (R z z0)) => + (X3 : + (__ : cupicef.Term (cupicef.type s1) (R z y) -> + __0 : cupicef.Term (cupicef.type s1) (R y x) -> + cupicef.Term (cupicef.type s1) (R z x)) => + (X4 : + (__ : cupicef.Term (cupicef.type s1) (R y x) -> + cupicef.Term (cupicef.type s1) (R z x)) => + (X5 : cupicef.Term (cupicef.type s1) (R z x) => X5) (X4 X)) + (X3 X0)) + (X2 x)) + (X1 y)) + (PER__Transitive0 z)))) + H. + +thm flip__Equivalence : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s2)) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Equivalence s0 s1 A R) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Equivalence s0 s1 A + (flip s0 s0 s2 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s2)) => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Equivalence s0 s1 A R) => + match____Equivalence s0 s1 + (cupicef.sup (cupicef.type s0) (cupicef.type s1)) A R + (e : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Equivalence s0 s1 A R) => + Equivalence s0 s1 A + (flip s0 s0 s2 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R)) + (Equivalence__Reflexive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Reflexive s0 s1 A R) => + Equivalence__Symmetric0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Symmetric s0 s1 A R) => + Equivalence__Transitive0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Transitive s0 s1 A R) => + Build__Equivalence s0 s1 A + (flip s0 s0 s2 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R) + Equivalence__Reflexive0 + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))) + (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))) + cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R x y) + (X : + cupicef.Term (cupicef.type s1) + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R x y) => + R x y)))) + (Symmetric s0 s1 A + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R)) + cupicef.I + (x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + X : + cupicef.Term (cupicef.type s1) + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R x y) => + (X0 : + (y0 : cupicef.Term (cupicef.type s0) A -> + z : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s1) (R y y0) -> + __0 : cupicef.Term (cupicef.type s1) (R y0 z) -> + cupicef.Term (cupicef.type s1) (R y z)) => + (X1 : + (y0 : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s1) (R y y0) -> + cupicef.Term (cupicef.type s1) (R y0 y)) => + (X2 : + (__ : cupicef.Term (cupicef.type s1) (R y x) -> + cupicef.Term (cupicef.type s1) (R x y)) => + (X3 : cupicef.Term (cupicef.type s1) (R x y) => X3) (X2 X)) + (X1 x)) + (Equivalence__Symmetric0 y)) + (Equivalence__Transitive0 y))) + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))))) + (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))))) + cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))) + cupicef.I A + (z : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))) + cupicef.I + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R x y) + (X : + cupicef.Term (cupicef.type s1) + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) + cupicef.I)) + R x y) => + cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) + cupicef.I)) + R y z) + (X0 : + cupicef.Term (cupicef.type s1) + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ + (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) + cupicef.I)) + R y z) => + R z x)))))) + (Transitive s0 s1 A + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R)) + cupicef.I + (x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + z : cupicef.Term (cupicef.type s0) A => + X : + cupicef.Term (cupicef.type s1) + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R x y) => + X0 : + cupicef.Term (cupicef.type s1) + (flip s0 s0 s2 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s2)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + cupicef.I) + (cupicef.univ (cupicef.type s2) + (cupicef.axiom (cupicef.type s2)) cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + R y z) => + (X1 : + (y0 : cupicef.Term (cupicef.type s0) A -> + z0 : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s1) (R z y0) -> + __0 : cupicef.Term (cupicef.type s1) (R y0 z0) -> + cupicef.Term (cupicef.type s1) (R z z0)) => + (X2 : + (z0 : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s1) (R z y) -> + __0 : cupicef.Term (cupicef.type s1) (R y z0) -> + cupicef.Term (cupicef.type s1) (R z z0)) => + (X3 : + (__ : cupicef.Term (cupicef.type s1) (R z y) -> + __0 : cupicef.Term (cupicef.type s1) (R y x) -> + cupicef.Term (cupicef.type s1) (R z x)) => + (X4 : + (__ : cupicef.Term (cupicef.type s1) (R y x) -> + cupicef.Term (cupicef.type s1) (R z x)) => + (X5 : cupicef.Term (cupicef.type s1) (R z x) => X5) (X4 X)) + (X3 X0)) + (X2 x)) + (X1 y)) + (Equivalence__Transitive0 z)))) + H. + +thm complement__Irreflexive : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (Reflexive s0 s3 A R) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Irreflexive s0 s1 s2 A (complement s0 s3 s2 A R)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (Reflexive s0 s3 A R) => + cupicef.cast (cupicef.rule (cupicef.type s0) (cupicef.type s1)) + (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.prod (cupicef.type s0) (cupicef.type s1) + (cupicef.rule (cupicef.type s0) (cupicef.type s1)) cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + complement s0 s2 s1 A (complement s0 s3 s2 A R) x x)) + (Irreflexive s0 s1 s2 A (complement s0 s3 s2 A R)) cupicef.I + (x : cupicef.Term (cupicef.type s0) A => + cupicef.cast (cupicef.rule (cupicef.type s2) cupicef.prop) + (cupicef.type s1) + (cupicef.prod (cupicef.type s2) cupicef.prop + (cupicef.rule (cupicef.type s2) cupicef.prop) cupicef.I + (complement s0 s3 s2 A R x x) + (X : cupicef.Term (cupicef.type s2) (complement s0 s3 s2 A R x x) => + Coq__Init__Logic.False)) + (complement s0 s2 s1 A (complement s0 s3 s2 A R) x x) cupicef.I + (X : cupicef.Term (cupicef.type s2) (complement s0 s3 s2 A R x x) => + (X0 : cupicef.Term (cupicef.type s3) (R x x) => + (H0 : cupicef.Term cupicef.prop Coq__Init__Logic.False => H0) + (X X0)) + (H x))). + +thm complement__Symmetric : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (Symmetric s0 s2 A R) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Symmetric s0 s1 A (complement s0 s2 s1 A R)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s2)) + (Symmetric s0 s2 A R) => + cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))) + (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))) + cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) cupicef.I + (complement s0 s2 s1 A R x y) + (X : + cupicef.Term (cupicef.type s1) (complement s0 s2 s1 A R x y) => + complement s0 s2 s1 A R y x)))) + (Symmetric s0 s1 A (complement s0 s2 s1 A R)) cupicef.I + (x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + X : cupicef.Term (cupicef.type s1) (complement s0 s2 s1 A R x y) => + cupicef.cast (cupicef.rule (cupicef.type s2) cupicef.prop) + (cupicef.type s1) + (cupicef.prod (cupicef.type s2) cupicef.prop + (cupicef.rule (cupicef.type s2) cupicef.prop) cupicef.I (R y x) + (X0 : cupicef.Term (cupicef.type s2) (R y x) => + Coq__Init__Logic.False)) + (complement s0 s2 s1 A R y x) cupicef.I + (X0 : cupicef.Term (cupicef.type s2) (R y x) => + (X1 : + (y0 : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s2) (R y y0) -> + cupicef.Term (cupicef.type s2) (R y0 y)) => + (X2 : + (__ : cupicef.Term (cupicef.type s2) (R y y) -> + cupicef.Term (cupicef.type s2) (R y y)) => + (X3 : + (__ : cupicef.Term (cupicef.type s2) (R y x) -> + cupicef.Term (cupicef.type s2) (R x y)) => + (X4 : cupicef.Term (cupicef.type s2) (R x y) => + (H0 : cupicef.Term cupicef.prop Coq__Init__Logic.False => H0) + (X X4)) + (X3 X0)) + (X1 x)) + (X1 y)) + (H y))). + +def RewriteRelation : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + cupicef.Univ cupicef.prop. + +#TYPE_CLASS RewriteRelation. + +def Build__RewriteRelation : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + cupicef.Term cupicef.prop (RewriteRelation s0 s1 A RA). + +[ ] cupicef.code_guarded? _ (Build__RewriteRelation _ _ _ _) --> cupicef.guarded. + +def match____RewriteRelation : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type s0) -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + P : + (cupicef.Term cupicef.prop (RewriteRelation s0 s1 A RA) -> cupicef.Univ s) -> + case__Build__RewriteRelation : + cupicef.Term s (P (Build__RewriteRelation s0 s1 A RA)) -> + x : cupicef.Term cupicef.prop (RewriteRelation s0 s1 A RA) -> + cupicef.Term s (P x). + +[ s0, s1, s, A, RA, P, case__Build__RewriteRelation] + match____RewriteRelation s0 s1 s A RA P case__Build__RewriteRelation + (Build__RewriteRelation s0 s1 A RA) + --> + case__Build__RewriteRelation. + +[ s0, s1, s, A, RA, P, s'] + match____RewriteRelation s0 s1 s A RA + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____RewriteRelation s0 s1 s' A RA (x => P x). + +def equivalence__rewrite__crelation : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + A : cupicef.Univ (cupicef.type s0) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Equivalence s0 s1 A eqA) -> + cupicef.Term cupicef.prop (RewriteRelation s0 s1 A eqA) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + A : cupicef.Univ (cupicef.type s0) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + H : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Equivalence s0 s1 A eqA) => + Build__RewriteRelation s0 s1 A eqA. + +#TYPE_CLASS_INSTANCE equivalence__rewrite__crelation. + +def eq__Reflexive : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) -> + A : cupicef.Univ (cupicef.type s0) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Reflexive s0 s1 A + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (crelation s0 s1 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_0 A)))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) => + A : cupicef.Univ (cupicef.type s0) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_0 A). + +#TYPE_CLASS_INSTANCE eq__Reflexive. + +def eq__Symmetric : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) -> + A : cupicef.Univ (cupicef.type s0) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Symmetric s0 s1 A + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (crelation s0 s1 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_0 A)))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) => + A : cupicef.Univ (cupicef.type s0) => + Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_1 A). + +#TYPE_CLASS_INSTANCE eq__Symmetric. + +def eq__Transitive : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) -> + A : cupicef.Univ (cupicef.type s0) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Transitive s0 s1 A + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (crelation s0 s1 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_0 A)))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) => + A : cupicef.Univ (cupicef.type s0) => + Coq__Init__Logic.eq__trans + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_1 A). + +#TYPE_CLASS_INSTANCE eq__Transitive. + +def eq__equivalence : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) -> + A : cupicef.Univ (cupicef.type s0) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Equivalence s0 s1 A + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (crelation s0 s1 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_0 A)))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.type s0) + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) => + A : cupicef.Univ (cupicef.type s0) => + Build__Equivalence s0 s1 A + (cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type s0) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type s0) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (crelation s0 s1 A) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type s0)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cstr_0 A))) + (eq__Reflexive s0 s1 cstr_0 A) (eq__Symmetric s0 s1 cstr_0 cstr_1 A) + (eq__Transitive s0 s1 cstr_0 cstr_1 A). + +#TYPE_CLASS_INSTANCE eq__equivalence. + +def RewriteRelation__instance__0 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + cstr_0 : + cupicef.eps (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0)) -> + cupicef.Term cupicef.prop + (RewriteRelation s0 s1 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (crelation s0 s1 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + cupicef.I Coq__Program__Basics.impl)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0)) => + Build__RewriteRelation s0 s1 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (crelation s0 s1 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + cupicef.I Coq__Program__Basics.impl). + +#TYPE_CLASS_INSTANCE RewriteRelation__instance__0. + +def RewriteRelation__instance__1 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + cstr_0 : + cupicef.eps (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0)) -> + cupicef.Term cupicef.prop + (RewriteRelation s0 s1 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (crelation s0 s1 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + cupicef.I Coq__Init__Logic.iff)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0)) => + Build__RewriteRelation s0 s1 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (crelation s0 s1 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + cupicef.I Coq__Init__Logic.iff). + +#TYPE_CLASS_INSTANCE RewriteRelation__instance__1. + +def impl__Reflexive__obligation__1 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + cstr_0 : + cupicef.eps (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0)) -> + x : cupicef.Univ cupicef.prop -> + cupicef.Term cupicef.prop (Coq__Program__Basics.impl x x) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0)) => + x : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop x => + H. + +def impl__Reflexive : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + cstr_0 : + cupicef.eps (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0)) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Reflexive s0 s1 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (crelation s0 s1 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + cupicef.I Coq__Program__Basics.impl)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0)) => + impl__Reflexive__obligation__1 s0 s1 cstr_0. + +#TYPE_CLASS_INSTANCE impl__Reflexive. + +def impl__Transitive__obligation__1 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + cstr_0 : + cupicef.eps (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0)) -> + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + z : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (Coq__Program__Basics.impl x y) -> + __0 : cupicef.Term cupicef.prop (Coq__Program__Basics.impl y z) -> + cupicef.Term cupicef.prop (Coq__Program__Basics.impl x z) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0)) => + x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + z : cupicef.Univ cupicef.prop => + H : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H0 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop z) => + H1 : cupicef.Term cupicef.prop x => + H0 (H H1). + +def impl__Transitive : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + cstr_0 : + cupicef.eps (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0)) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Transitive s0 s1 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (crelation s0 s1 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + cupicef.I Coq__Program__Basics.impl)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0)) => + impl__Transitive__obligation__1 s0 s1 cstr_0. + +#TYPE_CLASS_INSTANCE impl__Transitive. + +def iff__Reflexive : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + cstr_0 : + cupicef.eps (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0)) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Reflexive s0 s1 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (crelation s0 s1 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + cupicef.I Coq__Init__Logic.iff)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0)) => + Coq__Init__Logic.iff__refl. + +#TYPE_CLASS_INSTANCE iff__Reflexive. + +def iff__Symmetric : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + cstr_0 : + cupicef.eps (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0)) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Symmetric s0 s1 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (crelation s0 s1 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + cupicef.I Coq__Init__Logic.iff)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0)) => + Coq__Init__Logic.iff__sym. + +#TYPE_CLASS_INSTANCE iff__Symmetric. + +def iff__Transitive : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + cstr_0 : + cupicef.eps (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0)) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Transitive s0 s1 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (crelation s0 s1 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + cupicef.I Coq__Init__Logic.iff)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0)) => + Coq__Init__Logic.iff__trans. + +#TYPE_CLASS_INSTANCE iff__Transitive. + +def iff__equivalence : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + cstr_0 : + cupicef.eps (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0)) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Equivalence s0 s1 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (crelation s0 s1 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + cupicef.I Coq__Init__Logic.iff)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom cupicef.set) (cupicef.type s0)) => + Build__Equivalence s0 s1 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (crelation s0 s1 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type s0) + (cupicef.axiom (cupicef.type s0)) cupicef.I) + cstr_0 + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + cupicef.I Coq__Init__Logic.iff) + (iff__Reflexive s0 s1 cstr_0) (iff__Symmetric s0 s1 cstr_0) + (iff__Transitive s0 s1 cstr_0). + +#TYPE_CLASS_INSTANCE iff__equivalence. + +def arrow__Reflexive__obligation__1 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s0)) -> + x : cupicef.Univ (cupicef.type s1) -> + cupicef.Term (cupicef.type s1) (arrow s1 s1 x x) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s0)) => + x : cupicef.Univ (cupicef.type s1) => + X : cupicef.Term (cupicef.type s1) x => + X. + +def arrow__Reflexive : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s0)) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Reflexive s0 s1 + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I)) + (arrow s1 s1)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s0)) => + arrow__Reflexive__obligation__1 s0 s1 cstr_0. + +#TYPE_CLASS_INSTANCE arrow__Reflexive. + +def arrow__Transitive__obligation__1 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s0)) -> + x : cupicef.Univ (cupicef.type s1) -> + y : cupicef.Univ (cupicef.type s1) -> + z : cupicef.Univ (cupicef.type s1) -> + __ : cupicef.Term (cupicef.type s1) (arrow s1 s1 x y) -> + __0 : cupicef.Term (cupicef.type s1) (arrow s1 s1 y z) -> + cupicef.Term (cupicef.type s1) (arrow s1 s1 x z) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s0)) => + x : cupicef.Univ (cupicef.type s1) => + y : cupicef.Univ (cupicef.type s1) => + z : cupicef.Univ (cupicef.type s1) => + X : + (__ : cupicef.Term (cupicef.type s1) x -> + cupicef.Term (cupicef.type s1) y) => + X0 : + (__ : cupicef.Term (cupicef.type s1) y -> + cupicef.Term (cupicef.type s1) z) => + X1 : cupicef.Term (cupicef.type s1) x => + X0 (X X1). + +def arrow__Transitive : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s0)) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Transitive s0 s1 + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I)) + (arrow s1 s1)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s0)) => + arrow__Transitive__obligation__1 s0 s1 cstr_0. + +#TYPE_CLASS_INSTANCE arrow__Transitive. + +def iffT__Reflexive : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s0)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Reflexive s0 s1 + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I)) + (iffT s1 s1 cstr_1 cstr_2 cstr_1 cstr_2)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s0)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) => + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) => + cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type s1)) (cupicef.type s1)) + (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.prod (cupicef.axiom (cupicef.type s1)) (cupicef.type s1) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) (cupicef.type s1)) + cupicef.I + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (x : cupicef.Univ (cupicef.type s1) => + Coq__Init__Datatypes.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) cupicef.I x + (__ : cupicef.Term (cupicef.type s1) x => x)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) cupicef.I x + (__ : cupicef.Term (cupicef.type s1) x => x)))) + (Reflexive s0 s1 + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I)) + (iffT s1 s1 cstr_1 cstr_2 cstr_1 cstr_2)) + cupicef.I + (x : cupicef.Univ (cupicef.type s1) => + Coq__Init__Datatypes.pair + (cupicef.cast (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_1 + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) cupicef.I x + (__ : cupicef.Term (cupicef.type s1) x => x))) + (cupicef.cast (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_2 + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) cupicef.I x + (__ : cupicef.Term (cupicef.type s1) x => x))) + (X : cupicef.Term (cupicef.type s1) x => X) + (X : cupicef.Term (cupicef.type s1) x => X)). + +#TYPE_CLASS_INSTANCE iffT__Reflexive. + +def iffT__Symmetric : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s0)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_3 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Symmetric s0 s1 + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I)) + (iffT s1 s1 cstr_1 cstr_2 cstr_1 cstr_2)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s0)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) => + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) => + cstr_3 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) => + cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))) + (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))) + cupicef.I + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (x : cupicef.Univ (cupicef.type s1) => + cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))) + cupicef.I + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (y : cupicef.Univ (cupicef.type s1) => + cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) cupicef.I + (iffT s1 s1 cstr_1 cstr_2 cstr_1 cstr_2 x y) + (X : + cupicef.Term (cupicef.type s1) + (iffT s1 s1 cstr_1 cstr_2 cstr_1 cstr_2 x y) => + (X0 : + cupicef.Term (cupicef.type s1) + (Coq__Init__Datatypes.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I x + (__ : cupicef.Term (cupicef.type s1) x => y)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => x))) => + iffT s1 s1 cstr_1 cstr_2 cstr_1 cstr_2 y x) X)))) + (Symmetric s0 s1 + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I)) + (iffT s1 s1 cstr_1 cstr_2 cstr_1 cstr_2)) + cupicef.I + (x : cupicef.Univ (cupicef.type s1) => + y : cupicef.Univ (cupicef.type s1) => + X : + cupicef.Term (cupicef.type s1) + (iffT s1 s1 cstr_1 cstr_2 cstr_1 cstr_2 x y) => + Coq__Init__Datatypes.prod__rect + (cupicef.cast (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_1 + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) cupicef.I x + (__ : cupicef.Term (cupicef.type s1) x => y))) + (cupicef.cast (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_2 + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => x))) + (cupicef.cast + (cupicef.rule (cupicef.type s1) (cupicef.axiom (cupicef.type s1))) + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s1))) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I x (__ : cupicef.Term (cupicef.type s1) x => y)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y (__ : cupicef.Term (cupicef.type s1) y => x))) + (X0 : + cupicef.Term (cupicef.type s1) + (Coq__Init__Datatypes.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I x + (__ : cupicef.Term (cupicef.type s1) x => y)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => x))) => + cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + (cupicef.prod (cupicef.type s1) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I x (__ : cupicef.Term (cupicef.type s1) x => y)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y (__ : cupicef.Term (cupicef.type s1) y => x))) + (p : + cupicef.Term (cupicef.type s1) + (Coq__Init__Datatypes.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I x + (__ : cupicef.Term (cupicef.type s1) x => y)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => x))) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cstr_3 + (X0 : + cupicef.Term (cupicef.type s1) + (Coq__Init__Datatypes.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I x + (__ : cupicef.Term (cupicef.type s1) x => y)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => x))) => + iffT s1 s1 cstr_1 cstr_2 cstr_1 cstr_2 y x)) + (a : + (__ : cupicef.Term (cupicef.type s1) x -> + cupicef.Term (cupicef.type s1) y) => + b : + (__ : cupicef.Term (cupicef.type s1) y -> + cupicef.Term (cupicef.type s1) x) => + Coq__Init__Datatypes.pair + (cupicef.cast (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_1 + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) cupicef.I + y (__ : cupicef.Term (cupicef.type s1) y => x))) + (cupicef.cast (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_2 + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) cupicef.I + x (__ : cupicef.Term (cupicef.type s1) x => y))) + b a) + X). + +#TYPE_CLASS_INSTANCE iffT__Symmetric. + +def iffT__Transitive : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s0)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_3 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (Transitive s0 s1 + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I)) + (iffT s1 s1 cstr_1 cstr_2 cstr_1 cstr_2)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s1)) (cupicef.type s0)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) => + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) => + cstr_3 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) => + cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))))) + (cupicef.sup (cupicef.type s0) (cupicef.type s1)) + (cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))))) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))))) + cupicef.I + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (x : cupicef.Univ (cupicef.type s1) => + cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))))) + cupicef.I + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (y : cupicef.Univ (cupicef.type s1) => + cupicef.prod (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))) + (cupicef.rule (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)))) + cupicef.I + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (z : cupicef.Univ (cupicef.type s1) => + cupicef.prod (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1))) + cupicef.I (iffT s1 s1 cstr_1 cstr_2 cstr_1 cstr_2 x y) + (X : + cupicef.Term (cupicef.type s1) + (iffT s1 s1 cstr_1 cstr_2 cstr_1 cstr_2 x y) => + cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I (iffT s1 s1 cstr_1 cstr_2 cstr_1 cstr_2 y z) + (X0 : + cupicef.Term (cupicef.type s1) + (iffT s1 s1 cstr_1 cstr_2 cstr_1 cstr_2 y z) => + (X1 : + cupicef.Term (cupicef.type s1) + (Coq__Init__Datatypes.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.rule (cupicef.type s1) + (cupicef.type s1)) + cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => z)) + (cupicef.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.rule (cupicef.type s1) + (cupicef.type s1)) + cupicef.I z + (__ : cupicef.Term (cupicef.type s1) z => y))) => + iffT s1 s1 cstr_1 cstr_2 cstr_1 cstr_2 x z) X0)))))) + (Transitive s0 s1 + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s1))) + (cupicef.axiom (cupicef.type s0)) + (cupicef.univ (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.axiom (cupicef.type s1))) cupicef.I) + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + cstr_0 + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I)) + (iffT s1 s1 cstr_1 cstr_2 cstr_1 cstr_2)) + cupicef.I + (x : cupicef.Univ (cupicef.type s1) => + y : cupicef.Univ (cupicef.type s1) => + z : cupicef.Univ (cupicef.type s1) => + X : + cupicef.Term (cupicef.type s1) + (iffT s1 s1 cstr_1 cstr_2 cstr_1 cstr_2 x y) => + X0 : + cupicef.Term (cupicef.type s1) + (iffT s1 s1 cstr_1 cstr_2 cstr_1 cstr_2 y z) => + Coq__Init__Datatypes.prod__rect + (cupicef.cast (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_1 + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => z))) + (cupicef.cast (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_2 + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) cupicef.I z + (__ : cupicef.Term (cupicef.type s1) z => y))) + (cupicef.cast + (cupicef.rule (cupicef.type s1) (cupicef.axiom (cupicef.type s1))) + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s1))) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y (__ : cupicef.Term (cupicef.type s1) y => z)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I z (__ : cupicef.Term (cupicef.type s1) z => y))) + (X1 : + cupicef.Term (cupicef.type s1) + (Coq__Init__Datatypes.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => z)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I z + (__ : cupicef.Term (cupicef.type s1) z => y))) => + cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + (cupicef.prod (cupicef.type s1) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y (__ : cupicef.Term (cupicef.type s1) y => z)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I z (__ : cupicef.Term (cupicef.type s1) z => y))) + (p : + cupicef.Term (cupicef.type s1) + (Coq__Init__Datatypes.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => z)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I z + (__ : cupicef.Term (cupicef.type s1) z => y))) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cstr_3 + (X1 : + cupicef.Term (cupicef.type s1) + (Coq__Init__Datatypes.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => z)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I z + (__ : cupicef.Term (cupicef.type s1) z => y))) => + iffT s1 s1 cstr_1 cstr_2 cstr_1 cstr_2 x z)) + (a : + (__ : cupicef.Term (cupicef.type s1) y -> + cupicef.Term (cupicef.type s1) z) => + b : + (__ : cupicef.Term (cupicef.type s1) z -> + cupicef.Term (cupicef.type s1) y) => + Coq__Init__Datatypes.prod__rect + (cupicef.cast (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_1 + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) cupicef.I + x (__ : cupicef.Term (cupicef.type s1) x => y))) + (cupicef.cast (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_2 + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) cupicef.I + y (__ : cupicef.Term (cupicef.type s1) y => x))) + (cupicef.cast + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type s1))) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I x + (__ : cupicef.Term (cupicef.type s1) x => y)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => x))) + (X1 : + cupicef.Term (cupicef.type s1) + (Coq__Init__Datatypes.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I x + (__ : cupicef.Term (cupicef.type s1) x => y)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => x))) => + cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I)) + (cupicef.prod (cupicef.type s1) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s1) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I x + (__ : cupicef.Term (cupicef.type s1) x => y)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => x))) + (p : + cupicef.Term (cupicef.type s1) + (Coq__Init__Datatypes.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I x + (__ : cupicef.Term (cupicef.type s1) x => y)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => x))) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cstr_3 + (X1 : + cupicef.Term (cupicef.type s1) + (Coq__Init__Datatypes.prod (cupicef.type s1) + (cupicef.type s1) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I x + (__ : cupicef.Term (cupicef.type s1) x => y)) + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I y + (__ : cupicef.Term (cupicef.type s1) y => x))) => + iffT s1 s1 cstr_1 cstr_2 cstr_1 cstr_2 x z)) + (a0 : + (__ : cupicef.Term (cupicef.type s1) x -> + cupicef.Term (cupicef.type s1) y) => + b0 : + (__ : cupicef.Term (cupicef.type s1) y -> + cupicef.Term (cupicef.type s1) x) => + Coq__Init__Datatypes.pair + (cupicef.cast (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_1 + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I x (__ : cupicef.Term (cupicef.type s1) x => z))) + (cupicef.cast (cupicef.axiom (cupicef.type s1)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_2 + (cupicef.prod (cupicef.type s1) (cupicef.type s1) + (cupicef.rule (cupicef.type s1) (cupicef.type s1)) + cupicef.I z (__ : cupicef.Term (cupicef.type s1) z => x))) + (X1 : cupicef.Term (cupicef.type s1) x => + (X2 : cupicef.Term (cupicef.type s1) y => + (X3 : cupicef.Term (cupicef.type s1) x => + (X4 : cupicef.Term (cupicef.type s1) z => X4) (a X2)) + (b0 X2)) + (a0 X1)) + (X1 : cupicef.Term (cupicef.type s1) z => + (X2 : cupicef.Term (cupicef.type s1) y => + (X3 : cupicef.Term (cupicef.type s1) z => + (X4 : cupicef.Term (cupicef.type s1) x => X4) (b0 X2)) + (a X2)) + (b X1))) + X) + X0). + +#TYPE_CLASS_INSTANCE iffT__Transitive. + +def relation__equivalence : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s1)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) -> + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + cupicef.Term + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s2))) + (crelation s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) + (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (crelation s0 s3 A))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s1)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) => + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3)))))) + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s2)))) + (cupicef.prod (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3))))) + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3)))))) + cupicef.I + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (A : cupicef.Univ (cupicef.type s0) => + cupicef.prod + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3)))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3))))) + cupicef.I (crelation s0 s3 A) + (R : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3)))) + cupicef.I (crelation s0 s3 A) + (R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3))) + cupicef.I)))) + (cupicef.prod (cupicef.axiom (cupicef.type s0)) + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s2))) + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s2)))) + cupicef.I + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (A : cupicef.Univ (cupicef.type s0) => + crelation s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) + (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (crelation s0 s3 A)))) + (cupicef.pair + (cupicef.BoolSome (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_3 + (cupicef.BoolSome (cupicef.Cumul (cupicef.type s0) (cupicef.type s2))) + cstr_2 cupicef.BoolNone) + (A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) (cupicef.type s3) + (cupicef.rule (cupicef.type s0) (cupicef.type s3)) cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + iffT s3 s3 cstr_4 cstr_5 cstr_4 cstr_5 (R x y) (R_27_ x y)))). + +def RewriteRelation__instance__2 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s1)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) -> + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + cupicef.Term cupicef.prop + (RewriteRelation s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) + (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (crelation s0 s3 A)) + (relation__equivalence s0 s1 s2 s3 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 + cstr_5 A)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s1)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) => + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + Build__RewriteRelation s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) + (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (crelation s0 s3 A)) + (relation__equivalence s0 s1 s2 s3 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 + cstr_5 A). + +#TYPE_CLASS_INSTANCE RewriteRelation__instance__2. + +def relation__conjunction : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s3)) -> + cstr_3 : + cupicef.eps + (cupicef.Cumul (cupicef.type s2) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) -> + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + cstr_0 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s3)) => + cstr_3 : + cupicef.eps + (cupicef.Cumul (cupicef.type s2) (cupicef.type (cupicef.s cupicef.z))) => + cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))))))) + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))))) + (cupicef.prod (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) (cupicef.type s2))))))) + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))))))) + cupicef.I + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (A : cupicef.Univ (cupicef.type s0) => + cupicef.prod + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) (cupicef.type s2))))))) + cupicef.I (crelation s0 s1 A) + (R : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) (cupicef.type s2))))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))))) + cupicef.I (crelation s0 s2 A) + (R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) (cupicef.type s2))))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))) + cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + cupicef.univ + (cupicef.sup (cupicef.type s1) (cupicef.type s2)) + (cupicef.axiom + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + cupicef.I)))))) + (cupicef.prod (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))))) + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))))) + cupicef.I + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (A : cupicef.Univ (cupicef.type s0) => + cupicef.prod + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))))) + cupicef.I (crelation s0 s1 A) + (R : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + cupicef.I (crelation s0 s2 A) + (R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) => + crelation s0 s3 A)))) + (cupicef.pair + (cupicef.BoolSome (cupicef.Cumul (cupicef.type s2) (cupicef.type s3))) + cstr_2 + (cupicef.BoolSome (cupicef.Cumul (cupicef.type s1) (cupicef.type s3))) + cstr_0 cupicef.BoolNone) + (A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) => + x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + Coq__Init__Datatypes.prod (cupicef.type s1) (cupicef.type s2) ( + R x y) (R_27_ x y)). + +def relation__disjunction : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + cstr_0 : cupicef.eps (cupicef.Cumul cupicef.set (cupicef.type s3)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) -> + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s3)) -> + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s2) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) -> + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + cstr_0 : cupicef.eps (cupicef.Cumul cupicef.set (cupicef.type s3)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s1) (cupicef.type s3)) => + cstr_2 : + cupicef.eps + (cupicef.Cumul (cupicef.type s1) (cupicef.type (cupicef.s cupicef.z))) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s2) (cupicef.type s3)) => + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s2) (cupicef.type (cupicef.s cupicef.z))) => + cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s1) (cupicef.type s2))))))))) + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))))) + (cupicef.prod (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))))))) + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s1) (cupicef.type s2))))))))) + cupicef.I + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (A : cupicef.Univ (cupicef.type s0) => + cupicef.prod + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s1) (cupicef.type s2))))))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))))))) + cupicef.I (crelation s0 s1 A) + (R : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s1) (cupicef.type s2))))))) + cupicef.I (crelation s0 s2 A) + (R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s1) (cupicef.type s2))))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.axiom + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))) + (cupicef.rule (cupicef.type s0) + (cupicef.axiom + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s1) (cupicef.type s2))))) + cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + cupicef.univ + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s1) (cupicef.type s2))) + (cupicef.axiom + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type s1) (cupicef.type s2)))) + cupicef.I)))))) + (cupicef.prod (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))))) + (cupicef.rule (cupicef.axiom (cupicef.type s0)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))))) + cupicef.I + (cupicef.univ (cupicef.type s0) (cupicef.axiom (cupicef.type s0)) + cupicef.I) + (A : cupicef.Univ (cupicef.type s0) => + cupicef.prod + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))))) + cupicef.I (crelation s0 s1 A) + (R : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + cupicef.I (crelation s0 s2 A) + (R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) => + crelation s0 s3 A)))) + (cupicef.pair + (cupicef.BoolSome (cupicef.Cumul (cupicef.type s2) (cupicef.type s3))) + cstr_3 + (cupicef.BoolSome (cupicef.Cumul (cupicef.type s1) (cupicef.type s3))) + cstr_1 cupicef.BoolNone) + (A : cupicef.Univ (cupicef.type s0) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s1))) + (crelation s0 s1 A) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s2))) + (crelation s0 s2 A) => + x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + Coq__Init__Datatypes.sum (cupicef.type s1) (cupicef.type s2) (R x y) + (R_27_ x y)). + +def let__X : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s1)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) -> + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + x : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) -> + y : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) -> + z : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) -> + X : + (x0 : cupicef.Term (cupicef.type s0) A -> + y0 : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (x x0 y0) + (__ : cupicef.Term (cupicef.type s3) (x x0 y0) => y x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => x x0 y0)))) -> + X0 : + (x0 : cupicef.Term (cupicef.type s0) A -> + y0 : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => z x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (z x0 y0) + (__ : cupicef.Term (cupicef.type s3) (z x0 y0) => y x0 y0)))) -> + x0 : cupicef.Term (cupicef.type s0) A -> + y0 : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (x x0 y0) + (__ : cupicef.Term (cupicef.type s3) (x x0 y0) => y x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => x x0 y0))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s1)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) => + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + x : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + y : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + z : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + X : + (x0 : cupicef.Term (cupicef.type s0) A -> + y0 : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (x x0 y0) + (__ : cupicef.Term (cupicef.type s3) (x x0 y0) => y x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => x x0 y0)))) => + X0 : + (x0 : cupicef.Term (cupicef.type s0) A -> + y0 : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => z x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (z x0 y0) + (__ : cupicef.Term (cupicef.type s3) (z x0 y0) => y x0 y0)))) => + x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + X x0 y0. + +def let__X0 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s1)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) -> + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + x : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) -> + y : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) -> + z : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) -> + X : + (x0 : cupicef.Term (cupicef.type s0) A -> + y0 : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (x x0 y0) + (__ : cupicef.Term (cupicef.type s3) (x x0 y0) => y x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => x x0 y0)))) -> + X0 : + (x0 : cupicef.Term (cupicef.type s0) A -> + y0 : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => z x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (z x0 y0) + (__ : cupicef.Term (cupicef.type s3) (z x0 y0) => y x0 y0)))) -> + x0 : cupicef.Term (cupicef.type s0) A -> + y0 : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => z x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (z x0 y0) + (__ : cupicef.Term (cupicef.type s3) (z x0 y0) => y x0 y0))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s1)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) => + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + x : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + y : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + z : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + X : + (x0 : cupicef.Term (cupicef.type s0) A -> + y0 : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (x x0 y0) + (__ : cupicef.Term (cupicef.type s3) (x x0 y0) => y x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => x x0 y0)))) => + X0 : + (x0 : cupicef.Term (cupicef.type s0) A -> + y0 : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => z x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) cupicef.I + (z x0 y0) + (__ : cupicef.Term (cupicef.type s3) (z x0 y0) => y x0 y0)))) => + x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + X0 x0 y0. + +thm relation__equivalence__equivalence : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s1)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) -> + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s2)) + (Equivalence s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) + (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (crelation s0 s3 A)) + (relation__equivalence s0 s1 s2 s3 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 + cstr_5 A)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s1)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) => + cstr_4 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + Build__Equivalence s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) + (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (crelation s0 s3 A)) + (relation__equivalence s0 s1 s2 s3 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 + cstr_5 A) + (cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (cupicef.type s2)) + (cupicef.sup (cupicef.type s1) (cupicef.type s2)) + (cupicef.prod + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (cupicef.type s2) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.type s2)) + cupicef.I (crelation s0 s3 A) + (x : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + relation__equivalence s0 s1 s2 s3 cstr_0 cstr_1 cstr_2 cstr_3 + cstr_4 cstr_5 A x x)) + (Reflexive s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) + (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (crelation s0 s3 A)) + (relation__equivalence s0 s1 s2 s3 cstr_0 cstr_1 cstr_2 cstr_3 + cstr_4 cstr_5 A)) + cupicef.I + (cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3)))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.type s2)) + (cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3)))) + cupicef.I (crelation s0 s3 A) + (x : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) (cupicef.type s3) + (cupicef.rule (cupicef.type s0) (cupicef.type s3)) + cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (x x0 y) + (__ : cupicef.Term (cupicef.type s3) (x x0 y) => + x x0 y)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (x x0 y) + (__ : cupicef.Term (cupicef.type s3) (x x0 y) => + x x0 y)))))) + (cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.type s2) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.type s2)) + cupicef.I (crelation s0 s3 A) + (x : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + relation__equivalence s0 s1 s2 s3 cstr_0 cstr_1 cstr_2 cstr_3 + cstr_4 cstr_5 A x x)) + cupicef.I + (x : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + x0 : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + Coq__Init__Datatypes.pair + (cupicef.cast (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_4 + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (x x0 y) + (__ : cupicef.Term (cupicef.type s3) (x x0 y) => x x0 y))) + (cupicef.cast (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_5 + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (x x0 y) + (__ : cupicef.Term (cupicef.type s3) (x x0 y) => x x0 y))) + (X : cupicef.Term (cupicef.type s3) (x x0 y) => X) + (X : cupicef.Term (cupicef.type s3) (x x0 y) => X)))) + (cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)))) + (cupicef.sup (cupicef.type s1) (cupicef.type s2)) + (cupicef.prod + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s2) (cupicef.type s2))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)))) + cupicef.I (crelation s0 s3 A) + (x : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s2) (cupicef.type s2))) + cupicef.I (crelation s0 s3 A) + (y : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.prod (cupicef.type s2) (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)) cupicef.I + (relation__equivalence s0 s1 s2 s3 cstr_0 cstr_1 cstr_2 + cstr_3 cstr_4 cstr_5 A x y) + (__ : + cupicef.Term (cupicef.type s2) + (relation__equivalence s0 s1 s2 s3 cstr_0 cstr_1 cstr_2 + cstr_3 cstr_4 cstr_5 A x y) => + relation__equivalence s0 s1 s2 s3 cstr_0 cstr_1 cstr_2 + cstr_3 cstr_4 cstr_5 A y x)))) + (Symmetric s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) + (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (crelation s0 s3 A)) + (relation__equivalence s0 s1 s2 s3 cstr_0 cstr_1 cstr_2 cstr_3 + cstr_4 cstr_5 A)) + cupicef.I + (cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3)))))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)))) + (cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3))))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3)))))) + cupicef.I (crelation s0 s3 A) + (x : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3)))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3))))) + cupicef.I (crelation s0 s3 A) + (y : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.prod + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3)))) + cupicef.I + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) (cupicef.type s3) + (cupicef.rule (cupicef.type s0) (cupicef.type s3)) + cupicef.I A + (y0 : cupicef.Term (cupicef.type s0) A => + Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : + cupicef.Term (cupicef.type s3) (x x0 y0) => + y x0 y0)) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : + cupicef.Term (cupicef.type s3) (y x0 y0) => + x x0 y0))))) + (X : + (x0 : cupicef.Term (cupicef.type s0) A -> + y0 : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : + cupicef.Term (cupicef.type s3) (x x0 y0) => + y x0 y0)) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : + cupicef.Term (cupicef.type s3) (y x0 y0) => + x x0 y0)))) => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) (cupicef.type s3) + (cupicef.rule (cupicef.type s0) (cupicef.type s3)) + cupicef.I A + (y0 : cupicef.Term (cupicef.type s0) A => + Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : + cupicef.Term (cupicef.type s3) (y x0 y0) => + x x0 y0)) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : + cupicef.Term (cupicef.type s3) (x x0 y0) => + y x0 y0)))))))) + (cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s2) (cupicef.type s2))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)))) + cupicef.I (crelation s0 s3 A) + (x : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s2) (cupicef.type s2))) + cupicef.I (crelation s0 s3 A) + (y : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.prod (cupicef.type s2) (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)) + cupicef.I + (relation__equivalence s0 s1 s2 s3 cstr_0 cstr_1 cstr_2 + cstr_3 cstr_4 cstr_5 A x y) + (__ : + cupicef.Term (cupicef.type s2) + (relation__equivalence s0 s1 s2 s3 cstr_0 cstr_1 + cstr_2 cstr_3 cstr_4 cstr_5 A x y) => + relation__equivalence s0 s1 s2 s3 cstr_0 cstr_1 cstr_2 + cstr_3 cstr_4 cstr_5 A y x)))) + cupicef.I + (x : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + y : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + X : + (x0 : cupicef.Term (cupicef.type s0) A -> + y0 : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : cupicef.Term (cupicef.type s3) (x x0 y0) => + y x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => + x x0 y0)))) => + x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + Coq__Init__Datatypes.pair + (cupicef.cast (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_4 + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => x x0 y0))) + (cupicef.cast (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_5 + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : cupicef.Term (cupicef.type s3) (x x0 y0) => y x0 y0))) + (X0 : cupicef.Term (cupicef.type s3) (y x0 y0) => + (X1 : + (y1 : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (x x0 y1) + (__ : cupicef.Term (cupicef.type s3) (x x0 y1) => + y x0 y1)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (y x0 y1) + (__ : cupicef.Term (cupicef.type s3) (y x0 y1) => + x x0 y1)))) => + (X2 : + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : cupicef.Term (cupicef.type s3) (x x0 y0) => + y x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => + x x0 y0))) => + Coq__Init__Datatypes.prod__rect + (cupicef.cast (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_4 + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : cupicef.Term (cupicef.type s3) (x x0 y0) => + y x0 y0))) + (cupicef.cast (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_5 + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => + x x0 y0))) + (cupicef.cast + (cupicef.rule (cupicef.type s3) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s3) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + (cupicef.rule (cupicef.type s3) + (cupicef.axiom (cupicef.type s3))) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : cupicef.Term (cupicef.type s3) (x x0 y0) => + y x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => + x x0 y0))) + (X3 : + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : + cupicef.Term (cupicef.type s3) + (x x0 y0) => + y x0 y0)) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : + cupicef.Term (cupicef.type s3) + (y x0 y0) => + x x0 y0))) => + cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + (cupicef.prod (cupicef.type s3) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s3) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : cupicef.Term (cupicef.type s3) (x x0 y0) => + y x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => + x x0 y0))) + (p : + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : + cupicef.Term (cupicef.type s3) + (x x0 y0) => + y x0 y0)) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : + cupicef.Term (cupicef.type s3) + (y x0 y0) => + x x0 y0))) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cstr_6 + (X3 : + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : + cupicef.Term (cupicef.type s3) (x x0 y0) => + y x0 y0)) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : + cupicef.Term (cupicef.type s3) (y x0 y0) => + x x0 y0))) => + x x0 y0)) + (a : + (__ : cupicef.Term (cupicef.type s3) (x x0 y0) -> + cupicef.Term (cupicef.type s3) (y x0 y0)) => + b : + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) -> + cupicef.Term (cupicef.type s3) (x x0 y0)) => + (X3 : cupicef.Term (cupicef.type s3) (x x0 y0) => X3) + (b X0)) + X2) + (X1 y0)) + (X x0)) + (X0 : cupicef.Term (cupicef.type s3) (x x0 y0) => + (X1 : + (y1 : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (x x0 y1) + (__ : cupicef.Term (cupicef.type s3) (x x0 y1) => + y x0 y1)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (y x0 y1) + (__ : cupicef.Term (cupicef.type s3) (y x0 y1) => + x x0 y1)))) => + (X2 : + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : cupicef.Term (cupicef.type s3) (x x0 y0) => + y x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => + x x0 y0))) => + Coq__Init__Datatypes.prod__rect + (cupicef.cast (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_4 + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : cupicef.Term (cupicef.type s3) (x x0 y0) => + y x0 y0))) + (cupicef.cast (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_5 + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => + x x0 y0))) + (cupicef.cast + (cupicef.rule (cupicef.type s3) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s3) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + (cupicef.rule (cupicef.type s3) + (cupicef.axiom (cupicef.type s3))) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : cupicef.Term (cupicef.type s3) (x x0 y0) => + y x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => + x x0 y0))) + (X3 : + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : + cupicef.Term (cupicef.type s3) + (x x0 y0) => + y x0 y0)) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : + cupicef.Term (cupicef.type s3) + (y x0 y0) => + x x0 y0))) => + cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + (cupicef.prod (cupicef.type s3) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s3) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : cupicef.Term (cupicef.type s3) (x x0 y0) => + y x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => + x x0 y0))) + (p : + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : + cupicef.Term (cupicef.type s3) + (x x0 y0) => + y x0 y0)) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : + cupicef.Term (cupicef.type s3) + (y x0 y0) => + x x0 y0))) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cstr_6 + (X3 : + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : + cupicef.Term (cupicef.type s3) (x x0 y0) => + y x0 y0)) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : + cupicef.Term (cupicef.type s3) (y x0 y0) => + x x0 y0))) => + y x0 y0)) + (a : + (__ : cupicef.Term (cupicef.type s3) (x x0 y0) -> + cupicef.Term (cupicef.type s3) (y x0 y0)) => + b : + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) -> + cupicef.Term (cupicef.type s3) (x x0 y0)) => + (X3 : cupicef.Term (cupicef.type s3) (y x0 y0) => X3) + (a X0)) + X2) + (X1 y0)) + (X x0))))) + (cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)))))) + (cupicef.sup (cupicef.type s1) (cupicef.type s2)) + (cupicef.prod + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2))))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)))))) + cupicef.I (crelation s0 s3 A) + (x : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2))))) + cupicef.I (crelation s0 s3 A) + (y : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)))) + cupicef.I (crelation s0 s3 A) + (z : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.prod (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2))) + cupicef.I + (relation__equivalence s0 s1 s2 s3 cstr_0 cstr_1 cstr_2 + cstr_3 cstr_4 cstr_5 A x y) + (__ : + cupicef.Term (cupicef.type s2) + (relation__equivalence s0 s1 s2 s3 cstr_0 cstr_1 + cstr_2 cstr_3 cstr_4 cstr_5 A x y) => + cupicef.prod (cupicef.type s2) (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)) + cupicef.I + (relation__equivalence s0 s1 s2 s3 cstr_0 cstr_1 + cstr_2 cstr_3 cstr_4 cstr_5 A y z) + (__0 : + cupicef.Term (cupicef.type s2) + (relation__equivalence s0 s1 s2 s3 cstr_0 cstr_1 + cstr_2 cstr_3 cstr_4 cstr_5 A y z) => + relation__equivalence s0 s1 s2 s3 cstr_0 cstr_1 + cstr_2 cstr_3 cstr_4 cstr_5 A x z)))))) + (Transitive s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) + (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (crelation s0 s3 A)) + (relation__equivalence s0 s1 s2 s3 cstr_0 cstr_1 cstr_2 cstr_3 + cstr_4 cstr_5 A)) + cupicef.I + (cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3)))))))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)))))) + (cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3))))))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3)))))))) + cupicef.I (crelation s0 s3 A) + (x : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3)))))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3))))))) + cupicef.I (crelation s0 s3 A) + (y : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3))))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3)))))) + cupicef.I (crelation s0 s3 A) + (z : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.prod + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3)))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3))))) + cupicef.I + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s3)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) (cupicef.type s3) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3)) + cupicef.I A + (y0 : cupicef.Term (cupicef.type s0) A => + Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : + cupicef.Term (cupicef.type s3) + (x x0 y0) => + y x0 y0)) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : + cupicef.Term (cupicef.type s3) + (y x0 y0) => + x x0 y0))))) + (X : + (x0 : cupicef.Term (cupicef.type s0) A -> + y0 : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : + cupicef.Term (cupicef.type s3) + (x x0 y0) => + y x0 y0)) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : + cupicef.Term (cupicef.type s3) + (y x0 y0) => + x x0 y0)))) => + cupicef.prod + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3))) + (cupicef.rule + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3)))) + cupicef.I + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type s3) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3)) + cupicef.I A + (y0 : cupicef.Term (cupicef.type s0) A => + Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : + cupicef.Term (cupicef.type s3) + (y x0 y0) => + z x0 y0)) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (z x0 y0) + (__ : + cupicef.Term (cupicef.type s3) + (z x0 y0) => + y x0 y0))))) + (X0 : + (x0 : cupicef.Term (cupicef.type s0) A -> + y0 : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : + cupicef.Term (cupicef.type s3) + (y x0 y0) => + z x0 y0)) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (z x0 y0) + (__ : + cupicef.Term (cupicef.type s3) + (z x0 y0) => + y x0 y0)))) => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.type s3) + (cupicef.rule (cupicef.type s0) + (cupicef.type s3)) + cupicef.I A + (y0 : cupicef.Term (cupicef.type s0) A => + (X1 : + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod + (cupicef.type s3) (cupicef.type s3) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : + cupicef.Term (cupicef.type s3) + (y x0 y0) => + z x0 y0)) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (z x0 y0) + (__ : + cupicef.Term (cupicef.type s3) + (z x0 y0) => + y x0 y0))) => + Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : + cupicef.Term (cupicef.type s3) + (x x0 y0) => + z x0 y0)) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (z x0 y0) + (__ : + cupicef.Term (cupicef.type s3) + (z x0 y0) => + x x0 y0))) + (X0 x0 y0))))))))) + (cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2))))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)))))) + cupicef.I (crelation s0 s3 A) + (x : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2))))) + cupicef.I (crelation s0 s3 A) + (y : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)))) + cupicef.I (crelation s0 s3 A) + (z : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.prod (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)) + (cupicef.rule (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2))) + cupicef.I + (relation__equivalence s0 s1 s2 s3 cstr_0 cstr_1 + cstr_2 cstr_3 cstr_4 cstr_5 A x y) + (__ : + cupicef.Term (cupicef.type s2) + (relation__equivalence s0 s1 s2 s3 cstr_0 cstr_1 + cstr_2 cstr_3 cstr_4 cstr_5 A x y) => + cupicef.prod (cupicef.type s2) (cupicef.type s2) + (cupicef.rule (cupicef.type s2) (cupicef.type s2)) + cupicef.I + (relation__equivalence s0 s1 s2 s3 cstr_0 cstr_1 + cstr_2 cstr_3 cstr_4 cstr_5 A y z) + (__0 : + cupicef.Term (cupicef.type s2) + (relation__equivalence s0 s1 s2 s3 cstr_0 + cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 A y z) => + relation__equivalence s0 s1 s2 s3 cstr_0 cstr_1 + cstr_2 cstr_3 cstr_4 cstr_5 A x z)))))) + cupicef.I + (x : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + y : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + z : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + X : + (x0 : cupicef.Term (cupicef.type s0) A -> + y0 : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : cupicef.Term (cupicef.type s3) (x x0 y0) => + y x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => + x x0 y0)))) => + X0 : + (x0 : cupicef.Term (cupicef.type s0) A -> + y0 : cupicef.Term (cupicef.type s0) A -> + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => + z x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (z x0 y0) + (__ : cupicef.Term (cupicef.type s3) (z x0 y0) => + y x0 y0)))) => + x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + Coq__Init__Datatypes.prod__rect + (cupicef.cast (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_4 + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => z x0 y0))) + (cupicef.cast (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_5 + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (z x0 y0) + (__ : cupicef.Term (cupicef.type s3) (z x0 y0) => y x0 y0))) + (cupicef.cast + (cupicef.rule (cupicef.type s3) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s3) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + (cupicef.rule (cupicef.type s3) + (cupicef.axiom (cupicef.type s3))) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => + z x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (z x0 y0) + (__ : cupicef.Term (cupicef.type s3) (z x0 y0) => + y x0 y0))) + (X1 : + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => + z x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (z x0 y0) + (__ : cupicef.Term (cupicef.type s3) (z x0 y0) => + y x0 y0))) => + cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + (cupicef.prod (cupicef.type s3) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s3) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => + z x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (z x0 y0) + (__ : cupicef.Term (cupicef.type s3) (z x0 y0) => + y x0 y0))) + (p : + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => + z x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (z x0 y0) + (__ : cupicef.Term (cupicef.type s3) (z x0 y0) => + y x0 y0))) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cstr_6 + (X1 : + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => + z x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (z x0 y0) + (__ : cupicef.Term (cupicef.type s3) (z x0 y0) => + y x0 y0))) => + Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : cupicef.Term (cupicef.type s3) (x x0 y0) => + z x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (z x0 y0) + (__ : cupicef.Term (cupicef.type s3) (z x0 y0) => + x x0 y0)))) + (a : + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) -> + cupicef.Term (cupicef.type s3) (z x0 y0)) => + b : + (__ : cupicef.Term (cupicef.type s3) (z x0 y0) -> + cupicef.Term (cupicef.type s3) (y x0 y0)) => + Coq__Init__Datatypes.prod__rect + (cupicef.cast (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_4 + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : cupicef.Term (cupicef.type s3) (x x0 y0) => + y x0 y0))) + (cupicef.cast (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_5 + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => + x x0 y0))) + (cupicef.cast + (cupicef.rule (cupicef.type s3) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule (cupicef.type s3) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + (cupicef.rule (cupicef.type s3) + (cupicef.axiom (cupicef.type s3))) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : cupicef.Term (cupicef.type s3) (x x0 y0) => + y x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => + x x0 y0))) + (X1 : + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : + cupicef.Term (cupicef.type s3) (x x0 y0) => + y x0 y0)) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : + cupicef.Term (cupicef.type s3) (y x0 y0) => + x x0 y0))) => + cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + (cupicef.prod (cupicef.type s3) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type s3) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : cupicef.Term (cupicef.type s3) (x x0 y0) => + y x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => + x x0 y0))) + (p : + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : + cupicef.Term (cupicef.type s3) (x x0 y0) => + y x0 y0)) + (cupicef.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : + cupicef.Term (cupicef.type s3) (y x0 y0) => + x x0 y0))) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cstr_6 + (X1 : + cupicef.Term (cupicef.type s3) + (Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : cupicef.Term (cupicef.type s3) (x x0 y0) => + y x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) + (cupicef.type s3)) + cupicef.I (y x0 y0) + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) => + x x0 y0))) => + Coq__Init__Datatypes.prod (cupicef.type s3) + (cupicef.type s3) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : cupicef.Term (cupicef.type s3) (x x0 y0) => + z x0 y0)) + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (z x0 y0) + (__ : cupicef.Term (cupicef.type s3) (z x0 y0) => + x x0 y0)))) + (a0 : + (__ : cupicef.Term (cupicef.type s3) (x x0 y0) -> + cupicef.Term (cupicef.type s3) (y x0 y0)) => + b0 : + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) -> + cupicef.Term (cupicef.type s3) (x x0 y0)) => + Coq__Init__Datatypes.pair + (cupicef.cast (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_4 + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (x x0 y0) + (__ : cupicef.Term (cupicef.type s3) (x x0 y0) => + z x0 y0))) + (cupicef.cast (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_5 + (cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (z x0 y0) + (__ : cupicef.Term (cupicef.type s3) (z x0 y0) => + x x0 y0))) + (X1 : cupicef.Term (cupicef.type s3) (x x0 y0) => + (X2 : cupicef.Term (cupicef.type s3) (y x0 y0) => + (X3 : cupicef.Term (cupicef.type s3) (x x0 y0) => + (X4 : cupicef.Term (cupicef.type s3) (z x0 y0) => X4) + (a X2)) + (b0 X2)) + (a0 X1)) + (X1 : cupicef.Term (cupicef.type s3) (z x0 y0) => + (X2 : cupicef.Term (cupicef.type s3) (y x0 y0) => + (X3 : cupicef.Term (cupicef.type s3) (z x0 y0) => + (X4 : cupicef.Term (cupicef.type s3) (x x0 y0) => X4) + (b0 X2)) + (a X2)) + (b X1))) + (let__X s0 s1 s2 s3 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 + cstr_5 cstr_6 A x y z X X0 x0 y0)) + (let__X0 s0 s1 s2 s3 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 + cstr_6 A x y z X X0 x0 y0)))). + +#TYPE_CLASS_INSTANCE relation__equivalence__equivalence. + +thm relation__implication__preorder : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s1)) -> + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) -> + A : cupicef.Univ (cupicef.type s0) -> + cupicef.Term (cupicef.sup (cupicef.type s1) (cupicef.type s2)) + (PreOrder s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) + (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (crelation s0 s3 A)) + (cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3))))) + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s2))) + (cupicef.prod + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3)))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3))))) + cupicef.I (crelation s0 s3 A) + (R : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3)))) + cupicef.I (crelation s0 s3 A) + (R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3))) + cupicef.I))) + (crelation s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) + (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (crelation s0 s3 A))) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_3 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s2))) + cstr_2 cupicef.BoolNone) + (subrelation s0 s3 s3 A))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s1)) => + cstr_1 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) => + A : cupicef.Univ (cupicef.type s0) => + Build__PreOrder s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) (cupicef.axiom (cupicef.type s1)) + cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) + (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (crelation s0 s3 A)) + (cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3))))) + (cupicef.sup (cupicef.type s1) (cupicef.axiom (cupicef.type s2))) + (cupicef.prod + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3)))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3))))) + cupicef.I (crelation s0 s3 A) + (R : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3)))) + cupicef.I (crelation s0 s3 A) + (R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.univ (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3))) + cupicef.I))) + (crelation s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) + (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (crelation s0 s3 A))) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_3 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s2))) + cstr_2 cupicef.BoolNone) + (subrelation s0 s3 s3 A)) + (cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (cupicef.sup (cupicef.type s0) (cupicef.type s3))) + (cupicef.sup (cupicef.type s1) (cupicef.type s2)) + (cupicef.prod + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.sup (cupicef.type s0) (cupicef.type s3))) + cupicef.I (crelation s0 s3 A) + (x : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + subrelation s0 s3 s3 A x x)) + (Reflexive s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) + (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (crelation s0 s3 A)) + (cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3))))) + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2))) + (cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3)))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3))))) + cupicef.I (crelation s0 s3 A) + (R : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3)))) + cupicef.I (crelation s0 s3 A) + (R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3))) + cupicef.I))) + (crelation s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) + (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (crelation s0 s3 A))) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_3 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s2))) + cstr_2 cupicef.BoolNone) + (subrelation s0 s3 s3 A))) + cupicef.I + (x : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)))) + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s3) (cupicef.type s3))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s3) (cupicef.type s3))) + cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (x x0 y) + (X : cupicef.Term (cupicef.type s3) (x x0 y) => x x0 y)))) + (subrelation s0 s3 s3 A x x) cupicef.I + (x0 : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + X : cupicef.Term (cupicef.type s3) (x x0 y) => + X))) + (cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.sup (cupicef.type s0) (cupicef.type s3))))))) + (cupicef.sup (cupicef.type s1) (cupicef.type s2)) + (cupicef.prod + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.sup (cupicef.type s0) (cupicef.type s3)))))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.sup (cupicef.type s0) (cupicef.type s3))))))) + cupicef.I (crelation s0 s3 A) + (x : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.sup (cupicef.type s0) (cupicef.type s3))))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.sup (cupicef.type s0) (cupicef.type s3)))))) + cupicef.I (crelation s0 s3 A) + (y : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.sup (cupicef.type s0) (cupicef.type s3)))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.sup (cupicef.type s0) (cupicef.type s3))))) + cupicef.I (crelation s0 s3 A) + (z : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.prod + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.sup (cupicef.type s0) (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.sup (cupicef.type s0) (cupicef.type s3)))) + cupicef.I (subrelation s0 s3 s3 A x y) + (X : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (subrelation s0 s3 s3 A x y) => + cupicef.prod + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.rule + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.sup (cupicef.type s0) (cupicef.type s3))) + cupicef.I (subrelation s0 s3 s3 A y z) + (X0 : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (subrelation s0 s3 s3 A y z) => + subrelation s0 s3 s3 A x z)))))) + (Transitive s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) + (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (crelation s0 s3 A)) + (cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3))))) + (cupicef.sup (cupicef.type s1) + (cupicef.axiom (cupicef.type s2))) + (cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3)))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3))))) + cupicef.I (crelation s0 s3 A) + (R : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.prod + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3))) + (cupicef.rule + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3)))) + cupicef.I (crelation s0 s3 A) + (R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + cupicef.univ + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) (cupicef.type s3))) + cupicef.I))) + (crelation s1 s2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + (cupicef.axiom (cupicef.type s1)) + (cupicef.univ + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom + (cupicef.sup (cupicef.type s0) + (cupicef.axiom (cupicef.type s3)))) + cupicef.I) + (cupicef.univ (cupicef.type s1) + (cupicef.axiom (cupicef.type s1)) cupicef.I) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s1))) + cstr_1 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) + (cupicef.type s1))) + cstr_0 cupicef.BoolNone) + (crelation s0 s3 A))) + (cupicef.pair + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s3) (cupicef.type s2))) + cstr_3 + (cupicef.BoolSome + (cupicef.Cumul (cupicef.type s0) (cupicef.type s2))) + cstr_2 cupicef.BoolNone) + (subrelation s0 s3 s3 A))) + cupicef.I + (x : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + y : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + z : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + X : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (subrelation s0 s3 s3 A x y) => + X0 : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (subrelation s0 s3 s3 A y z) => + cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)))) + (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s3) (cupicef.type s3))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s3) (cupicef.type s3))) + cupicef.I A + (y0 : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s3) (cupicef.type s3) + (cupicef.rule (cupicef.type s3) (cupicef.type s3)) + cupicef.I (x x0 y0) + (X1 : cupicef.Term (cupicef.type s3) (x x0 y0) => z x0 y0)))) + (subrelation s0 s3 s3 A x z) cupicef.I + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + X1 : cupicef.Term (cupicef.type s3) (x x0 y0) => + (X2 : + (y1 : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s3) (y x0 y1) -> + cupicef.Term (cupicef.type s3) (z x0 y1)) => + (X3 : + (__ : cupicef.Term (cupicef.type s3) (y x0 y0) -> + cupicef.Term (cupicef.type s3) (z x0 y0)) => + (X4 : + (y1 : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s3) (x x0 y1) -> + cupicef.Term (cupicef.type s3) (y x0 y1)) => + X0 x0 y0 (X x0 y0 X1)) (X x0)) (X2 y0)) + (X0 x0)))). + +#TYPE_CLASS_INSTANCE relation__implication__preorder. + +def PartialOrder : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s1)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s4)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) -> + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) -> + equ : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (Equivalence s0 s3 A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) -> + preo : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (PreOrder s0 s3 A R) -> + cupicef.Univ (cupicef.type s2) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s1)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s4)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) => + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + equ : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (Equivalence s0 s3 A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + preo : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (PreOrder s0 s3 A R) => + relation__equivalence s0 s1 s2 s3 cstr_0 cstr_2 cstr_3 cstr_4 cstr_5 + cstr_6 A eqA + (relation__conjunction s0 s3 s3 s3 cupicef.I cstr_5 cupicef.I cstr_6 A R + (flip s0 s0 s4 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R)). + +#TYPE_CLASS PartialOrder. + +def partial__order__equivalence : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s1)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s4)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) -> + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) -> + equ : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (Equivalence s0 s3 A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) -> + preo : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (PreOrder s0 s3 A R) -> + PartialOrder0 : + cupicef.Term (cupicef.type s2) + (PartialOrder s0 s1 s2 s3 s4 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 + cstr_6 A eqA equ R preo) -> + cupicef.Term (cupicef.type s2) + (relation__equivalence s0 s1 s2 s3 cstr_0 cstr_2 cstr_3 cstr_4 cstr_5 + cstr_6 A eqA + (relation__conjunction s0 s3 s3 s3 cupicef.I cstr_5 cupicef.I cstr_6 A + R + (flip s0 s0 s4 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + (cupicef.axiom (cupicef.type s4)) + (cupicef.univ (cupicef.axiom (cupicef.type s3)) + (cupicef.axiom (cupicef.axiom (cupicef.type s3))) + cupicef.I) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I)) + R))) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s1)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s3)) (cupicef.type s4)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s3) (cupicef.type s2)) => + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s3) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + equ : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (Equivalence s0 s3 A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s3))) + (crelation s0 s3 A) => + preo : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s3)) + (PreOrder s0 s3 A R) => + PartialOrder0 : + cupicef.Term (cupicef.type s2) + (PartialOrder s0 s1 s2 s3 s4 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 + cstr_6 A eqA equ R preo) => + PartialOrder0. + +def let__X1 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s1)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) -> + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (crelation s0 s4 A) -> + equ : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Equivalence s0 s4 A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (crelation s0 s4 A) -> + preo : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (PreOrder s0 s4 A R) -> + H : + cupicef.Term (cupicef.type s2) + (PartialOrder s0 s1 s2 s4 s3 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 + cstr_6 A eqA equ R preo) -> + x : cupicef.Term (cupicef.type s0) A -> + y : cupicef.Term (cupicef.type s0) A -> + X : cupicef.Term (cupicef.type s4) (R x y) -> + X0 : cupicef.Term (cupicef.type s4) (R y x) -> + x0 : cupicef.Term (cupicef.type s0) A -> + y0 : cupicef.Term (cupicef.type s0) A -> + __ : + cupicef.Term (cupicef.type s4) + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I cstr_6 A + R + (flip s0 s0 s3 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0) -> + cupicef.Term (cupicef.type s4) (eqA x0 y0) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s1)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) => + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (crelation s0 s4 A) => + equ : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Equivalence s0 s4 A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (crelation s0 s4 A) => + preo : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (PreOrder s0 s4 A R) => + H : + cupicef.Term (cupicef.type s2) + (PartialOrder s0 s1 s2 s4 s3 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 + cstr_6 A eqA equ R preo) => + x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + X : cupicef.Term (cupicef.type s4) (R x y) => + X0 : cupicef.Term (cupicef.type s4) (R y x) => + x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + Coq__Init__Datatypes.match____prod (cupicef.type s4) + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) (cupicef.axiom (cupicef.type s4)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_5 + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA x0 y0) + (__ : cupicef.Term (cupicef.type s4) (eqA x0 y0) => + relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0))) + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) (cupicef.axiom (cupicef.type s4)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_6 + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0) + (__ : + cupicef.Term (cupicef.type s4) + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0) => + eqA x0 y0))) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA x0 y0) + (__ : cupicef.Term (cupicef.type s4) (eqA x0 y0) => + relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0)) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0) + (__ : + cupicef.Term (cupicef.type s4) + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 + cupicef.I cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0) => + eqA x0 y0))) => + cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I cstr_6 + A R + (flip s0 s0 s3 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0) + (__0 : + cupicef.Term (cupicef.type s4) + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0) => + eqA x0 y0)) + (__ : + (__ : cupicef.Term (cupicef.type s4) (eqA x0 y0) -> + cupicef.Term (cupicef.type s4) + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0)) => + __0 : + (__0 : + cupicef.Term (cupicef.type s4) + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0) -> + cupicef.Term (cupicef.type s4) (eqA x0 y0)) => + __0) + (H x0 y0). + +thm partial__order__antisym : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s1)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) -> + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (crelation s0 s4 A) -> + equ : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Equivalence s0 s4 A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (crelation s0 s4 A) -> + preo : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (PreOrder s0 s4 A R) -> + H : + cupicef.Term (cupicef.type s2) + (PartialOrder s0 s1 s2 s4 s3 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 + cstr_6 A eqA equ R preo) -> + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Antisymmetric s0 s4 s4 A eqA equ R) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s1)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) => + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (crelation s0 s4 A) => + equ : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Equivalence s0 s4 A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (crelation s0 s4 A) => + preo : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (PreOrder s0 s4 A R) => + H : + cupicef.Term (cupicef.type s2) + (PartialOrder s0 s1 s2 s4 s3 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 + cstr_6 A eqA equ R preo) => + cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4))))) + (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4))))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4))) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)))) + cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) + (cupicef.rule (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4))) + cupicef.I (R x y) + (X : cupicef.Term (cupicef.type s4) (R x y) => + cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (R y x) + (X0 : cupicef.Term (cupicef.type s4) (R y x) => eqA x y))))) + (Antisymmetric s0 s4 s4 A eqA equ R) cupicef.I + (x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + X : cupicef.Term (cupicef.type s4) (R x y) => + X0 : cupicef.Term (cupicef.type s4) (R y x) => + let__X1 s0 s1 s2 s3 s4 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 cstr_6 + A eqA equ R preo H x y X X0 x y + (Coq__Init__Datatypes.pair + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_5 (R x y)) + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_6 + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R x y)) + X X0)). + +#TYPE_CLASS_INSTANCE partial__order__antisym. + +def let__X2 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s1)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) -> + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (crelation s0 s4 A) -> + equ : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Equivalence s0 s4 A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (crelation s0 s4 A) -> + preo : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (PreOrder s0 s4 A R) -> + H : + cupicef.Term (cupicef.type s2) + (PartialOrder s0 s1 s2 s4 s3 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 + cstr_6 A eqA equ R preo) -> + x : cupicef.Term (cupicef.type s0) A -> + y : cupicef.Term (cupicef.type s0) A -> + X : cupicef.Term (cupicef.type s4) (eqA x y) -> + x0 : cupicef.Term (cupicef.type s0) A -> + y0 : cupicef.Term (cupicef.type s0) A -> + __ : cupicef.Term (cupicef.type s4) (eqA x0 y0) -> + cupicef.Term (cupicef.type s4) + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s1)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) => + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (crelation s0 s4 A) => + equ : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Equivalence s0 s4 A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (crelation s0 s4 A) => + preo : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (PreOrder s0 s4 A R) => + H : + cupicef.Term (cupicef.type s2) + (PartialOrder s0 s1 s2 s4 s3 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 + cstr_6 A eqA equ R preo) => + x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + X : cupicef.Term (cupicef.type s4) (eqA x y) => + x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + Coq__Init__Datatypes.match____prod (cupicef.type s4) + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) (cupicef.axiom (cupicef.type s4)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_5 + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA x0 y0) + (__ : cupicef.Term (cupicef.type s4) (eqA x0 y0) => + relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0))) + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) (cupicef.axiom (cupicef.type s4)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_6 + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0) + (__ : + cupicef.Term (cupicef.type s4) + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0) => + eqA x0 y0))) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA x0 y0) + (__ : cupicef.Term (cupicef.type s4) (eqA x0 y0) => + relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0)) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0) + (__ : + cupicef.Term (cupicef.type s4) + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 + cupicef.I cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0) => + eqA x0 y0))) => + cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA x0 y0) + (__0 : cupicef.Term (cupicef.type s4) (eqA x0 y0) => + relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I cstr_6 + A R + (flip s0 s0 s3 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0)) + (__ : + (__ : cupicef.Term (cupicef.type s4) (eqA x0 y0) -> + cupicef.Term (cupicef.type s4) + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0)) => + __0 : + (__0 : + cupicef.Term (cupicef.type s4) + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0) -> + cupicef.Term (cupicef.type s4) (eqA x0 y0)) => + __) + (H x0 y0). + +def let__X3 : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s1)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) -> + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (crelation s0 s4 A) -> + equ : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Equivalence s0 s4 A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (crelation s0 s4 A) -> + preo : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (PreOrder s0 s4 A R) -> + H : + cupicef.Term (cupicef.type s2) + (PartialOrder s0 s1 s2 s4 s3 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 + cstr_6 A eqA equ R preo) -> + x : cupicef.Term (cupicef.type s0) A -> + y : cupicef.Term (cupicef.type s0) A -> + X : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) ( + R y x) (R x y)) -> + H1 : cupicef.Term (cupicef.type s4) (R y x) -> + H2 : cupicef.Term (cupicef.type s4) (R x y) -> + x0 : cupicef.Term (cupicef.type s0) A -> + y0 : cupicef.Term (cupicef.type s0) A -> + __ : + cupicef.Term (cupicef.type s4) + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I cstr_6 A + R + (flip s0 s0 s3 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0) -> + cupicef.Term (cupicef.type s4) (eqA x0 y0) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s1)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) => + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (crelation s0 s4 A) => + equ : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Equivalence s0 s4 A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (crelation s0 s4 A) => + preo : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (PreOrder s0 s4 A R) => + H : + cupicef.Term (cupicef.type s2) + (PartialOrder s0 s1 s2 s4 s3 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 + cstr_6 A eqA equ R preo) => + x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + X : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) ( + R y x) (R x y)) => + H1 : cupicef.Term (cupicef.type s4) (R y x) => + H2 : cupicef.Term (cupicef.type s4) (R x y) => + x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + Coq__Init__Datatypes.match____prod (cupicef.type s4) + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) (cupicef.axiom (cupicef.type s4)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_5 + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA x0 y0) + (__ : cupicef.Term (cupicef.type s4) (eqA x0 y0) => + relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0))) + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) (cupicef.axiom (cupicef.type s4)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_6 + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0) + (__ : + cupicef.Term (cupicef.type s4) + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0) => + eqA x0 y0))) + (__ : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA x0 y0) + (__ : cupicef.Term (cupicef.type s4) (eqA x0 y0) => + relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0)) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0) + (__ : + cupicef.Term (cupicef.type s4) + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 + cupicef.I cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0) => + eqA x0 y0))) => + cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I cstr_6 + A R + (flip s0 s0 s3 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0) + (__0 : + cupicef.Term (cupicef.type s4) + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0) => + eqA x0 y0)) + (__ : + (__ : cupicef.Term (cupicef.type s4) (eqA x0 y0) -> + cupicef.Term (cupicef.type s4) + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0)) => + __0 : + (__0 : + cupicef.Term (cupicef.type s4) + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + x0 y0) -> + cupicef.Term (cupicef.type s4) (eqA x0 y0)) => + __0) + (H x0 y0). + +thm PartialOrder__inverse : + s0 : cupicef.Nat -> + s1 : cupicef.Nat -> + s2 : cupicef.Nat -> + s3 : cupicef.Nat -> + s4 : cupicef.Nat -> + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s1)) -> + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) -> + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) -> + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) -> + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) -> + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) -> + A : cupicef.Univ (cupicef.type s0) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (crelation s0 s4 A) -> + equ : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Equivalence s0 s4 A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (crelation s0 s4 A) -> + preo : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (PreOrder s0 s4 A R) -> + H : + cupicef.Term (cupicef.type s2) + (PartialOrder s0 s1 s2 s4 s3 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 + cstr_6 A eqA equ R preo) -> + cupicef.Term (cupicef.type s2) + (PartialOrder s0 s1 s2 s4 s3 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 + cstr_6 A eqA equ + (flip s0 s0 s3 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + (flip__PreOrder s0 s4 s3 cstr_1 A R preo)) + +:= s0 : cupicef.Nat => + s1 : cupicef.Nat => + s2 : cupicef.Nat => + s3 : cupicef.Nat => + s4 : cupicef.Nat => + cstr_0 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s1)) => + cstr_1 : + cupicef.eps + (cupicef.Cumul (cupicef.axiom (cupicef.type s4)) (cupicef.type s3)) => + cstr_2 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s1)) => + cstr_3 : cupicef.eps (cupicef.Cumul (cupicef.type s0) (cupicef.type s2)) => + cstr_4 : cupicef.eps (cupicef.Cumul (cupicef.type s4) (cupicef.type s2)) => + cstr_5 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + cstr_6 : + cupicef.eps + (cupicef.Cumul (cupicef.type s4) (cupicef.type (cupicef.s cupicef.z))) => + A : cupicef.Univ (cupicef.type s0) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (crelation s0 s4 A) => + equ : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (Equivalence s0 s4 A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type s0) (cupicef.axiom (cupicef.type s4))) + (crelation s0 s4 A) => + preo : + cupicef.Term (cupicef.sup (cupicef.type s0) (cupicef.type s4)) + (PreOrder s0 s4 A R) => + H : + cupicef.Term (cupicef.type s2) + (PartialOrder s0 s1 s2 s4 s3 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 + cstr_6 A eqA equ R preo) => + cupicef.cast + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s4))) + (cupicef.type s2) + (cupicef.prod (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s4)) + (cupicef.rule (cupicef.type s0) + (cupicef.rule (cupicef.type s0) (cupicef.type s4))) + cupicef.I A + (x : cupicef.Term (cupicef.type s0) A => + cupicef.prod (cupicef.type s0) (cupicef.type s4) + (cupicef.rule (cupicef.type s0) (cupicef.type s4)) cupicef.I A + (y : cupicef.Term (cupicef.type s0) A => + Coq__Init__Datatypes.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA x y) + (__ : cupicef.Term (cupicef.type s4) (eqA x y) => + relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 + cupicef.I cstr_6 A + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + R y0 x0) + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + R y0 x0)) + x y)) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 + cupicef.I cstr_6 A + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + R y0 x0) + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + R y0 x0)) + x y) + (__ : + cupicef.Term (cupicef.type s4) + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 + cupicef.I cstr_6 A + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + R y0 x0) + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + R y0 x0)) + x y) => + eqA x y))))) + (PartialOrder s0 s1 s2 s4 s3 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 + cstr_6 A eqA equ + (flip s0 s0 s3 A A + (cupicef.cast (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + (flip__PreOrder s0 s4 s3 cstr_1 A R preo)) + cupicef.I + (x : cupicef.Term (cupicef.type s0) A => + y : cupicef.Term (cupicef.type s0) A => + Coq__Init__Datatypes.pair + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) (cupicef.axiom (cupicef.type s4)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_5 + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA x y) + (__ : cupicef.Term (cupicef.type s4) (eqA x y) => + relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + R y0 x0) + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + R y0 x0)) + x y))) + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) (cupicef.axiom (cupicef.type s4)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cstr_6 + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + R y0 x0) + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + R y0 x0)) + x y) + (__ : + cupicef.Term (cupicef.type s4) + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 + cupicef.I cstr_6 A + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + R y0 x0) + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + R y0 x0)) + x y) => + eqA x y))) + (cupicef.cast (cupicef.rule (cupicef.type s4) (cupicef.type s4)) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA x y) + (X : cupicef.Term (cupicef.type s4) (eqA x y) => + relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A R + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + R) + y x)) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (eqA x y) + (__ : cupicef.Term (cupicef.type s4) (eqA x y) => + relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + R y0 x0) + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + R y0 x0)) + x y)) + cupicef.I + (X : cupicef.Term (cupicef.type s4) (eqA x y) => + let__X2 s0 s1 s2 s3 s4 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 cstr_5 + cstr_6 A eqA equ R preo H x y X y x + (symmetry s0 s4 A eqA (Equivalence__Symmetric s0 s4 A eqA equ) + x y X))) + (cupicef.cast (cupicef.rule (cupicef.type s4) (cupicef.type s4)) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + R y0 x0) + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + R x0 y0) + x y) + (__ : + cupicef.Term (cupicef.type s4) + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 + cupicef.I cstr_6 A + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + R y0 x0) + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + R x0 y0) + x y) => + eqA x y)) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 cupicef.I + cstr_6 A + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + R y0 x0) + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + R y0 x0)) + x y) + (__ : + cupicef.Term (cupicef.type s4) + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 + cupicef.I cstr_6 A + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + R y0 x0) + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I)) + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + R y0 x0)) + x y) => + eqA x y)) + cupicef.I + (cupicef.cast (cupicef.rule (cupicef.type s4) (cupicef.type s4)) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R y x) (R x y)) + (X : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R y x) (R x y)) => + eqA x y)) + (cupicef.prod (cupicef.type s4) (cupicef.type s4) + (cupicef.rule (cupicef.type s4) (cupicef.type s4)) cupicef.I + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 + cupicef.I cstr_6 A + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + R y0 x0) + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + R x0 y0) + x y) + (__ : + cupicef.Term (cupicef.type s4) + (relation__conjunction s0 s4 s4 s4 cupicef.I cstr_5 + cupicef.I cstr_6 A + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + R y0 x0) + (x0 : cupicef.Term (cupicef.type s0) A => + y0 : cupicef.Term (cupicef.type s0) A => + R x0 y0) + x y) => + eqA x y)) + cupicef.I + (X : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R y x) (R x y)) => + Coq__Init__Datatypes.match____prod (cupicef.type s4) + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_5 (R y x)) + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_6 (R x y)) + (p : + cupicef.Term (cupicef.type s4) + (Coq__Init__Datatypes.prod (cupicef.type s4) + (cupicef.type s4) (R y x) (R x y)) => + eqA x y) + (H1 : cupicef.Term (cupicef.type s4) (R y x) => + H2 : cupicef.Term (cupicef.type s4) (R x y) => + let__X3 s0 s1 s2 s3 s4 cstr_0 cstr_1 cstr_2 cstr_3 cstr_4 + cstr_5 cstr_6 A eqA equ R preo H x y X H1 H2 x y + (Coq__Init__Datatypes.pair + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_5 (R x y)) + (cupicef.cast (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cstr_6 + (flip s0 s0 s3 A A + (cupicef.cast + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + (cupicef.axiom (cupicef.type s3)) + (cupicef.univ + (cupicef.axiom (cupicef.type s4)) + (cupicef.axiom + (cupicef.axiom (cupicef.type s4))) + cupicef.I) + (cupicef.univ (cupicef.type s3) + (cupicef.axiom (cupicef.type s3)) + cupicef.I) + cstr_1 + (cupicef.univ (cupicef.type s4) + (cupicef.axiom (cupicef.type s4)) + cupicef.I)) + R x y)) + H2 H1)) + X)))). + +(; End of translation. ;) diff --git a/out/Coq__Classes__Equivalence.dk b/out/Coq__Classes__Equivalence.dk new file mode 100644 index 000000000..2318839a3 --- /dev/null +++ b/out/Coq__Classes__Equivalence.dk @@ -0,0 +1,5430 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def equiv : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Equivalence A R) -> + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A R) => + R. + +def pequiv : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) -> + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) => + R. + +def equiv__reflexive__obligation__1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + sa : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Equivalence A R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (equiv A R sa x x) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + sa : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Classes__RelationClasses.match____Equivalence cupicef.prop A R + (__ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A R) => + R x x) + (Equivalence__Reflexive : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Reflexive A R) => + Equivalence__Symmetric : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Symmetric A R) => + Equivalence__Transitive : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Transitive A R) => + Equivalence__Reflexive x) + sa. + +def equiv__reflexive : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + sa : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Equivalence A R) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Reflexive A (equiv A R sa)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + sa : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A R) => + (A0 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A0) => + sa0 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A0 R0) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A0 => + equiv__reflexive__obligation__1 A0 R0 sa0 x) A R sa. + +#TYPE_CLASS_INSTANCE equiv__reflexive. + +def equiv__symmetric__obligation__1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + sa : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Equivalence A R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H : cupicef.Term cupicef.prop (equiv A R sa x y) -> + cupicef.Term cupicef.prop (equiv A R sa y x) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + sa : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : cupicef.Term cupicef.prop (equiv A R sa x y) => + Coq__Classes__RelationClasses.symmetry A R + (Coq__Classes__RelationClasses.Equivalence__Symmetric A R sa) x y H. + +def equiv__symmetric : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + sa : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Equivalence A R) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Symmetric A (equiv A R sa)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + sa : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A R) => + (A0 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A0) => + sa0 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A0 R0) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A0 => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A0 => + H : cupicef.Term cupicef.prop (equiv A0 R0 sa0 x y) => + equiv__symmetric__obligation__1 A0 R0 sa0 x y H) A R sa. + +#TYPE_CLASS_INSTANCE equiv__symmetric. + +thm equiv__transitive__obligation__1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + sa : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Equivalence A R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hxy : cupicef.Term cupicef.prop (equiv A R sa x y) -> + Hyz : cupicef.Term cupicef.prop (equiv A R sa y z) -> + cupicef.Term cupicef.prop (equiv A R sa x z) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + sa : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hxy : cupicef.Term cupicef.prop (equiv A R sa x y) => + Hyz : cupicef.Term cupicef.prop (equiv A R sa y z) => + Coq__Classes__RelationClasses.transitivity A (equiv A R sa) + (Coq__Classes__RelationClasses.Equivalence__Transitive A (equiv A R sa) + sa) + x y z Hxy Hyz. + +def equiv__transitive : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + sa : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Equivalence A R) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Transitive A (equiv A R sa)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + sa : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A R) => + (A0 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A0) => + sa0 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A0 R0) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A0 => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A0 => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A0 => + Hxy : cupicef.Term cupicef.prop (equiv A0 R0 sa0 x y) => + Hyz : cupicef.Term cupicef.prop (equiv A0 R0 sa0 y z) => + equiv__transitive__obligation__1 A0 R0 sa0 x y z Hxy Hyz) A R sa. + +#TYPE_CLASS_INSTANCE equiv__transitive. + +def respecting : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + eqa : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A + (cupicef.cast + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) + (Coq__Relations__Relation__Definitions.relation A) cupicef.I R)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + eqb : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence B + (cupicef.cast + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) + (Coq__Relations__Relation__Definitions.relation B) cupicef.I + R_27_)) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) + +:= cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.rule + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))))))))) + (cupicef.rule (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.rule + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))))))))) + (cupicef.prod (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.rule + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))))))) + (cupicef.rule (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.rule + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))))))))) + cupicef.I + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.rule + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))))))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.rule + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))))))) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod cupicef.prop + (cupicef.rule + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))))) + (cupicef.rule cupicef.prop + (cupicef.rule + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))))))) + cupicef.I + (Coq__Classes__RelationClasses.Equivalence A + (cupicef.cast + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) + (Coq__Relations__Relation__Definitions.relation A) + cupicef.I R)) + (eqa : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A + (cupicef.cast + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) + (Coq__Relations__Relation__Definitions.relation A) + cupicef.I R)) => + cupicef.prod + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))))) + (cupicef.rule + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))))) + cupicef.I + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))))) + cupicef.I + (Coq__Relations__Relation__Definitions.relation B) + (R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + cupicef.prod cupicef.prop + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule cupicef.prop + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I + (Coq__Classes__RelationClasses.Equivalence B + (cupicef.cast + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation + B) + (Coq__Relations__Relation__Definitions.relation + B) + cupicef.I R_27_)) + (eqb : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence B + (cupicef.cast + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation + B) + (Coq__Relations__Relation__Definitions.relation + B) + cupicef.I R_27_)) => + cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I))))))) + (cupicef.prod (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.rule + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))))))) + (cupicef.rule (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.rule + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))))))))) + cupicef.I + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.rule + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))))))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.rule + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))))))) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod cupicef.prop + (cupicef.rule + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))))) + (cupicef.rule cupicef.prop + (cupicef.rule + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))))))) + cupicef.I + (Coq__Classes__RelationClasses.Equivalence A + (cupicef.cast + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) + (Coq__Relations__Relation__Definitions.relation A) + cupicef.I R)) + (eqa : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A + (cupicef.cast + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) + (Coq__Relations__Relation__Definitions.relation A) + cupicef.I R)) => + cupicef.prod + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.rule + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))))) + cupicef.I + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I + (Coq__Relations__Relation__Definitions.relation B) + (R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + cupicef.prod cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Classes__RelationClasses.Equivalence B + (cupicef.cast + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation + B) + (Coq__Relations__Relation__Definitions.relation + B) + cupicef.I R_27_)) + (eqb : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence B + (cupicef.cast + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation + B) + (Coq__Relations__Relation__Definitions.relation + B) + cupicef.I R_27_)) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I))))))) + cupicef.I + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + eqa : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A + (cupicef.cast + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) + (Coq__Relations__Relation__Definitions.relation A) cupicef.I + R)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + eqb : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence B + (cupicef.cast + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) + (Coq__Relations__Relation__Definitions.relation B) cupicef.I + R_27_)) => + Coq__Init__Specif.sig + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (morph : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph)). + +def respecting__equiv__obligation__1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + eqa : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Equivalence A R) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + eqb : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence B R_27_) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Reflexive (respecting A R eqa B R_27_ eqb) + (f : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (respecting A R eqa B R_27_ eqb) => + g : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (respecting A R eqa B R_27_ eqb) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x y) + (__ : cupicef.Term cupicef.prop (R x y) => + R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + f x) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + g y)))))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + eqa : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A R) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + eqb : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence B R_27_) => + x : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sig + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (morph : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph)) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : cupicef.Term cupicef.prop (R x0 y) => + Coq__Init__Specif.match____sig cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B))) + (morph : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph) + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sig + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B)) + (morph : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph)) => + R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph) + s x0) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph) + s y)) + (x1 : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + H0 : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.respectful A B R R_27_ x1 x1) => + H0 x0 y H) + x. + +def respecting__equiv__obligation__2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + eqa : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Equivalence A R) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + eqb : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence B R_27_) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Symmetric (respecting A R eqa B R_27_ eqb) + (f : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (respecting A R eqa B R_27_ eqb) => + g : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (respecting A R eqa B R_27_ eqb) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x y) + (__ : cupicef.Term cupicef.prop (R x y) => + R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + f x) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + g y)))))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + eqa : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A R) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + eqb : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence B R_27_) => + x : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sig + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (morph : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph)) => + y : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sig + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (morph : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph)) => + H : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x0 y0) -> + cupicef.Term cupicef.prop + (R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph) + x x0) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph) + y y0))) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x0 y0) => + Coq__Init__Specif.match____sig cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B))) + (morph : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph) + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sig + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B)) + (morph : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph)) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + R x1 y1) + (__ : cupicef.Term cupicef.prop (R x1 y1) => + R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + x x1) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + s y1))))) + (H1 : + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x1 y1) -> + cupicef.Term cupicef.prop + (R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + x x1) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + s y1))) => + R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph) + s x0) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph) + x y0))) + (y1 : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + H1 : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.respectful A B R R_27_ y1 y1) => + H2 : + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x1 y2) -> + cupicef.Term cupicef.prop + (R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph) + x x1) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph) + (Coq__Init__Specif.exist + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + y1 H1) + y2))) => + Coq__Init__Specif.match____sig cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B))) + (morph : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph) + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sig + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B)) + (morph : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph)) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (R x1 y2) + (__ : cupicef.Term cupicef.prop (R x1 y2) => + R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ + morph morph) + s x1) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ + morph morph) + (Coq__Init__Specif.exist + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ + morph morph) + y1 H1) + y2))))) + (H3 : + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x1 y2) -> + cupicef.Term cupicef.prop + (R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + s x1) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + (Coq__Init__Specif.exist + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ + morph morph) + y1 H1) + y2))) => + R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph) + (Coq__Init__Specif.exist + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + y1 H1) + x0) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph) + s y0))) + (x1 : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + H3 : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.respectful A B R R_27_ x1 x1) => + H4 : + (x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x2 y2) -> + cupicef.Term cupicef.prop + (R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + (Coq__Init__Specif.exist + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + x1 H3) + x2) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + (Coq__Init__Specif.exist + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + y1 H1) + y2))) => + Coq__Classes__RelationClasses.match____Equivalence cupicef.prop A R + (__ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A R) => + R_27_ (y1 x0) (x1 y0)) + (Equivalence__Reflexive : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Reflexive A R) => + Equivalence__Symmetric : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Symmetric A R) => + Equivalence__Transitive : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Transitive A R) => + Coq__Classes__RelationClasses.match____Equivalence cupicef.prop B + R_27_ + (__ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence B R_27_) => + R_27_ (y1 x0) (x1 y0)) + (Equivalence__Reflexive0 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Reflexive B R_27_) => + Equivalence__Symmetric0 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Symmetric B R_27_) => + Equivalence__Transitive0 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Transitive B R_27_) => + Equivalence__Symmetric0 (x1 y0) (y1 x0) + (H4 y0 x0 (Equivalence__Symmetric x0 y0 H0))) + eqb) + eqa) + x H2) + y H. + +thm respecting__equiv__obligation__3 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + eqa : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Equivalence A R) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + eqb : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence B R_27_) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Transitive + (respecting A R eqa B R_27_ eqb) + (f : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (respecting A R eqa B R_27_ eqb) => + g : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (respecting A R eqa B R_27_ eqb) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x y) + (__ : cupicef.Term cupicef.prop (R x y) => + R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + f x) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + g y)))))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + eqa : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A R) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + eqb : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence B R_27_) => + f : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (respecting A R eqa B R_27_ eqb) => + g : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (respecting A R eqa B R_27_ eqb) => + h : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (respecting A R eqa B R_27_ eqb) => + H : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop + (R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph) + f x) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph) + g y))) => + H_27_ : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop + (R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph) + g x) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph) + h y))) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Rxy : cupicef.Term cupicef.prop (R x y) => + Coq__Init__Specif.match____sig cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B))) + (morph : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph) + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sig + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B)) + (morph : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph)) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + R x0 y0) + (__ : cupicef.Term cupicef.prop (R x0 y0) => + R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + g x0) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + s y0))))) + (H_27_0 : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x0 y0) -> + cupicef.Term cupicef.prop + (R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + g x0) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + s y0))) => + R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph) + f x) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph) + s y))) + (h0 : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + H0 : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.respectful A B R R_27_ h0 h0) => + H_27_0 : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x0 y0) -> + cupicef.Term cupicef.prop + (R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph) + g x0) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph) + (Coq__Init__Specif.exist + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + h0 H0) + y0))) => + Coq__Init__Specif.match____sig cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B))) + (morph : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph) + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sig + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B)) + (morph : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph)) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop)) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (R x0 y0) + (__ : cupicef.Term cupicef.prop (R x0 y0) => + R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ + morph morph) + f x0) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ + morph morph) + s y0))))) + (H1 : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x0 y0) -> + cupicef.Term cupicef.prop + (R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + f x0) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + s y0))) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (R x0 y0) + (__ : cupicef.Term cupicef.prop (R x0 y0) => + R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ + morph morph) + s x0) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ + morph morph) + (Coq__Init__Specif.exist + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type + (cupicef.s cupicef.z)) + A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R + R_27_ morph morph) + h0 H0) + y0))))) + (H_27_1 : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x0 y0) -> + cupicef.Term cupicef.prop + (R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ + morph morph) + s x0) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ + morph morph) + (Coq__Init__Specif.exist + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ + morph morph) + h0 H0) + y0))) => + R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + f x) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + (Coq__Init__Specif.exist + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + h0 H0) + y)))) + (g0 : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + H1 : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.respectful A B R R_27_ g0 g0) => + H2 : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x0 y0) -> + cupicef.Term cupicef.prop + (R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + f x0) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + (Coq__Init__Specif.exist + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + g0 H1) + y0))) => + H_27_1 : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x0 y0) -> + cupicef.Term cupicef.prop + (R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + (Coq__Init__Specif.exist + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + g0 H1) + x0) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + (Coq__Init__Specif.exist + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + h0 H0) + y0))) => + Coq__Init__Specif.match____sig cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph morph) + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sig + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B)) + (morph : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph)) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (R x0 y0) + (__ : cupicef.Term cupicef.prop (R x0 y0) => + R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ + morph morph) + s x0) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ + morph morph) + (Coq__Init__Specif.exist + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type + (cupicef.s cupicef.z)) + A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R + R_27_ morph morph) + g0 H1) + y0))))) + (H3 : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x0 y0) -> + cupicef.Term cupicef.prop + (R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ + morph morph) + s x0) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ + morph morph) + (Coq__Init__Specif.exist + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ + morph morph) + g0 H1) + y0))) => + R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + s x) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + (Coq__Init__Specif.exist + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + h0 H0) + y))) + (f0 : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + H3 : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.respectful A B R R_27_ f0 f0) => + H4 : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x0 y0) -> + cupicef.Term cupicef.prop + (R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + (Coq__Init__Specif.exist + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ + morph morph) + f0 H3) + x0) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + (Coq__Init__Specif.exist + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ + morph morph) + g0 H1) + y0))) => + Coq__Classes__RelationClasses.transitivity B R_27_ + (Coq__Classes__RelationClasses.Equivalence__Transitive B R_27_ + eqb) + (f0 x) (g0 y) (h0 y) (H4 x y Rxy) + (Coq__Classes__RelationClasses.match____Equivalence + cupicef.prop B R_27_ + (__ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence B R_27_) => + R_27_ (g0 y) (h0 y)) + (Equivalence__Reflexive : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Reflexive B R_27_) => + Equivalence__Symmetric : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Symmetric B R_27_) => + Equivalence__Transitive : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Transitive B R_27_) => + Coq__Classes__RelationClasses.match____Equivalence + cupicef.prop A R + (__ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A R) => + R_27_ (g0 y) (h0 y)) + (Equivalence__Reflexive0 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Reflexive A R) => + Equivalence__Symmetric0 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Symmetric A R) => + Equivalence__Transitive0 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Transitive A R) => + H_27_1 y y (Equivalence__Reflexive0 y)) + eqa) + eqb)) + f H2) + g H H_27_0) + h H_27_. + +def respecting__equiv : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + eqa : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Equivalence A R) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + eqb : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence B R_27_) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence + (respecting A R eqa B R_27_ eqb) + (f : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (respecting A R eqa B R_27_ eqb) => + g : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (respecting A R eqa B R_27_ eqb) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x y) + (__ : cupicef.Term cupicef.prop (R x y) => + R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + f x) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + g y)))))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + eqa : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A R) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + eqb : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence B R_27_) => + Coq__Classes__RelationClasses.Build__Equivalence + (respecting A R eqa B R_27_ eqb) + (f : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (respecting A R eqa B R_27_ eqb) => + g : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (respecting A R eqa B R_27_ eqb) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x y) + (__ : cupicef.Term cupicef.prop (R x y) => + R_27_ + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + f x) + (Coq__Init__Specif.proj1__sig + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (morph : + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.respectful A B R R_27_ morph + morph) + g y))))) + ((A0 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A0) => + eqa0 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A0 R0) => + B0 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R_27_0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B0) => + eqb0 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence B0 R_27_0) => + respecting__equiv__obligation__1 A0 R0 eqa0 B0 R_27_0 eqb0) A R eqa B + R_27_ eqb) + ((A0 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A0) => + eqa0 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A0 R0) => + B0 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R_27_0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B0) => + eqb0 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence B0 R_27_0) => + respecting__equiv__obligation__2 A0 R0 eqa0 B0 R_27_0 eqb0) A R eqa B + R_27_ eqb) + ((A0 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A0) => + eqa0 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A0 R0) => + B0 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R_27_0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B0) => + eqb0 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence B0 R_27_0) => + respecting__equiv__obligation__3 A0 R0 eqa0 B0 R_27_0 eqb0) A R eqa B + R_27_ eqb). + +thm pointwise__reflexive : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqB : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + reflb : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Reflexive B eqB) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Reflexive + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B))) + (Coq__Classes__Morphisms.pointwise__relation A B eqB)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqB : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + reflb : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Reflexive B eqB) => + cupicef.cast + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop) + cupicef.prop + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Classes__Morphisms.pointwise__relation A B eqB x x)) + (Coq__Classes__RelationClasses.Reflexive + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B))) + (Coq__Classes__Morphisms.pointwise__relation A B eqB)) + cupicef.I + (x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eqB (x a) (x a))) + (Coq__Classes__Morphisms.pointwise__relation A B eqB x x) cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (H : cupicef.Term cupicef.prop (eqB (x a) (x a)) => H) (reflb (x a)))). + +#TYPE_CLASS_INSTANCE pointwise__reflexive. + +thm pointwise__symmetric : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqB : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + symb : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Symmetric B eqB) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Symmetric + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B))) + (Coq__Classes__Morphisms.pointwise__relation A B eqB)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqB : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + symb : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Symmetric B eqB) => + cupicef.cast + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (y : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Classes__Morphisms.pointwise__relation A B eqB x y) + (H : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.pointwise__relation A B eqB x y) => + Coq__Classes__Morphisms.pointwise__relation A B eqB y x)))) + (Coq__Classes__RelationClasses.Symmetric + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B))) + (Coq__Classes__Morphisms.pointwise__relation A B eqB)) + cupicef.I + (x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + y : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.pointwise__relation A B eqB x y) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eqB (y a) (x a))) + (Coq__Classes__Morphisms.pointwise__relation A B eqB y x) cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (H0 : + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __ : cupicef.Term cupicef.prop (eqB (x a) y0) -> + cupicef.Term cupicef.prop (eqB y0 (x a))) => + (H1 : cupicef.Term cupicef.prop (eqB (x a) (y a)) => + (H2 : + (__ : cupicef.Term cupicef.prop (eqB (x a) (y a)) -> + cupicef.Term cupicef.prop (eqB (y a) (x a))) => + (H3 : cupicef.Term cupicef.prop (eqB (y a) (x a)) => H3) (H2 H1)) + (H0 (y a))) + (H a)) + (symb (x a)))). + +#TYPE_CLASS_INSTANCE pointwise__symmetric. + +thm pointwise__transitive : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqB : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + transb : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Transitive B eqB) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Transitive + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B))) + (Coq__Classes__Morphisms.pointwise__relation A B eqB)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqB : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + transb : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Transitive B eqB) => + cupicef.cast + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.prop + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (y : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B)) + (z : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Classes__Morphisms.pointwise__relation A B eqB x y) + (H : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.pointwise__relation A B eqB x + y) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Classes__Morphisms.pointwise__relation A B eqB y z) + (H0 : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.pointwise__relation A B eqB + y z) => + Coq__Classes__Morphisms.pointwise__relation A B eqB x z)))))) + (Coq__Classes__RelationClasses.Transitive + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B))) + (Coq__Classes__Morphisms.pointwise__relation A B eqB)) + cupicef.I + (x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + y : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + z : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.pointwise__relation A B eqB x y) => + H0 : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.pointwise__relation A B eqB y z) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eqB (x a) (z a))) + (Coq__Classes__Morphisms.pointwise__relation A B eqB x z) cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (H1 : + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + z0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __ : cupicef.Term cupicef.prop (eqB (x a) y0) -> + __0 : cupicef.Term cupicef.prop (eqB y0 z0) -> + cupicef.Term cupicef.prop (eqB (x a) z0)) => + (H2 : cupicef.Term cupicef.prop (eqB (x a) (y a)) => + (H3 : + (z0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __ : cupicef.Term cupicef.prop (eqB (x a) (y a)) -> + __0 : cupicef.Term cupicef.prop (eqB (y a) z0) -> + cupicef.Term cupicef.prop (eqB (x a) z0)) => + H3 (z a) H2 (H0 a)) (H1 (y a))) (H a)) + (transb (x a)))). + +#TYPE_CLASS_INSTANCE pointwise__transitive. + +thm pointwise__equivalence : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqB : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + eqb : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence B eqB) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B))) + (Coq__Classes__Morphisms.pointwise__relation A B eqB)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqB : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + eqb : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence B eqB) => + Coq__Classes__RelationClasses.Build__Equivalence + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B))) + (Coq__Classes__Morphisms.pointwise__relation A B eqB) + (pointwise__reflexive A B eqB + (Coq__Classes__RelationClasses.Equivalence__Reflexive B eqB eqb)) + (pointwise__symmetric A B eqB + (Coq__Classes__RelationClasses.Equivalence__Symmetric B eqB eqb)) + (pointwise__transitive A B eqB + (Coq__Classes__RelationClasses.Equivalence__Transitive B eqB eqb)). + +#TYPE_CLASS_INSTANCE pointwise__equivalence. + +(; End of translation. ;) diff --git a/out/Coq__Classes__Init.dk b/out/Coq__Classes__Init.dk new file mode 100644 index 000000000..fe250f765 --- /dev/null +++ b/out/Coq__Classes__Init.dk @@ -0,0 +1,30 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def Unconvertible : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.set + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Datatypes.unit. + +#TYPE_CLASS Unconvertible. + +def unconvertible : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Unconvertible0 : cupicef.Term cupicef.set (Unconvertible A a b) -> + cupicef.Term cupicef.set Coq__Init__Datatypes.unit + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Unconvertible0 : cupicef.Term cupicef.set (Unconvertible A a b) => + Unconvertible0. + +(; End of translation. ;) diff --git a/out/Coq__Classes__Morphisms.dk b/out/Coq__Classes__Morphisms.dk new file mode 100644 index 000000000..263b34a1b --- /dev/null +++ b/out/Coq__Classes__Morphisms.dk @@ -0,0 +1,20641 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def let__U : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +def Proper : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + m : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) let__U => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + m : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + R m m. + +#TYPE_CLASS Proper. + +def let__U0 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +def proper__prf : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + m : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Proper0 : cupicef.Term cupicef.prop (Proper A R m) -> + cupicef.Term cupicef.prop (R m m) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U0 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + m : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Proper0 : cupicef.Term cupicef.prop (Proper A R m) => + Proper0. + +def let__U1 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +def ProperProxy : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + m : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U1 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + m : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + R m m. + +#TYPE_CLASS ProperProxy. + +def let__U2 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +def proper__proxy : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + m : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + ProperProxy0 : cupicef.Term cupicef.prop (ProperProxy A R m) -> + cupicef.Term cupicef.prop (R m m) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U2 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + m : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + ProperProxy0 : cupicef.Term cupicef.prop (ProperProxy A R m) => + ProperProxy0. + +def let__U3 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm eq__proper__proxy : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (ProperProxy A + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + x) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U3 => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U3 + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x. + +def let__U4 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm reflexive__proper__proxy : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Reflexive A R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (ProperProxy A R x) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U4 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Reflexive A R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (H0 : cupicef.Term cupicef.prop (R x x) => H0) (H x). + +def let__U5 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm proper__proper__proxy : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Proper A R x) -> + cupicef.Term cupicef.prop (ProperProxy A R x) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U5 => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (Proper A R x) => + H. + +def respectful__hetero : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + C : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + D : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) -> + R_27_ : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (C x) -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (D y) -> + cupicef.Univ cupicef.prop) -> + __ : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (C x)) -> + __0 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (D x)) -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + C : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + D : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) => + R_27_ : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (C x) -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (D y) -> + cupicef.Univ cupicef.prop) => + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (C x)) => + g : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (D x)) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I B + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x y) + (__ : cupicef.Term cupicef.prop (R x y) => R_27_ x y (f x) (g y)))). + +def let__U6 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +def respectful : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B))) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U6 => + B : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U6 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + g : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x y) + (__ : cupicef.Term cupicef.prop (R x y) => R_27_ (f x) (g y)))). + +def let__U7 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +def forall__def : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U7 => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U7) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x). + +def let__U8 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +def forall__relation : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + sig : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation (P a))) -> + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x))) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U8 => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U8) => + sig : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation (P a))) => + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x)) => + g : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x)) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + sig a (f a) (g a)). + +def let__U9 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +def pointwise__relation : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B))) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U9 => + B : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U9 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + g : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + R (f a) (g a)). + +def let__U10 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm pointwise__pointwise : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.relation__equivalence + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (pointwise__relation A B R) + (respectful A B + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + R)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U10 => + B : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U10 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + x0 : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (pointwise__relation A B R x x0) + (__ : cupicef.Term cupicef.prop (pointwise__relation A B R x x0) => + respectful A B + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U10 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + R x x0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (respectful A B + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U10 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + R x x0) + (__ : + cupicef.Term cupicef.prop + (respectful A B + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U10 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + R x x0) => + pointwise__relation A B R x x0)) + (cupicef.cast + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I (pointwise__relation A B R x x0) + (H : cupicef.Term cupicef.prop (pointwise__relation A B R x x0) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U10 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x1 y) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + let__U10 + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x1 y) => + (x2 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + R (x x2) (x0 y)) x1))))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (pointwise__relation A B R x x0) + (__ : cupicef.Term cupicef.prop (pointwise__relation A B R x x0) => + respectful A B + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U10 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + R x x0)) + cupicef.I + (H : cupicef.Term cupicef.prop (pointwise__relation A B R x x0) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U10 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x1 y) => + Coq__Init__Logic.eq__ind__r A y + (x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + R (x x2) (x0 y)) + ((H1 : cupicef.Term cupicef.prop (R (x y) (x0 y)) => H1) (H y)) x1 + H0)) + (cupicef.cast + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop)) + cupicef.I + (respectful A B + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U10 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + R x x0) + (H : + cupicef.Term cupicef.prop + (respectful A B + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U10 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + R x x0) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + R (x a) (x0 a)))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (respectful A B + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U10 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + R x x0) + (__ : + cupicef.Term cupicef.prop + (respectful A B + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U10 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + R x x0) => + pointwise__relation A B R x x0)) + cupicef.I + (H : + cupicef.Term cupicef.prop + (respectful A B + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U10 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + R x x0) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (H0 : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U10 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a y) -> + cupicef.Term cupicef.prop (R (x a) (x0 y))) => + (H1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U10 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a a) -> + cupicef.Term cupicef.prop (R (x a) (x0 a))) => + (H2 : cupicef.Term cupicef.prop (R (x a) (x0 a)) => H2) + (H1 + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U10 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a))) + (H0 a)) + (H a))). + +def let__U11 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm subrelation__id__proper : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + RA_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation A RA RA_27_) -> + cupicef.Term cupicef.prop + (Proper + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => A)) + (respectful A A RA RA_27_) (Coq__Init__Datatypes.id A)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U11 => + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + RA_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation A RA RA_27_) => + H. + +#TYPE_CLASS_INSTANCE subrelation__id__proper. + +def let__U12 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm subrelation__respectful : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + RA_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + subl : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation A RA_27_ RA) -> + RB : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + RB_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + subr : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation B RB RB_27_) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (respectful A B RA RB) (respectful A B RA_27_ RB_27_)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U12 => + B : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U12 => + RA_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + subl : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation A RA_27_ RA) => + RB : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + RB_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + subr : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation B RB RB_27_) => + cupicef.cast + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (y : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (respectful A B RA RB x y) + (H : cupicef.Term cupicef.prop (respectful A B RA RB x y) => + respectful A B RA_27_ RB_27_ x y)))) + (Coq__Classes__RelationClasses.subrelation + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (respectful A B RA RB) (respectful A B RA_27_ RB_27_)) + cupicef.I + (x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + y : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + H : cupicef.Term cupicef.prop (respectful A B RA RB x y) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (RA_27_ x0 y0) + (H0 : cupicef.Term cupicef.prop (RA_27_ x0 y0) => + RB_27_ (x x0) (y y0))))) + (respectful A B RA_27_ RB_27_ x y) cupicef.I + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (RA_27_ x0 y0) => + (H1 : + (y1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __ : cupicef.Term cupicef.prop (RB (x x0) y1) -> + cupicef.Term cupicef.prop (RB_27_ (x x0) y1)) => + (H2 : + (y1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (RA x0 y1) -> + cupicef.Term cupicef.prop (RB (x x0) (y y1))) => + (H3 : + (__ : cupicef.Term cupicef.prop (RB (x x0) (y y0)) -> + cupicef.Term cupicef.prop (RB_27_ (x x0) (y y0))) => + subr (x x0) (y y0) (H x0 y0 (subl x0 y0 H0))) (H1 (y y0))) ( + H x0)) + (subr (x x0)))). + +def let__U13 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm subrelation__refl : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.subrelation A R R) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U13 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x y) + (H : cupicef.Term cupicef.prop (R x y) => R x y)))) + (Coq__Classes__RelationClasses.subrelation A R R) cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : cupicef.Term cupicef.prop (R x y) => + H). + +def let__U14 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm subrelation__proper : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + m : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + mor : cupicef.Term cupicef.prop (Proper A R_27_ m) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + unc : + cupicef.Term cupicef.set + (Coq__Classes__Init.Unconvertible + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + R R_27_) -> + sub : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation A R_27_ R) -> + cupicef.Term cupicef.prop (Proper A R m) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U14 => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + m : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + mor : cupicef.Term cupicef.prop (Proper A R_27_ m) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + unc : + cupicef.Term cupicef.set + (Coq__Classes__Init.Unconvertible + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + R R_27_) => + sub : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation A R_27_ R) => + sub m m mor. + +def let__U15 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm proper__subrelation__proper : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__RelationClasses.subrelation A) + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + Coq__Program__Basics.impl)) + (Proper A)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U15 => + cupicef.cast cupicef.prop cupicef.prop + (respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (Coq__Classes__RelationClasses.subrelation A) + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U15 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + Coq__Program__Basics.impl) + (Proper A) (Proper A)) + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__RelationClasses.subrelation A) + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U15 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + Coq__Program__Basics.impl)) + (Proper A)) + cupicef.I + (cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (x : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (y : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Classes__RelationClasses.subrelation A x y) + (H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation A x y) => + respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U15 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + Coq__Program__Basics.impl (Proper A x) (Proper A y))))) + (respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__RelationClasses.subrelation A) + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U15 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + Coq__Program__Basics.impl) + (Proper A) (Proper A)) + cupicef.I + (x : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + y : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation A x y) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U15 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y0) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + let__U15 + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y0) => + Coq__Program__Basics.impl (Proper A x x0) + (Proper A y y0))))) + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U15 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + Coq__Program__Basics.impl (Proper A x) (Proper A y)) + cupicef.I + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U15 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y0) => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Proper A x x0) + (H1 : cupicef.Term cupicef.prop (Proper A x x0) => + Proper A y y0)) + (Coq__Program__Basics.impl (Proper A x x0) (Proper A y y0)) + cupicef.I + (H1 : cupicef.Term cupicef.prop (Proper A x x0) => + cupicef.cast cupicef.prop cupicef.prop (y y0 y0) + (Proper A y y0) cupicef.I + (Coq__Init__Logic.eq__ind__r A y0 + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Proper A x x1) + (H2 : cupicef.Term cupicef.prop (Proper A x x1) => + y y0 y0)) + (H2 : cupicef.Term cupicef.prop (Proper A x y0) => + (H3 : + (y1 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A -> + __ : cupicef.Term cupicef.prop (x y0 y1) -> + cupicef.Term cupicef.prop (y y0 y1)) => + (H4 : + (__ : cupicef.Term cupicef.prop (x y0 y0) -> + cupicef.Term cupicef.prop (y y0 y0)) => + (H5 : cupicef.Term cupicef.prop (y y0 y0) => H5) + (H4 H2)) + (H3 y0)) + (H y0)) + x0 H0 H1))))). + +#TYPE_CLASS_INSTANCE proper__subrelation__proper. + +def let__U16 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm pointwise__subrelation : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + sub : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation B R R_27_) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (pointwise__relation A B R) (pointwise__relation A B R_27_)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U16 => + B : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U16 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + sub : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation B R R_27_) => + cupicef.cast + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (y : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (pointwise__relation A B R x y) + (H : cupicef.Term cupicef.prop (pointwise__relation A B R x y) => + pointwise__relation A B R_27_ x y)))) + (Coq__Classes__RelationClasses.subrelation + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (pointwise__relation A B R) (pointwise__relation A B R_27_)) + cupicef.I + (x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + y : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + H : cupicef.Term cupicef.prop (pointwise__relation A B R x y) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + R_27_ (x a) (y a))) + (pointwise__relation A B R_27_ x y) cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.cast cupicef.prop cupicef.prop (R_27_ (x a) (y a)) + (R_27_ (x a) (y a)) cupicef.I (sub (x a) (y a) (H a)))). + +#TYPE_CLASS_INSTANCE pointwise__subrelation. + +def let__U17 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm forall__subrelation : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + R : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation (P x))) -> + S : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation (P x))) -> + __ : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation (P a) (R a) (S a))) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x)) + (forall__relation A P R) (forall__relation A P S)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U17 => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U17) => + R : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation (P x))) => + S : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation (P x))) => + cupicef.cast + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.prop) + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Classes__RelationClasses.subrelation (P a) (R a) (S a))) + (H : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation (P a) (R a) (S a))) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x)) + (x : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x)) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x0)) + (y : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x0)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (forall__relation A P R x y) + (H0 : cupicef.Term cupicef.prop (forall__relation A P R x y) => + forall__relation A P S x y))))) + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Classes__RelationClasses.subrelation (P a) (R a) (S a))) + (__ : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation (P a) (R a) (S a))) => + Coq__Classes__RelationClasses.subrelation + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x)) + (forall__relation A P R) (forall__relation A P S))) + cupicef.I + (H : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation (P a) (R a) (S a))) => + x : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x)) => + y : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x0)) => + H0 : cupicef.Term cupicef.prop (forall__relation A P R x y) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + S a (x a) (y a))) + (forall__relation A P S x y) cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H a (x a) (y a) (H0 a))). + +def apply__subrelation : + cupicef.Univ cupicef.prop. + +def do__subrelation : + cupicef.Term cupicef.prop apply__subrelation. + +thm iff__impl__subrelation : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff Coq__Program__Basics.impl) + +:= cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (x : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (y : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.iff x y) + (H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + Coq__Program__Basics.impl x y)))) + (Coq__Classes__RelationClasses.subrelation + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff Coq__Program__Basics.impl) + cupicef.I + (x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I x + (H0 : cupicef.Term cupicef.prop x => y)) + (Coq__Program__Basics.impl x y) cupicef.I + (H0 : cupicef.Term cupicef.prop x => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I x + (__ : cupicef.Term cupicef.prop x => y)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I y + (__ : cupicef.Term cupicef.prop y => x)) + y + (H1 : + (__ : cupicef.Term cupicef.prop x -> + cupicef.Term cupicef.prop y) => + H2 : + (__ : cupicef.Term cupicef.prop y -> + cupicef.Term cupicef.prop x) => + (H3 : cupicef.Term cupicef.prop y => H3) (H1 H0)) + H)). + +#TYPE_CLASS_INSTANCE iff__impl__subrelation. + +thm iff__flip__impl__subrelation : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff + (Coq__Program__Basics.flip + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl)) + +:= cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (x : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (y : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.iff x y) + (H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + Coq__Program__Basics.flip + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Program__Basics.impl x y)))) + (Coq__Classes__RelationClasses.subrelation + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff + (Coq__Program__Basics.flip + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl)) + cupicef.I + (x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I y + (H0 : cupicef.Term cupicef.prop y => x)) + (Coq__Program__Basics.flip + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl x y) + cupicef.I + (H0 : cupicef.Term cupicef.prop y => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I x + (__ : cupicef.Term cupicef.prop x => y)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I y + (__ : cupicef.Term cupicef.prop y => x)) + x + (H1 : + (__ : cupicef.Term cupicef.prop x -> + cupicef.Term cupicef.prop y) => + H2 : + (__ : cupicef.Term cupicef.prop y -> + cupicef.Term cupicef.prop x) => + (H3 : cupicef.Term cupicef.prop x => H3) (H2 H0)) + H)). + +#TYPE_CLASS_INSTANCE iff__flip__impl__subrelation. + +def let__U18 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +def respectful__per__obligation__1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + H0 : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER B R_27_) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Symmetric + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (respectful A B R R_27_)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U18 => + B : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U18 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + H0 : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER B R_27_) => + x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + y : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + H1 : cupicef.Term cupicef.prop (respectful A B R R_27_ x y) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H2 : cupicef.Term cupicef.prop (R x0 y0) => + Coq__Classes__RelationClasses.match____PER cupicef.prop A R + (__ : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) => + R_27_ (y x0) (x y0)) + (PER__Symmetric : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Symmetric A R) => + PER__Transitive : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Transitive A R) => + Coq__Classes__RelationClasses.match____PER cupicef.prop B R_27_ + (__ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.PER B R_27_) => + R_27_ (y x0) (x y0)) + (PER__Symmetric0 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Symmetric B R_27_) => + PER__Transitive0 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Transitive B R_27_) => + PER__Symmetric0 (x y0) (y x0) (H1 y0 x0 (PER__Symmetric x0 y0 H2))) + H0) + H. + +def let__U19 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +def let__H4 : + A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U19 -> + B : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U19 -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + H0 : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER B R_27_) -> + x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + y : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + z : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + H1 : cupicef.Term cupicef.prop (respectful A B R R_27_ x y) -> + H2 : cupicef.Term cupicef.prop (respectful A B R R_27_ y z) -> + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H3 : cupicef.Term cupicef.prop (R x0 y0) -> + cupicef.Term cupicef.prop (R x0 x0) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U19 => + B : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U19 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + H0 : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER B R_27_) => + x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + y : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + z : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + H1 : cupicef.Term cupicef.prop (respectful A B R R_27_ x y) => + H2 : cupicef.Term cupicef.prop (respectful A B R R_27_ y z) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H3 : cupicef.Term cupicef.prop (R x0 y0) => + Coq__Classes__RelationClasses.transitivity A R + (Coq__Classes__RelationClasses.PER__Transitive A R H) x0 y0 x0 H3 + (Coq__Classes__RelationClasses.symmetry A R + (Coq__Classes__RelationClasses.PER__Symmetric A R H) x0 y0 H3). + +thm respectful__per__obligation__2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + H0 : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER B R_27_) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Transitive + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (respectful A B R R_27_)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U19 => + B : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U19 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + H0 : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER B R_27_) => + x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + y : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + z : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + H1 : cupicef.Term cupicef.prop (respectful A B R R_27_ x y) => + H2 : cupicef.Term cupicef.prop (respectful A B R R_27_ y z) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H3 : cupicef.Term cupicef.prop (R x0 y0) => + Coq__Classes__RelationClasses.transitivity B R_27_ + (Coq__Classes__RelationClasses.PER__Transitive B R_27_ H0) (x x0) ( + y x0) (z y0) (H1 x0 x0 (let__H4 A B R H R_27_ H0 x y z H1 H2 x0 y0 H3)) + (H2 x0 y0 H3). + +def let__U20 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +def respectful__per : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + H0 : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER B R_27_) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.PER + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (respectful A B R R_27_)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U20 => + B : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U20 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + H0 : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER B R_27_) => + Coq__Classes__RelationClasses.Build__PER + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (respectful A B R R_27_) + ((R0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H1 : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R0) => + R_27_0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + H2 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.PER B R_27_0) => + respectful__per__obligation__1 A B R0 H1 R_27_0 H2) R H R_27_ H0) + ((R0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H1 : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R0) => + R_27_0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + H2 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.PER B R_27_0) => + respectful__per__obligation__2 A B R0 H1 R_27_0 H2) R H R_27_ H0). + +def let__U21 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +def let__i : + A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U21 -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + mR : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + RA + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + RA Coq__Init__Logic.iff)) + R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H : cupicef.Term cupicef.prop (RA x y) -> + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H0 : cupicef.Term cupicef.prop (RA x0 y0) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.iff (R x x0) (R y y0)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U21 => + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + mR : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + RA + (respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + RA Coq__Init__Logic.iff)) + R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : cupicef.Term cupicef.prop (RA x y) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (RA x0 y0) => + mR x y H x0 y0 H0. + +def let__H5 : + A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U21 -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + mR : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + RA + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + RA Coq__Init__Logic.iff)) + R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H : cupicef.Term cupicef.prop (RA x y) -> + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H0 : cupicef.Term cupicef.prop (RA x0 y0) -> + H1 : + (__ : cupicef.Term cupicef.prop (R x x0) -> + cupicef.Term cupicef.prop (R y y0)) -> + H2 : + (__ : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop (R x x0)) -> + H3 : + (__ : cupicef.Term cupicef.prop (R x x0) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H4 : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop (R x x0) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U21 => + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + mR : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + RA + (respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + RA Coq__Init__Logic.iff)) + R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : cupicef.Term cupicef.prop (RA x y) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (RA x0 y0) => + H1 : + (__ : cupicef.Term cupicef.prop (R x x0) -> + cupicef.Term cupicef.prop (R y y0)) => + H2 : + (__ : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop (R x x0)) => + H3 : + (__ : cupicef.Term cupicef.prop (R x x0) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop (R y y0) => + H2 H4. + +def let__H2 : + A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U21 -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + mR : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + RA + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + RA Coq__Init__Logic.iff)) + R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H : cupicef.Term cupicef.prop (RA x y) -> + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H0 : cupicef.Term cupicef.prop (RA x0 y0) -> + H1 : + (__ : cupicef.Term cupicef.prop (R x x0) -> + cupicef.Term cupicef.prop (R y y0)) -> + H2 : + (__ : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop (R x x0)) -> + H3 : + (__ : cupicef.Term cupicef.prop (R x x0) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H4 : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop (R y y0) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U21 => + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + mR : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + RA + (respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + RA Coq__Init__Logic.iff)) + R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : cupicef.Term cupicef.prop (RA x y) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (RA x0 y0) => + H1 : + (__ : cupicef.Term cupicef.prop (R x x0) -> + cupicef.Term cupicef.prop (R y y0)) => + H2 : + (__ : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop (R x x0)) => + H3 : + (__ : cupicef.Term cupicef.prop (R x x0) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop (R y y0) => + H1 (let__H5 A RA R mR x y H x0 y0 H0 H1 H2 H3 H4). + +def let__H1 : + A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U21 -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + mR : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + RA + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + RA Coq__Init__Logic.iff)) + R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H : cupicef.Term cupicef.prop (RA x y) -> + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H0 : cupicef.Term cupicef.prop (RA x0 y0) -> + H1 : + (__ : cupicef.Term cupicef.prop (R x x0) -> + cupicef.Term cupicef.prop (R y y0)) -> + H2 : + (__ : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop (R x x0)) -> + H3 : + (__ : cupicef.Term cupicef.prop (R x x0) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H4 : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U21 => + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + mR : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + RA + (respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + RA Coq__Init__Logic.iff)) + R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : cupicef.Term cupicef.prop (RA x y) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (RA x0 y0) => + H1 : + (__ : cupicef.Term cupicef.prop (R x x0) -> + cupicef.Term cupicef.prop (R y y0)) => + H2 : + (__ : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop (R x x0)) => + H3 : + (__ : cupicef.Term cupicef.prop (R x x0) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop (R y y0) => + H3 (let__H5 A RA R mR x y H x0 y0 H0 H1 H2 H3 H4). + +def let__H6 : + A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U21 -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + mR : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + RA + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + RA Coq__Init__Logic.iff)) + R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H : cupicef.Term cupicef.prop (RA x y) -> + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H0 : cupicef.Term cupicef.prop (RA x0 y0) -> + H1 : + (__ : cupicef.Term cupicef.prop (R x x0) -> + cupicef.Term cupicef.prop (R y y0)) -> + H2 : + (__ : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop (R x x0)) -> + H3 : + (__ : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H4 : cupicef.Term cupicef.prop (R x x0) -> + cupicef.Term cupicef.prop (R y y0) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U21 => + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + mR : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + RA + (respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + RA Coq__Init__Logic.iff)) + R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : cupicef.Term cupicef.prop (RA x y) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (RA x0 y0) => + H1 : + (__ : cupicef.Term cupicef.prop (R x x0) -> + cupicef.Term cupicef.prop (R y y0)) => + H2 : + (__ : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop (R x x0)) => + H3 : + (__ : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop (R x x0) => + H1 H4. + +def let__H3 : + A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U21 -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + mR : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + RA + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + RA Coq__Init__Logic.iff)) + R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H : cupicef.Term cupicef.prop (RA x y) -> + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H0 : cupicef.Term cupicef.prop (RA x0 y0) -> + H1 : + (__ : cupicef.Term cupicef.prop (R x x0) -> + cupicef.Term cupicef.prop (R y y0)) -> + H2 : + (__ : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop (R x x0)) -> + H3 : + (__ : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H4 : cupicef.Term cupicef.prop (R x x0) -> + cupicef.Term cupicef.prop (R x x0) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U21 => + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + mR : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + RA + (respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + RA Coq__Init__Logic.iff)) + R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : cupicef.Term cupicef.prop (RA x y) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (RA x0 y0) => + H1 : + (__ : cupicef.Term cupicef.prop (R x x0) -> + cupicef.Term cupicef.prop (R y y0)) => + H2 : + (__ : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop (R x x0)) => + H3 : + (__ : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop (R x x0) => + H2 (let__H6 A RA R mR x y H x0 y0 H0 H1 H2 H3 H4). + +def let__H7 : + A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U21 -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + mR : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + RA + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + RA Coq__Init__Logic.iff)) + R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H : cupicef.Term cupicef.prop (RA x y) -> + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H0 : cupicef.Term cupicef.prop (RA x0 y0) -> + H1 : + (__ : cupicef.Term cupicef.prop (R x x0) -> + cupicef.Term cupicef.prop (R y y0)) -> + H2 : + (__ : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop (R x x0)) -> + H3 : + (__ : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H4 : cupicef.Term cupicef.prop (R x x0) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U21 => + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + mR : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + RA + (respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + RA Coq__Init__Logic.iff)) + R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : cupicef.Term cupicef.prop (RA x y) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (RA x0 y0) => + H1 : + (__ : cupicef.Term cupicef.prop (R x x0) -> + cupicef.Term cupicef.prop (R y y0)) => + H2 : + (__ : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop (R x x0)) => + H3 : + (__ : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop (R x x0) => + H3 (let__H6 A RA R mR x y H x0 y0 H0 H1 H2 H3 H4). + +thm complement__proper__obligation__1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + mR : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + RA + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + RA Coq__Init__Logic.iff)) + R) -> + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + RA + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + RA Coq__Init__Logic.iff)) + (Coq__Classes__RelationClasses.complement A R)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U21 => + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + mR : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + RA + (respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + RA Coq__Init__Logic.iff)) + R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : cupicef.Term cupicef.prop (RA x y) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (RA x0 y0) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x x0) + (__ : cupicef.Term cupicef.prop (R x x0) => R y y0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R y y0) + (__ : cupicef.Term cupicef.prop (R y y0) => R x x0)) + (Coq__Init__Logic.iff + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x x0) + (__ : cupicef.Term cupicef.prop (R x x0) => Coq__Init__Logic.False)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R y y0) + (__ : cupicef.Term cupicef.prop (R y y0) => Coq__Init__Logic.False))) + (H1 : + (__ : cupicef.Term cupicef.prop (R x x0) -> + cupicef.Term cupicef.prop (R y y0)) => + H2 : + (__ : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop (R x x0)) => + Coq__Init__Logic.conj + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x x0) + (__ : cupicef.Term cupicef.prop (R x x0) => + Coq__Init__Logic.False)) + (__ : + (__ : cupicef.Term cupicef.prop (R x x0) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R y y0) + (__0 : cupicef.Term cupicef.prop (R y y0) => + Coq__Init__Logic.False))) + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R y y0) + (__ : cupicef.Term cupicef.prop (R y y0) => + Coq__Init__Logic.False)) + (__ : + (__ : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x x0) + (__0 : cupicef.Term cupicef.prop (R x x0) => + Coq__Init__Logic.False))) + (H3 : + (__ : cupicef.Term cupicef.prop (R x x0) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop (R y y0) => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H1 A RA R mR x y H x0 y0 H0 H1 H2 H3 H4)) + (H3 : + (__ : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop (R x x0) => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H7 A RA R mR x y H x0 y0 H0 H1 H2 H3 H4))) + (let__i A RA R mR x y H x0 y0 H0). + +def let__U22 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +def complement__proper : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + mR : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + RA + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + RA Coq__Init__Logic.iff)) + R) -> + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + RA + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + RA Coq__Init__Logic.iff)) + (Coq__Classes__RelationClasses.complement A R)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U22 => + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + mR : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + RA + (respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + RA Coq__Init__Logic.iff)) + R) => + (RA0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R0 : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + mR0 : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + RA0 + (respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + RA0 Coq__Init__Logic.iff)) + R0) => + complement__proper__obligation__1 A RA0 R0 mR0) RA R mR. + +def let__U23 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm flip__proper__obligation__1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + C : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + RB : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + RC : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation C) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C) -> + mor : + cupicef.Term cupicef.prop + (Proper + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + C))) + (respectful A + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => C)) + RA (respectful B C RB RC)) + f) -> + cupicef.Term cupicef.prop + (Proper + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I B + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C))) + (respectful B + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C)) + RB (respectful A C RA RC)) + (Coq__Program__Basics.flip A B C f)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U23 => + B : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U23 => + C : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U23 => + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + RB : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + RC : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation C) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C) => + mor : + cupicef.Term cupicef.prop + (Proper + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + C))) + (respectful A + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + C)) + RA (respectful B C RB RC)) + f) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + H : cupicef.Term cupicef.prop (RB x y) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (RA x0 y0) => + mor x0 y0 H0 x y H. + +def let__U24 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +def flip__proper : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + C : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + RB : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + RC : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation C) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C) -> + mor : + cupicef.Term cupicef.prop + (Proper + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + C))) + (respectful A + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => C)) + RA (respectful B C RB RC)) + f) -> + cupicef.Term cupicef.prop + (Proper + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I B + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C))) + (respectful B + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C)) + RB (respectful A C RA RC)) + (Coq__Program__Basics.flip A B C f)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U24 => + B : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U24 => + C : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U24 => + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + RB : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + RC : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation C) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C) => + mor : + cupicef.Term cupicef.prop + (Proper + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + C))) + (respectful A + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + C)) + RA (respectful B C RB RC)) + f) => + (RA0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + RB0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + RC0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation C) => + f0 : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C) => + mor0 : + cupicef.Term cupicef.prop + (Proper + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + C))) + (respectful A + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + C)) + RA0 (respectful B C RB0 RC0)) + f0) => + flip__proper__obligation__1 A B C RA0 RB0 RC0 f0 mor0) RA RB RC f mor. + +def let__U25 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm trans__contra__co__morphism__obligation__1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Transitive A R) -> + cupicef.Term cupicef.prop + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R Coq__Program__Basics.impl) + R R) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U25 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Transitive A R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R y x) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H1 : cupicef.Term cupicef.prop (R x0 y0) => + H2 : cupicef.Term cupicef.prop (R x x0) => + Coq__Classes__RelationClasses.transitivity A R H y x y0 H0 + (Coq__Classes__RelationClasses.transitivity A R H x x0 y0 H2 H1). + +def let__U26 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +def trans__contra__co__morphism : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Transitive A R) -> + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R Coq__Program__Basics.impl)) + R) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U26 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Transitive A R) => + (R0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H0 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Transitive A R0) => + trans__contra__co__morphism__obligation__1 A R0 H0) R H. + +#TYPE_CLASS_INSTANCE trans__contra__co__morphism. + +def let__U27 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm trans__contra__inv__impl__morphism__obligation__1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Transitive A R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + (Coq__Program__Basics.flip + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl) + (R x) (R x)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U27 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Transitive A R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R y x0) => + H1 : cupicef.Term cupicef.prop (R x y) => + Coq__Classes__RelationClasses.transitivity A R H x y x0 H1 H0. + +def let__U28 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +def trans__contra__inv__impl__morphism : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Transitive A R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + (Coq__Program__Basics.flip + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Program__Basics.impl)) + (R x)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U28 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Transitive A R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (R0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H0 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Transitive A R0) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + trans__contra__inv__impl__morphism__obligation__1 A R0 H0 x0) R H x. + +#TYPE_CLASS_INSTANCE trans__contra__inv__impl__morphism. + +def let__U29 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm trans__co__impl__morphism__obligation__1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Transitive A R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R Coq__Program__Basics.impl (R x) (R x)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U29 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Transitive A R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x0 y) => + H1 : cupicef.Term cupicef.prop (R x x0) => + Coq__Classes__RelationClasses.transitivity A R H x x0 y H1 H0. + +def let__U30 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +def trans__co__impl__morphism : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Transitive A R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R Coq__Program__Basics.impl) + (R x)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U30 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Transitive A R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (R0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H0 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Transitive A R0) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + trans__co__impl__morphism__obligation__1 A R0 H0 x0) R H x. + +#TYPE_CLASS_INSTANCE trans__co__impl__morphism. + +def let__U31 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm trans__sym__co__inv__impl__morphism__obligation__1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R + (Coq__Program__Basics.flip + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl) + (R x) (R x)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U31 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x0 y) => + H1 : cupicef.Term cupicef.prop (R x y) => + Coq__Classes__RelationClasses.transitivity A R + (Coq__Classes__RelationClasses.PER__Transitive A R H) x y x0 H1 + (Coq__Classes__RelationClasses.symmetry A R + (Coq__Classes__RelationClasses.PER__Symmetric A R H) x0 y H0). + +def let__U32 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +def trans__sym__co__inv__impl__morphism : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R + (Coq__Program__Basics.flip + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Program__Basics.impl)) + (R x)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U32 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (R0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H0 : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R0) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + trans__sym__co__inv__impl__morphism__obligation__1 A R0 H0 x0) R H x. + +#TYPE_CLASS_INSTANCE trans__sym__co__inv__impl__morphism. + +def let__U33 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm trans__sym__contra__impl__morphism__obligation__1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + Coq__Program__Basics.impl (R x) (R x)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U33 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R y x0) => + H1 : cupicef.Term cupicef.prop (R x x0) => + Coq__Classes__RelationClasses.transitivity A R + (Coq__Classes__RelationClasses.PER__Transitive A R H) x x0 y H1 + (Coq__Classes__RelationClasses.symmetry A R + (Coq__Classes__RelationClasses.PER__Symmetric A R H) y x0 H0). + +def let__U34 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +def trans__sym__contra__impl__morphism : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + Coq__Program__Basics.impl) + (R x)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U34 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (R0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H0 : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R0) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + trans__sym__contra__impl__morphism__obligation__1 A R0 H0 x0) R H x. + +#TYPE_CLASS_INSTANCE trans__sym__contra__impl__morphism. + +def let__U35 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm per__partial__app__morphism__obligation__1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R Coq__Init__Logic.iff (R x) (R x)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U35 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x0 y) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x x0) + (__ : cupicef.Term cupicef.prop (R x x0) => R x y)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x y) + (__ : cupicef.Term cupicef.prop (R x y) => R x x0)) + (H1 : cupicef.Term cupicef.prop (R x x0) => + Coq__Classes__RelationClasses.transitivity A R + (Coq__Classes__RelationClasses.PER__Transitive A R H) x x0 y H1 H0) + (H1 : cupicef.Term cupicef.prop (R x y) => + Coq__Classes__RelationClasses.transitivity A R + (Coq__Classes__RelationClasses.PER__Transitive A R H) x y x0 H1 + (Coq__Classes__RelationClasses.symmetry A R + (Coq__Classes__RelationClasses.PER__Symmetric A R H) x0 y H0)). + +def let__U36 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +def per__partial__app__morphism : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R Coq__Init__Logic.iff) + (R x)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U36 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (R0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H0 : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R0) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + per__partial__app__morphism__obligation__1 A R0 H0 x0) R H x. + +#TYPE_CLASS_INSTANCE per__partial__app__morphism. + +def let__U37 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm trans__co__eq__inv__impl__morphism__obligation__1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Transitive A R) -> + cupicef.Term cupicef.prop + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + R + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + (Coq__Program__Basics.flip + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Program__Basics.impl)) + R R) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U37 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Transitive A R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x y) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U37 + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y0) => + H2 : cupicef.Term cupicef.prop (R y y0) => + Coq__Init__Logic.eq__ind__r A y0 + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => R x x1) + (Coq__Classes__RelationClasses.transitivity A R H x y y0 H0 H2) x0 H1. + +def let__U38 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +def trans__co__eq__inv__impl__morphism : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Transitive A R) -> + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + R + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + (Coq__Program__Basics.flip + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Program__Basics.impl))) + R) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U38 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Transitive A R) => + (R0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H0 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Transitive A R0) => + trans__co__eq__inv__impl__morphism__obligation__1 A R0 H0) R H. + +#TYPE_CLASS_INSTANCE trans__co__eq__inv__impl__morphism. + +def let__U39 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm PER__morphism__obligation__1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) -> + cupicef.Term cupicef.prop + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + R + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R Coq__Init__Logic.iff) + R R) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U39 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x y) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H1 : cupicef.Term cupicef.prop (R x0 y0) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x x0) + (__ : cupicef.Term cupicef.prop (R x x0) => R y y0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R y y0) + (__ : cupicef.Term cupicef.prop (R y y0) => R x x0)) + (H2 : cupicef.Term cupicef.prop (R x x0) => + Coq__Classes__RelationClasses.transitivity A R + (Coq__Classes__RelationClasses.PER__Transitive A R H) y x0 y0 + (Coq__Classes__RelationClasses.transitivity A R + (Coq__Classes__RelationClasses.PER__Transitive A R H) y x x0 + (Coq__Classes__RelationClasses.symmetry A R + (Coq__Classes__RelationClasses.PER__Symmetric A R H) x y H0) + H2) + H1) + (H2 : cupicef.Term cupicef.prop (R y y0) => + Coq__Classes__RelationClasses.transitivity A R + (Coq__Classes__RelationClasses.PER__Transitive A R H) x y x0 H0 + (Coq__Classes__RelationClasses.transitivity A R + (Coq__Classes__RelationClasses.PER__Transitive A R H) y y0 x0 H2 + (Coq__Classes__RelationClasses.symmetry A R + (Coq__Classes__RelationClasses.PER__Symmetric A R H) x0 y0 H1))). + +def let__U40 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +def PER__morphism : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) -> + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + R + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R Coq__Init__Logic.iff)) + R) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U40 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R) => + (R0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H0 : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PER A R0) => + PER__morphism__obligation__1 A R0 H0) R H. + +#TYPE_CLASS_INSTANCE PER__morphism. + +def let__U41 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm symmetric__equiv__flip : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Symmetric A R) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.relation__equivalence A R + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U41 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Symmetric A R) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Classes__RelationClasses.pointwise__lifting + Coq__Init__Logic.iff + (Coq__Classes__RelationClasses.Tcons A + Coq__Classes__RelationClasses.Tnil) + (R x) + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x))) + (Coq__Classes__RelationClasses.relation__equivalence A R + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R)) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + R x x0) + (__ : cupicef.Term cupicef.prop (R x x0) => + Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x x0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x x0) + (__ : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x x0) => + R x x0)))) + (Coq__Classes__RelationClasses.pointwise__lifting + Coq__Init__Logic.iff + (Coq__Classes__RelationClasses.Tcons A + Coq__Classes__RelationClasses.Tnil) + (R x) + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x)) + cupicef.I + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x x0) + (__ : cupicef.Term cupicef.prop (R x x0) => + Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x x0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x x0) + (__ : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x x0) => + R x x0)) + (H0 : cupicef.Term cupicef.prop (R x x0) => + (H1 : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (R y x)) => + (H2 : + (__ : cupicef.Term cupicef.prop (R x x0) -> + cupicef.Term cupicef.prop (R x0 x)) => + (H3 : cupicef.Term cupicef.prop (R x0 x) => H3) (H2 H0)) + (H1 x0)) + (H x)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x x0) => + (H1 : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x0 y) -> + cupicef.Term cupicef.prop (R y x0)) => + (H2 : + (__ : cupicef.Term cupicef.prop (R x0 x0) -> + cupicef.Term cupicef.prop (R x0 x0)) => + (H3 : + (__ : cupicef.Term cupicef.prop (R x0 x) -> + cupicef.Term cupicef.prop (R x x0)) => + (H4 : cupicef.Term cupicef.prop (R x x0) => H4) (H3 H0)) + (H1 x)) + (H1 x0)) + (H x0)))). + +def let__U42 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm compose__proper__obligation__1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + C : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + RB : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + RC : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation C) -> + cupicef.Term cupicef.prop + (respectful + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => C)) + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (__ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C))) + (respectful B C RB RC) + (respectful + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C)) + (respectful A B RA RB) (respectful A C RA RC)) + (Coq__Program__Basics.compose A B C) + (Coq__Program__Basics.compose A B C)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U42 => + B : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U42 => + C : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U42 => + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + RB : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + RC : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation C) => + x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C) => + y : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C) => + H : cupicef.Term cupicef.prop (respectful B C RB RC x y) => + x0 : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + y0 : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + H0 : cupicef.Term cupicef.prop (respectful A B RA RB x0 y0) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H1 : cupicef.Term cupicef.prop (RA x1 y1) => + H (x0 x1) (y0 y1) (H0 x1 y1 H1). + +def let__U43 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +def compose__proper : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + C : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + RB : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + RC : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation C) -> + cupicef.Term cupicef.prop + (Proper + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => C)) + (__ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B)) + (__0 : + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C)))) + (respectful + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => C)) + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B)) + (__ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C))) + (respectful B C RB RC) + (respectful + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C)) + (respectful A B RA RB) (respectful A C RA RC))) + (Coq__Program__Basics.compose A B C)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U43 => + B : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U43 => + C : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U43 => + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + RB : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + RC : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation C) => + (RA0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + RB0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + RC0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation C) => + compose__proper__obligation__1 A B C RA0 RB0 RC0) RA RB RC. + +#TYPE_CLASS_INSTANCE compose__proper. + +def let__U44 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm reflexive__eq__dom__reflexive : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Reflexive B R_27_) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Reflexive + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (respectful A B + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + R_27_)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U44 => + B : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U44 => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Reflexive B R_27_) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Classes__RelationClasses.Reflexive + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (respectful A B + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U44 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + R_27_)) + (Coq__Classes__RelationClasses.Reflexive + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (respectful A B + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U44 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + R_27_)) + cupicef.I + (cupicef.cast + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop) + cupicef.prop + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + respectful A B + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U44 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + R_27_ x x)) + (Coq__Classes__RelationClasses.Reflexive + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (respectful A B + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U44 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + R_27_)) + cupicef.I + (x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U44 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + let__U44 + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y) => + R_27_ (x x0) (x y))))) + (respectful A B + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U44 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + R_27_ x x) + cupicef.I + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U44 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y) => + cupicef.cast cupicef.prop cupicef.prop (R_27_ (x x0) (x y)) + (R_27_ (x x0) (x y)) cupicef.I + (cupicef.cast cupicef.prop cupicef.prop + ((x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + R_27_ (x x1) (x y)) x0) + (R_27_ (x x0) (x y)) cupicef.I + (Coq__Init__Logic.eq__ind__r A y + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + R_27_ (x x1) (x y)) + (H (x y)) x0 H0))))). + +#TYPE_CLASS_INSTANCE reflexive__eq__dom__reflexive. + +def let__U45 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm respectful__morphism : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (Coq__Relations__Relation__Definitions.relation B) + (__0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + Coq__Relations__Relation__Definitions.relation + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__1 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B)))))) + (respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (Coq__Relations__Relation__Definitions.relation B) + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + Coq__Relations__Relation__Definitions.relation + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))))) + (Coq__Classes__RelationClasses.relation__equivalence A) + (respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation B)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Relations__Relation__Definitions.relation + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B)))) + (Coq__Classes__RelationClasses.relation__equivalence B) + (Coq__Classes__RelationClasses.relation__equivalence + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))))) + (respectful A B)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U45 => + B : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U45 => + cupicef.cast cupicef.prop cupicef.prop + (respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (Coq__Relations__Relation__Definitions.relation B) + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + Coq__Relations__Relation__Definitions.relation + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))))) + (Coq__Classes__RelationClasses.relation__equivalence A) + (respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation B)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Relations__Relation__Definitions.relation + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B)))) + (Coq__Classes__RelationClasses.relation__equivalence B) + (Coq__Classes__RelationClasses.relation__equivalence + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B)))) + (respectful A B) (respectful A B)) + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (Coq__Relations__Relation__Definitions.relation B) + (__0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + Coq__Relations__Relation__Definitions.relation + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__1 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B)))))) + (respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (Coq__Relations__Relation__Definitions.relation B) + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + Coq__Relations__Relation__Definitions.relation + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))))) + (Coq__Classes__RelationClasses.relation__equivalence A) + (respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation B)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Relations__Relation__Definitions.relation + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B)))) + (Coq__Classes__RelationClasses.relation__equivalence B) + (Coq__Classes__RelationClasses.relation__equivalence + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))))) + (respectful A B)) + cupicef.I + (cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (x : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (y : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Classes__RelationClasses.relation__equivalence A x y) + (H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.relation__equivalence A + x y) => + respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Relations__Relation__Definitions.relation B)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Relations__Relation__Definitions.relation + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B)))) + (Coq__Classes__RelationClasses.relation__equivalence B) + (Coq__Classes__RelationClasses.relation__equivalence + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (respectful A B x) (respectful A B y))))) + (respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (Coq__Relations__Relation__Definitions.relation B) + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + Coq__Relations__Relation__Definitions.relation + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))))) + (Coq__Classes__RelationClasses.relation__equivalence A) + (respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation B)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Relations__Relation__Definitions.relation + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B)))) + (Coq__Classes__RelationClasses.relation__equivalence B) + (Coq__Classes__RelationClasses.relation__equivalence + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B)))) + (respectful A B) (respectful A B)) + cupicef.I + (x : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + y : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.relation__equivalence A x y) => + cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I (Coq__Relations__Relation__Definitions.relation B) + (x0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (Coq__Relations__Relation__Definitions.relation B) + (y0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Classes__RelationClasses.relation__equivalence B x0 + y0) + (H0 : + cupicef.Term cupicef.prop + 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cupicef.I (Coq__Relations__Relation__Definitions.relation B)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Relations__Relation__Definitions.relation + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B)))) + (Coq__Classes__RelationClasses.relation__equivalence B) + (Coq__Classes__RelationClasses.relation__equivalence + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (respectful A B x) (respectful A B y)) + cupicef.I + (x0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + y0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + H0 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.relation__equivalence B x0 y0) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Classes__RelationClasses.predicate__equivalence + (Coq__Classes__RelationClasses.Tcons + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B)) + (Coq__Classes__RelationClasses.Tcons + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B)) + Coq__Classes__RelationClasses.Tnil)) + (respectful A B x x0) (respectful A B y y0)) + (Coq__Classes__RelationClasses.relation__equivalence + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B)) + (respectful A B x x0) (respectful A B y y0)) + cupicef.I + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Classes__RelationClasses.pointwise__lifting + Coq__Init__Logic.iff + (Coq__Classes__RelationClasses.Tcons + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B)) + (Coq__Classes__RelationClasses.Tcons + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B)) + Coq__Classes__RelationClasses.Tnil)) + (respectful A B x x0) (respectful A B y y0)) + (Coq__Classes__RelationClasses.predicate__equivalence + (Coq__Classes__RelationClasses.Tcons + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type 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(cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop)) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B)) + (x1 : + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + B) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop + (cupicef.rule + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B)) + (x2 : + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Init__Logic.iff (respectful A B x x0 x1 x2) + (respectful A B y y0 x1 x2)))) + (Coq__Classes__RelationClasses.pointwise__lifting + Coq__Init__Logic.iff + (Coq__Classes__RelationClasses.Tcons + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B)) + (Coq__Classes__RelationClasses.Tcons + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B)) + Coq__Classes__RelationClasses.Tnil)) + (respectful A B x x0) (respectful A B y y0)) + cupicef.I + (x1 : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + x2 : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.iff + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x3 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (x x3 y1) + (__ : cupicef.Term cupicef.prop (x x3 y1) => + x0 (x1 x3) (x2 y1))))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x3 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (y x3 y1) + (__ : cupicef.Term cupicef.prop (y x3 y1) => + y0 (x1 x3) (x2 y1)))))) + (Coq__Init__Logic.iff (respectful A B x x0 x1 x2) + (respectful A B y y0 x1 x2)) + cupicef.I + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop)))) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop))) + cupicef.I A + (x3 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop)) + cupicef.I A + (y1 : + cupicef.Term + (cupicef.type + (cupicef.s cupicef.z)) + A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop + cupicef.prop) + cupicef.I (x x3 y1) + (__ : + cupicef.Term cupicef.prop + (x x3 y1) => + x0 (x1 x3) (x2 y1))))) + (__ : + (x3 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A -> + y1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A -> + __ : cupicef.Term cupicef.prop (x x3 y1) -> + cupicef.Term cupicef.prop + (x0 (x1 x3) (x2 y1))) => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop))) + cupicef.I A + (x3 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop)) + cupicef.I A + (y1 : + cupicef.Term + (cupicef.type + (cupicef.s cupicef.z)) + A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop + cupicef.prop) + cupicef.I (y x3 y1) + (__0 : + cupicef.Term cupicef.prop + (y x3 y1) => + y0 (x1 x3) (x2 y1)))))) + (cupicef.prod + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop)))) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop))) + cupicef.I A + (x3 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop)) + cupicef.I A + (y1 : + cupicef.Term + (cupicef.type + (cupicef.s cupicef.z)) + A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop + cupicef.prop) + cupicef.I (y x3 y1) + (__ : + cupicef.Term cupicef.prop + (y x3 y1) => + y0 (x1 x3) (x2 y1))))) + (__ : + (x3 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A -> + y1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A -> + __ : cupicef.Term cupicef.prop (y x3 y1) -> + cupicef.Term cupicef.prop + (y0 (x1 x3) (x2 y1))) => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop))) + cupicef.I A + (x3 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop)) + cupicef.I A + (y1 : + cupicef.Term + (cupicef.type + (cupicef.s cupicef.z)) + A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop + cupicef.prop) + cupicef.I (x x3 y1) + (__0 : + cupicef.Term cupicef.prop + (x x3 y1) => + x0 (x1 x3) (x2 y1))))))) + (Coq__Init__Logic.iff + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x3 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (x x3 y1) + (__ : + cupicef.Term cupicef.prop (x x3 y1) => + x0 (x1 x3) (x2 y1))))) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x3 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (y x3 y1) + (__ : + cupicef.Term cupicef.prop (y x3 y1) => + y0 (x1 x3) (x2 y1)))))) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop)))) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop))) + cupicef.I A + (x3 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop)) + cupicef.I A + (y1 : + cupicef.Term + (cupicef.type + (cupicef.s cupicef.z)) + A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop + cupicef.prop) + cupicef.I (x x3 y1) + (__ : + cupicef.Term cupicef.prop + (x x3 y1) => + x0 (x1 x3) (x2 y1))))) + (__ : + (x3 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A -> + y1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A -> + __ : cupicef.Term cupicef.prop (x x3 y1) -> + cupicef.Term cupicef.prop + (x0 (x1 x3) (x2 y1))) => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop))) + cupicef.I A + (x3 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop)) + cupicef.I A + (y1 : + cupicef.Term + (cupicef.type + (cupicef.s cupicef.z)) + A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop + cupicef.prop) + cupicef.I (y x3 y1) + (__0 : + cupicef.Term cupicef.prop + (y x3 y1) => + y0 (x1 x3) (x2 y1)))))) + (cupicef.prod + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop)))) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop))) + cupicef.I A + (x3 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop)) + cupicef.I A + (y1 : + cupicef.Term + (cupicef.type + (cupicef.s cupicef.z)) + A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop + cupicef.prop) + cupicef.I (y x3 y1) + (__ : + cupicef.Term cupicef.prop + (y x3 y1) => + y0 (x1 x3) (x2 y1))))) + (__ : + (x3 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A -> + y1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A -> + __ : cupicef.Term cupicef.prop (y x3 y1) -> + cupicef.Term cupicef.prop + (y0 (x1 x3) (x2 y1))) => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop))) + cupicef.I A + (x3 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop)) + cupicef.I A + (y1 : + cupicef.Term + (cupicef.type + (cupicef.s cupicef.z)) + A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop + cupicef.prop) + cupicef.I (x x3 y1) + (__0 : + cupicef.Term cupicef.prop + (x x3 y1) => + x0 (x1 x3) (x2 y1)))))) + (H1 : + (x3 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + y1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (x x3 y1) -> + cupicef.Term cupicef.prop + (x0 (x1 x3) (x2 y1))) => + x3 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + y1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + H2 : cupicef.Term cupicef.prop (y x3 y1) => + (lemma : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (x0 (x1 x3) (x2 y1)) + (y0 (x1 x3) (x2 y1))) => + iff__flip__impl__subrelation + (y0 (x1 x3) (x2 y1)) (x0 (x1 x3) (x2 y1)) + (Coq__Classes__RelationClasses.symmetry + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff + Coq__Classes__RelationClasses.iff__Symmetric + (x0 (x1 x3) (x2 y1)) (y0 (x1 x3) (x2 y1)) + lemma)) + (H0 (x1 x3) (x2 y1)) + (H1 x3 y1 + ((lemma : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (x x3 y1) + (y x3 y1)) => + iff__flip__impl__subrelation (x x3 y1) + (y x3 y1) lemma) + (H x3 y1) H2))) + (H1 : + (x3 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + y1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (y x3 y1) -> + cupicef.Term cupicef.prop + (y0 (x1 x3) (x2 y1))) => + x3 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + y1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + H2 : cupicef.Term cupicef.prop (x x3 y1) => + (lemma : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (x0 (x1 x3) (x2 y1)) + (y0 (x1 x3) (x2 y1))) => + iff__flip__impl__subrelation + (x0 (x1 x3) (x2 y1)) (y0 (x1 x3) (x2 y1)) + lemma) + (H0 (x1 x3) (x2 y1)) + (H1 x3 y1 + ((lemma : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (x x3 y1) + (y x3 y1)) => + iff__flip__impl__subrelation (y x3 y1) + (x x3 y1) + (Coq__Classes__RelationClasses.symmetry + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff + Coq__Classes__RelationClasses.iff__Symmetric + (x x3 y1) (y x3 y1) lemma)) + (H x3 y1) H2))))))))))). + +#TYPE_CLASS_INSTANCE respectful__morphism. + +def let__U46 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm Reflexive__partial__app__morphism : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + m : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + H : + cupicef.Term cupicef.prop + (Proper + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (respectful A B R R_27_) m) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H0 : cupicef.Term cupicef.prop (ProperProxy A R x) -> + cupicef.Term cupicef.prop (Proper B R_27_ (m x)) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U46 => + B : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U46 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + m : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + H : + cupicef.Term cupicef.prop + (Proper + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (respectful A B R R_27_) m) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (ProperProxy A R x) => + cupicef.cast cupicef.prop cupicef.prop (Proper B R_27_ (m x)) + (Proper B R_27_ (m x)) cupicef.I + (cupicef.cast cupicef.prop cupicef.prop (R_27_ (m x) (m x)) + (Proper B R_27_ (m x)) cupicef.I + (cupicef.cast cupicef.prop cupicef.prop (R_27_ (m x) (m x)) + (R_27_ (m x) (m x)) cupicef.I + (cupicef.cast cupicef.prop cupicef.prop (R_27_ (m x) (m x)) + (R_27_ (m x) (m x)) cupicef.I (H x x H0)))). + +def let__U47 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm flip__respectful : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.relation__equivalence + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (Coq__Program__Basics.flip + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (respectful A B R R_27_)) + (respectful A B + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + (Coq__Program__Basics.flip B B + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27_))) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U47 => + B : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U47 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + cupicef.cast + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop)) + cupicef.prop + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop)) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (x0 : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Init__Logic.and + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (R x1 y) + (__ : cupicef.Term cupicef.prop (R x1 y) => + R_27_ (x0 x1) (x y))))) + (__ : + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x1 y) -> + cupicef.Term cupicef.prop (R_27_ (x0 x1) (x y))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (R y x1) + (__0 : cupicef.Term cupicef.prop (R y x1) => + R_27_ (x0 y) (x x1)))))) + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (R y x1) + (__ : cupicef.Term cupicef.prop (R y x1) => + R_27_ (x0 y) (x x1))))) + (__ : + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x1) -> + cupicef.Term cupicef.prop (R_27_ (x0 y) (x x1))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (R x1 y) + (__0 : cupicef.Term cupicef.prop (R x1 y) => + R_27_ (x0 x1) (x y))))))))) + (Coq__Classes__RelationClasses.relation__equivalence + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (Coq__Program__Basics.flip + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (respectful A B R R_27_)) + (respectful A B + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + (Coq__Program__Basics.flip B B + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27_))) + cupicef.I + (x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + x0 : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Init__Logic.conj + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (R x1 y) + (__ : cupicef.Term cupicef.prop (R x1 y) => + R_27_ (x0 x1) (x y))))) + (__ : + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x1 y) -> + cupicef.Term cupicef.prop (R_27_ (x0 x1) (x y))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (R y x1) + (__0 : cupicef.Term cupicef.prop (R y x1) => + R_27_ (x0 y) (x x1)))))) + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (R y x1) + (__ : cupicef.Term cupicef.prop (R y x1) => + R_27_ (x0 y) (x x1))))) + (__ : + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x1) -> + cupicef.Term cupicef.prop (R_27_ (x0 y) (x x1))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (R x1 y) + (__0 : cupicef.Term cupicef.prop (R x1 y) => + R_27_ (x0 x1) (x y)))))) + (H : + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x1 y) -> + cupicef.Term cupicef.prop (R_27_ (x0 x1) (x y))) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R y x1) => + cupicef.cast cupicef.prop cupicef.prop (R_27_ (x0 y) (x x1)) + (R_27_ (x0 y) (x x1)) cupicef.I (H y x1 H0)) + (H : + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x1) -> + cupicef.Term cupicef.prop (R_27_ (x0 y) (x x1))) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x1 y) => + cupicef.cast cupicef.prop cupicef.prop (R_27_ (x0 x1) (x y)) + (R_27_ (x0 x1) (x y)) cupicef.I (H y x1 H0))). + +def let__U48 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm flip1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation A R_27_ R) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation A + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27_)) + R) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U48 => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation A R_27_ R) => + H. + +def let__U49 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm flip2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation A R R_27_) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation A R + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27_))) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U49 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation A R R_27_) => + H. + +def let__U50 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm eq__subrelation : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Reflexive A R) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.subrelation A + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + R) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U50 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Reflexive A R) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Classes__RelationClasses.subrelation A + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U50 + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + R) + (Coq__Classes__RelationClasses.subrelation A + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U50 + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + R) + cupicef.I + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U50 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + let__U50 + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + R x y)))) + (Coq__Classes__RelationClasses.subrelation A + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U50 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + R) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U50 + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + cupicef.cast cupicef.prop cupicef.prop (R x y) (R x y) cupicef.I + (cupicef.cast cupicef.prop cupicef.prop + ((x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + R x0 y) x) + (R x y) cupicef.I + (Coq__Init__Logic.eq__ind__r A y + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + R x0 y) + (H y) x H0)))). + +def let__U51 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +def proper__flip__proper : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + m : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + mor : cupicef.Term cupicef.prop (Proper A R m) -> + cupicef.Term cupicef.prop + (Proper A + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + m) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U51 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + m : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + mor : cupicef.Term cupicef.prop (Proper A R m) => + mor. + +def let__U52 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm reflexive__proper : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Reflexive A R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (Proper A R x) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U52 => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Reflexive A R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (H0 : cupicef.Term cupicef.prop (R x x) => H0) (H x). + +def let__U53 : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I. + +thm proper__eq : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Proper A + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + x) + +:= A : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + let__U53 => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + reflexive__proper A + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + let__U53 + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + (Coq__Classes__RelationClasses.eq__Reflexive A) x. + +def PartialApplication : + cupicef.Univ cupicef.prop. + +#TYPE_CLASS PartialApplication. + +def Build__PartialApplication : + cupicef.Term cupicef.prop PartialApplication. + +[ ] cupicef.code_guarded? _ Build__PartialApplication --> cupicef.guarded. + +def match____PartialApplication : + s : cupicef.Sort -> + P : (cupicef.Term cupicef.prop PartialApplication -> cupicef.Univ s) -> + case__Build__PartialApplication : + cupicef.Term s (P Build__PartialApplication) -> + x : cupicef.Term cupicef.prop PartialApplication -> + cupicef.Term s (P x). + +[ s, P, case__Build__PartialApplication] + match____PartialApplication s P case__Build__PartialApplication + Build__PartialApplication + --> + case__Build__PartialApplication. + +[ s, P, s'] + match____PartialApplication s + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____PartialApplication s' (x => P x). + +def normalization__done : + cupicef.Univ cupicef.prop. + +def did__normalization : + cupicef.Term cupicef.prop normalization__done. + +def Params : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + of : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + arity : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop. + +#TYPE_CLASS Params. + +def Build__Params : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + of : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + arity : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (Params A of arity). + +[ ] cupicef.code_guarded? _ (Build__Params _ _ _) --> cupicef.guarded. + +def match____Params : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + of : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + arity : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + P : (cupicef.Term cupicef.prop (Params A of arity) -> cupicef.Univ s) -> + case__Build__Params : cupicef.Term s (P (Build__Params A of arity)) -> + x : cupicef.Term cupicef.prop (Params A of arity) -> + cupicef.Term s (P x). + +[ s, A, of, arity, P, case__Build__Params] + match____Params s A of arity P case__Build__Params + (Build__Params A of arity) + --> + case__Build__Params. + +[ s, A, of, arity, P, s'] + match____Params s A of arity + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____Params s' A of arity (x => P x). + +thm proper__proper : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__RelationClasses.relation__equivalence A) + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + Coq__Init__Logic.iff)) + (Proper A)) + +:= cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop) + (cupicef.rule (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop) + (cupicef.prod (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop + (cupicef.rule (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop) + cupicef.I + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__RelationClasses.relation__equivalence A) + (respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + Coq__Init__Logic.iff)) + (Proper A))) + (cupicef.prod (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop + (cupicef.rule (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop) + cupicef.I + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__RelationClasses.relation__equivalence A) + (respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + Coq__Init__Logic.iff)) + (Proper A))) + cupicef.I + (cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop) + (cupicef.rule (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop) + (cupicef.prod (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop + (cupicef.rule (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop) + cupicef.I + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__RelationClasses.relation__equivalence A) + (respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + Coq__Init__Logic.iff) + (Proper A) (Proper A))) + (cupicef.prod (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop + (cupicef.rule (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop) + cupicef.I + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Relations__Relation__Definitions.relation A) + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__RelationClasses.relation__equivalence A) + (respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + Coq__Init__Logic.iff)) + (Proper A))) + cupicef.I + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (x : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (y : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Classes__RelationClasses.relation__equivalence A x + y) + (H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.relation__equivalence + A x y) => + respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + Coq__Init__Logic.iff (Proper A x) (Proper A y))))) + (respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__RelationClasses.relation__equivalence A) + (respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + Coq__Init__Logic.iff) + (Proper A) (Proper A)) + cupicef.I + (x : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + y : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.relation__equivalence A x y) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y0) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y0) => + Coq__Init__Logic.iff (Proper A x x0) (Proper A y y0))))) + (respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + Coq__Init__Logic.iff (Proper A x) (Proper A y)) + cupicef.I + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y0) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.iff (Proper A x x0) (Proper A y y0)) + (Coq__Init__Logic.iff (Proper A x x0) (Proper A y y0)) + cupicef.I + (cupicef.cast cupicef.prop cupicef.prop + ((x1 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.iff (Proper A x x1) (Proper A y y0)) + x0) + (Coq__Init__Logic.iff (Proper A x x0) (Proper A y y0)) + cupicef.I + (Coq__Init__Logic.eq__ind__r A y0 + (x1 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.iff (Proper A x x1) (Proper A y y0)) + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (Proper A x y0) + (__ : cupicef.Term cupicef.prop (Proper A x y0) => + Proper A y y0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (Proper A y y0) + (__ : cupicef.Term cupicef.prop (Proper A y y0) => + Proper A x y0)) + (cupicef.cast + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (Proper A x y0) + (H1 : + cupicef.Term cupicef.prop (Proper A x y0) => + y y0 y0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (Proper A x y0) + (__ : + cupicef.Term cupicef.prop (Proper A x y0) => + Proper A y y0)) + cupicef.I + (H1 : cupicef.Term cupicef.prop (Proper A x y0) => + iff__flip__impl__subrelation (y y0 y0) + (x y0 y0) + (Coq__Classes__RelationClasses.symmetry + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff + Coq__Classes__RelationClasses.iff__Symmetric + (x y0 y0) (y y0 y0) (H y0 y0)) + H1)) + (cupicef.cast + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (Proper A y y0) + (H1 : + cupicef.Term cupicef.prop (Proper A y y0) => + x y0 y0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (Proper A y y0) + (__ : + cupicef.Term cupicef.prop (Proper A y y0) => + Proper A x y0)) + cupicef.I + (H1 : cupicef.Term cupicef.prop (Proper A y y0) => + iff__flip__impl__subrelation (x y0 y0) + (y y0 y0) (H y0 y0) H1))) + x0 H0)))))). + +#TYPE_CLASS_INSTANCE proper__proper. + +def Normalizes : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + m : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + m_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + m : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + m_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + Coq__Classes__RelationClasses.relation__equivalence A m m_27_. + +#TYPE_CLASS Normalizes. + +def normalizes : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + m : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + m_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + Normalizes0 : cupicef.Term cupicef.prop (Normalizes A m m_27_) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.relation__equivalence A m m_27_) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + m : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + m_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + Normalizes0 : cupicef.Term cupicef.prop (Normalizes A m m_27_) => + Normalizes0. + +thm proper__normalizes__proper : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + R1 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Normalizes A R0 R1) -> + m : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H0 : cupicef.Term cupicef.prop (Proper A R1 m) -> + cupicef.Term cupicef.prop (Proper A R0 m) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R1 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (Normalizes A R0 R1) => + m : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (Proper A R1 m) => + (lemma : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.relation__equivalence A R0 R1) => + subrelation__proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (m0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__RelationClasses.relation__equivalence A) + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + Coq__Init__Logic.iff)) + (Proper A) (proper__proper A) + (respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (m0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__RelationClasses.relation__equivalence A) + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + (Coq__Program__Basics.flip + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Program__Basics.impl))) + Coq__Init__Datatypes.tt + (subrelation__respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (m0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__RelationClasses.relation__equivalence A) + (Coq__Classes__RelationClasses.relation__equivalence A) + (subrelation__refl + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (Coq__Classes__RelationClasses.relation__equivalence A)) + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + Coq__Init__Logic.iff) + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + (Coq__Program__Basics.flip + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Program__Basics.impl)) + (subrelation__respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + (subrelation__refl A + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A))) + Coq__Init__Logic.iff + (Coq__Program__Basics.flip + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Program__Basics.impl) + iff__flip__impl__subrelation)) + R0 R1 lemma m m (eq__proper__proxy A m)) + H H0. + +thm flip__atom : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Term cupicef.prop + (Normalizes A R + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Classes__RelationClasses.pointwise__lifting + Coq__Init__Logic.iff + (Coq__Classes__RelationClasses.Tcons A + Coq__Classes__RelationClasses.Tnil) + (R x) + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + x))) + (Normalizes A R + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R))) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + R x x0) + (__ : cupicef.Term cupicef.prop (R x x0) => + Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + x x0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + x x0) + (__ : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x x0) => + R x x0)))) + (Coq__Classes__RelationClasses.pointwise__lifting + Coq__Init__Logic.iff + (Coq__Classes__RelationClasses.Tcons A + Coq__Classes__RelationClasses.Tnil) + (R x) + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + x)) + cupicef.I + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x x0) + (__ : cupicef.Term cupicef.prop (R x x0) => + Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + x x0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + x x0) + (__ : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + x x0) => + R x x0)) + (H : cupicef.Term cupicef.prop (R x x0) => H) + (H : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + x x0) => + H))). + +def let__H8 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + R_27__27__27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + NA : + cupicef.Term cupicef.prop + (Normalizes A R + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27__27__27_)) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + R_27__27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) -> + NB : + cupicef.Term cupicef.prop + (Normalizes B R_27_ + (Coq__Program__Basics.flip B B + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27__27_)) -> + x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + x0 : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + H : + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x1 y) -> + cupicef.Term cupicef.prop (R_27_ (x x1) (x0 y))) -> + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H0 : cupicef.Term cupicef.prop (R_27__27__27_ x1 y) -> + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + x3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __ : cupicef.Term cupicef.prop (R_27_ x2 x3) -> + cupicef.Term cupicef.prop (R_27__27_ x3 x2) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R_27__27__27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + NA : + cupicef.Term cupicef.prop + (Normalizes A R + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27__27__27_)) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + R_27__27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) => + NB : + cupicef.Term cupicef.prop + (Normalizes B R_27_ + (Coq__Program__Basics.flip B B + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27__27_)) => + x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + x0 : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + H : + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x1 y) -> + cupicef.Term cupicef.prop (R_27_ (x x1) (x0 y))) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R_27__27__27_ x1 y) => + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + x3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R_27_ x2 x3) + (__ : cupicef.Term cupicef.prop (R_27_ x2 x3) => + Coq__Program__Basics.flip B B + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R_27__27_ x2 x3)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Program__Basics.flip B B + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R_27__27_ x2 x3) + (__ : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip B B + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27__27_ x2 x3) => + R_27_ x2 x3)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (R_27_ x2 x3) + (__ : cupicef.Term cupicef.prop (R_27_ x2 x3) => + R_27__27_ x3 x2)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (R_27__27_ x3 x2) + (__ : cupicef.Term cupicef.prop (R_27__27_ x3 x2) => + R_27_ x2 x3))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R_27_ x2 x3) + (__0 : cupicef.Term cupicef.prop (R_27_ x2 x3) => R_27__27_ x3 x2)) + (__ : + (__ : cupicef.Term cupicef.prop (R_27_ x2 x3) -> + cupicef.Term cupicef.prop (R_27__27_ x3 x2)) => + __0 : + (__0 : cupicef.Term cupicef.prop (R_27__27_ x3 x2) -> + cupicef.Term cupicef.prop (R_27_ x2 x3)) => + __) + (NB x2 x3). + +def let__H9 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + R_27__27__27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + NA : + cupicef.Term cupicef.prop + (Normalizes A R + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27__27__27_)) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + R_27__27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) -> + NB : + cupicef.Term cupicef.prop + (Normalizes B R_27_ + (Coq__Program__Basics.flip B B + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27__27_)) -> + x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + x0 : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + H : + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x1 y) -> + cupicef.Term cupicef.prop (R_27_ (x x1) (x0 y))) -> + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H0 : cupicef.Term cupicef.prop (R_27__27__27_ x1 y) -> + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R_27__27__27_ x3 x2) -> + cupicef.Term cupicef.prop (R x2 x3) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R_27__27__27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + NA : + cupicef.Term cupicef.prop + (Normalizes A R + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27__27__27_)) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + R_27__27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) => + NB : + cupicef.Term cupicef.prop + (Normalizes B R_27_ + (Coq__Program__Basics.flip B B + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27__27_)) => + x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + x0 : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + H : + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x1 y) -> + cupicef.Term cupicef.prop (R_27_ (x x1) (x0 y))) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R_27__27__27_ x1 y) => + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x2 x3) + (__ : cupicef.Term cupicef.prop (R x2 x3) => + Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R_27__27__27_ x2 x3)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R_27__27__27_ x2 x3) + (__ : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27__27__27_ x2 x3) => + R x2 x3)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x2 x3) + (__ : cupicef.Term cupicef.prop (R x2 x3) => + R_27__27__27_ x3 x2)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (R_27__27__27_ x3 x2) + (__ : cupicef.Term cupicef.prop (R_27__27__27_ x3 x2) => + R x2 x3))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (R_27__27__27_ x3 x2) + (__0 : cupicef.Term cupicef.prop (R_27__27__27_ x3 x2) => R x2 x3)) + (__ : + (__ : cupicef.Term cupicef.prop (R x2 x3) -> + cupicef.Term cupicef.prop (R_27__27__27_ x3 x2)) => + __0 : + (__0 : cupicef.Term cupicef.prop (R_27__27__27_ x3 x2) -> + cupicef.Term cupicef.prop (R x2 x3)) => + __0) + (NA x2 x3). + +def let__H10 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + R_27__27__27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + NA : + cupicef.Term cupicef.prop + (Normalizes A R + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27__27__27_)) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + R_27__27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) -> + NB : + cupicef.Term cupicef.prop + (Normalizes B R_27_ + (Coq__Program__Basics.flip B B + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27__27_)) -> + x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + x0 : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + H : + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R_27__27__27_ x1 y) -> + cupicef.Term cupicef.prop (R_27__27_ (x0 x1) (x y))) -> + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H0 : cupicef.Term cupicef.prop (R x1 y) -> + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + x3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __ : cupicef.Term cupicef.prop (R_27__27_ x3 x2) -> + cupicef.Term cupicef.prop (R_27_ x2 x3) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R_27__27__27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + NA : + cupicef.Term cupicef.prop + (Normalizes A R + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27__27__27_)) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + R_27__27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) => + NB : + cupicef.Term cupicef.prop + (Normalizes B R_27_ + (Coq__Program__Basics.flip B B + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27__27_)) => + x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + x0 : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + H : + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R_27__27__27_ x1 y) -> + cupicef.Term cupicef.prop (R_27__27_ (x0 x1) (x y))) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x1 y) => + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + x3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R_27_ x2 x3) + (__ : cupicef.Term cupicef.prop (R_27_ x2 x3) => + Coq__Program__Basics.flip B B + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R_27__27_ x2 x3)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Program__Basics.flip B B + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R_27__27_ x2 x3) + (__ : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip B B + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27__27_ x2 x3) => + R_27_ x2 x3)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (R_27_ x2 x3) + (__ : cupicef.Term cupicef.prop (R_27_ x2 x3) => + R_27__27_ x3 x2)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (R_27__27_ x3 x2) + (__ : cupicef.Term cupicef.prop (R_27__27_ x3 x2) => + R_27_ x2 x3))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R_27__27_ x3 x2) + (__0 : cupicef.Term cupicef.prop (R_27__27_ x3 x2) => R_27_ x2 x3)) + (__ : + (__ : cupicef.Term cupicef.prop (R_27_ x2 x3) -> + cupicef.Term cupicef.prop (R_27__27_ x3 x2)) => + __0 : + (__0 : cupicef.Term cupicef.prop (R_27__27_ x3 x2) -> + cupicef.Term cupicef.prop (R_27_ x2 x3)) => + __0) + (NB x2 x3). + +def let__H11 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + R_27__27__27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + NA : + cupicef.Term cupicef.prop + (Normalizes A R + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27__27__27_)) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + R_27__27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) -> + NB : + cupicef.Term cupicef.prop + (Normalizes B R_27_ + (Coq__Program__Basics.flip B B + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27__27_)) -> + x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + x0 : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + H : + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R_27__27__27_ x1 y) -> + cupicef.Term cupicef.prop (R_27__27_ (x0 x1) (x y))) -> + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H0 : cupicef.Term cupicef.prop (R x1 y) -> + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x2 x3) -> + cupicef.Term cupicef.prop (R_27__27__27_ x3 x2) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R_27__27__27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + NA : + cupicef.Term cupicef.prop + (Normalizes A R + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27__27__27_)) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + R_27__27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) => + NB : + cupicef.Term cupicef.prop + (Normalizes B R_27_ + (Coq__Program__Basics.flip B B + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27__27_)) => + x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + x0 : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + H : + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R_27__27__27_ x1 y) -> + cupicef.Term cupicef.prop (R_27__27_ (x0 x1) (x y))) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x1 y) => + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x2 x3) + (__ : cupicef.Term cupicef.prop (R x2 x3) => + Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R_27__27__27_ x2 x3)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R_27__27__27_ x2 x3) + (__ : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27__27__27_ x2 x3) => + R x2 x3)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x2 x3) + (__ : cupicef.Term cupicef.prop (R x2 x3) => + R_27__27__27_ x3 x2)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (R_27__27__27_ x3 x2) + (__ : cupicef.Term cupicef.prop (R_27__27__27_ x3 x2) => + R x2 x3))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x2 x3) + (__0 : cupicef.Term cupicef.prop (R x2 x3) => R_27__27__27_ x3 x2)) + (__ : + (__ : cupicef.Term cupicef.prop (R x2 x3) -> + cupicef.Term cupicef.prop (R_27__27__27_ x3 x2)) => + __0 : + (__0 : cupicef.Term cupicef.prop (R_27__27__27_ x3 x2) -> + cupicef.Term cupicef.prop (R x2 x3)) => + __) + (NA x2 x3). + +thm flip__arrow : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + R_27__27__27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + NA : + cupicef.Term cupicef.prop + (Normalizes A R + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27__27__27_)) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + R_27__27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) -> + NB : + cupicef.Term cupicef.prop + (Normalizes B R_27_ + (Coq__Program__Basics.flip B B + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27__27_)) -> + cupicef.Term cupicef.prop + (Normalizes + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B))) + (respectful A B R R_27_) + (Coq__Program__Basics.flip + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (respectful A B R_27__27__27_ R_27__27_))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R_27__27__27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + NA : + cupicef.Term cupicef.prop + (Normalizes A R + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27__27__27_)) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + R_27__27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) => + NB : + cupicef.Term cupicef.prop + (Normalizes B R_27_ + (Coq__Program__Basics.flip B B + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27__27_)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Classes__RelationClasses.relation__equivalence + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B))) + (respectful A B R R_27_) + (Coq__Program__Basics.flip + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (respectful A B R_27__27__27_ R_27__27_))) + (Normalizes + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B))) + (respectful A B R R_27_) + (Coq__Program__Basics.flip + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (respectful A B R_27__27__27_ R_27__27_))) + cupicef.I + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Classes__RelationClasses.predicate__equivalence + (Coq__Classes__RelationClasses.Tcons + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (Coq__Classes__RelationClasses.Tcons + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + Coq__Classes__RelationClasses.Tnil)) + (respectful A B R R_27_) + (Coq__Program__Basics.flip + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + 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cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + B))) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (respectful A B R_27__27__27_ R_27__27_) x x0)))) + cupicef.I + (cupicef.cast + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop)) + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop)) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B)) + (x : + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + B) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop + (cupicef.rule + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B)) + (x0 : + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Init__Logic.and + (cupicef.prod + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop)))) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop))) + cupicef.I A + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop)) + cupicef.I A + (y : + cupicef.Term + (cupicef.type + (cupicef.s cupicef.z)) + A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop + cupicef.prop) + cupicef.I (R x1 y) + (__ : + cupicef.Term cupicef.prop + (R x1 y) => + R_27_ (x x1) (x0 y))))) + (__ : + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A -> + y : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A -> + __ : cupicef.Term cupicef.prop (R x1 y) -> + cupicef.Term cupicef.prop + (R_27_ (x x1) (x0 y))) => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop))) + cupicef.I A + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop)) + cupicef.I A + (y : + cupicef.Term + (cupicef.type + (cupicef.s cupicef.z)) + A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop + cupicef.prop) + cupicef.I (R_27__27__27_ x1 y) + (__0 : + cupicef.Term cupicef.prop + (R_27__27__27_ x1 y) => + R_27__27_ (x0 x1) (x y)))))) + (cupicef.prod + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop)))) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop))) + cupicef.I A + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop)) + cupicef.I A + (y : + cupicef.Term + (cupicef.type + (cupicef.s cupicef.z)) + A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop + cupicef.prop) + cupicef.I (R_27__27__27_ x1 y) + (__ : + cupicef.Term cupicef.prop + (R_27__27__27_ x1 y) => + R_27__27_ (x0 x1) (x y))))) + (__ : + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A -> + y : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A -> + __ : + cupicef.Term cupicef.prop + (R_27__27__27_ x1 y) -> + cupicef.Term cupicef.prop + (R_27__27_ (x0 x1) (x y))) => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop))) + cupicef.I A + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop)) + cupicef.I A + (y : + cupicef.Term + (cupicef.type + (cupicef.s cupicef.z)) + A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop + cupicef.prop) + cupicef.I (R x1 y) + (__0 : + cupicef.Term cupicef.prop + (R x1 y) => + R_27_ (x x1) (x0 y))))))))) + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop)) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B)) + (x : + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + B) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop + (cupicef.rule + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + B)) + (x0 : + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Init__Logic.iff + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (R x1 y) + (__ : + cupicef.Term cupicef.prop (R x1 y) => + R_27_ (x x1) (x0 y))))) + (Coq__Program__Basics.flip + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type + (cupicef.s cupicef.z)) + A => + B))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type + (cupicef.s cupicef.z)) + A => + B))) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (f : + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) B) => + g : + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) B) => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop))) + cupicef.I A + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + cupicef.prop)) + cupicef.I A + (y : + cupicef.Term + (cupicef.type + (cupicef.s cupicef.z)) + A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop + cupicef.prop) + cupicef.I (R_27__27__27_ x1 y) + (__ : + cupicef.Term cupicef.prop + (R_27__27__27_ x1 y) => + R_27__27_ (f x1) (g y))))) + x x0)))) + cupicef.I + (x : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + x0 : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Init__Logic.conj + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (R x1 y) + (__ : cupicef.Term cupicef.prop (R x1 y) => + R_27_ (x x1) (x0 y))))) + (__ : + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + y : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x1 y) -> + cupicef.Term cupicef.prop (R_27_ (x x1) (x0 y))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (R_27__27__27_ x1 y) + (__0 : + cupicef.Term cupicef.prop + (R_27__27__27_ x1 y) => + R_27__27_ (x0 x1) (x y)))))) + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (R_27__27__27_ x1 y) + (__ : + cupicef.Term cupicef.prop + (R_27__27__27_ x1 y) => + R_27__27_ (x0 x1) (x y))))) + (__ : + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + y : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (R_27__27__27_ x1 y) -> + cupicef.Term cupicef.prop + (R_27__27_ (x0 x1) (x y))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (R x1 y) + (__0 : cupicef.Term cupicef.prop (R x1 y) => + R_27_ (x x1) (x0 y)))))) + (H : + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + y : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x1 y) -> + cupicef.Term cupicef.prop (R_27_ (x x1) (x0 y))) => + x1 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R_27__27__27_ x1 y) => + let__H8 A B R R_27__27__27_ NA R_27_ R_27__27_ NB x + x0 H x1 y H0 (x y) (x0 x1) + (H y x1 + (let__H9 A B R R_27__27__27_ NA R_27_ R_27__27_ + NB x x0 H x1 y H0 y x1 H0))) + (H : + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + y : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop (R_27__27__27_ x1 y) -> + cupicef.Term cupicef.prop + (R_27__27_ (x0 x1) (x y))) => + x1 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x1 y) => + let__H10 A B R R_27__27__27_ NA R_27_ R_27__27_ NB x + x0 H x1 y H0 (x x1) (x0 y) + (H y x1 + (let__H11 A B R R_27__27__27_ NA R_27_ R_27__27_ + NB x x0 H x1 y H0 x1 y H0))))))))). + +thm proper__sym__flip : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R1 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Symmetric A R1) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R2 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + H0 : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B))) + (respectful A B R1 R2) f) -> + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B))) + (respectful A B R1 + (Coq__Program__Basics.flip B B + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R2)) + f) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R1 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + Sym : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Symmetric A R1) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R2 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Hf : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B))) + (respectful A B R1 R2) f) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R1 x x_27_) + (Hxx_27_ : cupicef.Term cupicef.prop (R1 x x_27_) => + R2 (f x_27_) (f x))))) + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B))) + (respectful A B R1 + (Coq__Program__Basics.flip B B + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R2)) + f) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hxx_27_ : cupicef.Term cupicef.prop (R1 x x_27_) => + Hf x_27_ x (Sym x x_27_ Hxx_27_)). + +thm proper__sym__flip__2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R1 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Symmetric A R1) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R2 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + H0 : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Symmetric B R2) -> + C : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R3 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation C) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C) -> + H1 : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + C)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + C))) + R1 (respectful B C R2 R3)) + f) -> + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + C)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + C))) + R1 + (respectful B C R2 + (Coq__Program__Basics.flip C C + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R3))) + f) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R1 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + Sym1 : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Symmetric A R1) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R2 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + Sym2 : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Symmetric B R2) => + C : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R3 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation C) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C) => + Hf : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + C)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + C))) + R1 (respectful B C R2 R3)) + f) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R1 x x_27_) + (Hxx_27_ : cupicef.Term cupicef.prop (R1 x x_27_) => + respectful B C R2 + (Coq__Program__Basics.flip C C + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R3) + (f x) (f x_27_))))) + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + C)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + C))) + R1 + (respectful B C R2 + (Coq__Program__Basics.flip C C + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R3))) + f) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hxx_27_ : cupicef.Term cupicef.prop (R1 x x_27_) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I B + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I B + (y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (R2 y y_27_) + (Hyy_27_ : cupicef.Term cupicef.prop (R2 y y_27_) => + R3 (f x_27_ y_27_) (f x y))))) + (respectful B C R2 + (Coq__Program__Basics.flip C C + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R3) + (f x) (f x_27_)) + cupicef.I + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + Hyy_27_ : cupicef.Term cupicef.prop (R2 y y_27_) => + Hf x_27_ x (Sym1 x x_27_ Hxx_27_) y_27_ y (Sym2 y y_27_ Hyy_27_))). + +thm proper__sym__impl__iff : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Symmetric A R) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + H0 : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R Coq__Program__Basics.impl) + f) -> + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R Coq__Init__Logic.iff) + f) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + Sym : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Symmetric A R) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Hf : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R Coq__Program__Basics.impl) + f) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x x_27_) + (Hxx_27_ : cupicef.Term cupicef.prop (R x x_27_) => + Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + f x) (__ : cupicef.Term cupicef.prop (f x) => f x_27_)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (f x_27_) + (__ : cupicef.Term cupicef.prop (f x_27_) => f x)))))) + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R Coq__Init__Logic.iff) + f) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hxx_27_ : cupicef.Term cupicef.prop (R x x_27_) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (f x) + (__ : cupicef.Term cupicef.prop (f x) => f x_27_)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (f x_27_) + (__ : cupicef.Term cupicef.prop (f x_27_) => f x)) + (H : cupicef.Term cupicef.prop (f x) => Hf x x_27_ Hxx_27_ H) + (H : cupicef.Term cupicef.prop (f x_27_) => + Hf x_27_ x (Sym x x_27_ Hxx_27_) H)). + +thm proper__sym__impl__iff__2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Symmetric A R) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Symmetric B R_27_) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) -> + H1 : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + R + (respectful B + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27_ Coq__Program__Basics.impl)) + f) -> + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + R + (respectful B + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27_ Coq__Init__Logic.iff)) + f) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + Sym : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Symmetric A R) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + Sym_27_ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Symmetric B R_27_) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) => + Hf : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + R + (respectful B + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27_ Coq__Program__Basics.impl)) + f) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x x_27_) + (Hxx_27_ : cupicef.Term cupicef.prop (R x x_27_) => + respectful B + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27_ Coq__Init__Logic.iff (f x) (f x_27_))))) + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I B + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + R + (respectful B + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R_27_ Coq__Init__Logic.iff)) + f) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hxx_27_ : cupicef.Term cupicef.prop (R x x_27_) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I B + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I B + (y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (R_27_ y y_27_) + (Hyy_27_ : cupicef.Term cupicef.prop (R_27_ y y_27_) => + Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (f x y) + (__ : cupicef.Term cupicef.prop (f x y) => + f x_27_ y_27_)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (f x_27_ y_27_) + (__ : cupicef.Term cupicef.prop (f x_27_ y_27_) => + f x y)))))) + (respectful B + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R_27_ Coq__Init__Logic.iff (f x) (f x_27_)) + cupicef.I + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + Hyy_27_ : cupicef.Term cupicef.prop (R_27_ y y_27_) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (f x y) + (__ : cupicef.Term cupicef.prop (f x y) => f x_27_ y_27_)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (f x_27_ y_27_) + (__ : cupicef.Term cupicef.prop (f x_27_ y_27_) => f x y)) + (H : cupicef.Term cupicef.prop (f x y) => + Hf x x_27_ Hxx_27_ y y_27_ Hyy_27_ H) + (H : cupicef.Term cupicef.prop (f x_27_ y_27_) => + Hf x_27_ x (Sym x x_27_ Hxx_27_) y_27_ y + (Sym_27_ y y_27_ Hyy_27_) H))). + +def let__H0 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + equ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + preo : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PreOrder A R) -> + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.PartialOrder A eqA equ R preo) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hx : cupicef.Term cupicef.prop (eqA x x_27_) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hy : cupicef.Term cupicef.prop (eqA y y_27_) -> + Hr : cupicef.Term cupicef.prop (R x y) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eqA x x_27_) + (__ : cupicef.Term cupicef.prop (eqA x x_27_) => + Coq__Init__Logic.and (R x x_27_) (R x_27_ x))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) => + eqA x x_27_))) -> + H1 : + (__ : cupicef.Term cupicef.prop (eqA x x_27_) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.and (R x x_27_) (R x_27_ x))) -> + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) -> + cupicef.Term cupicef.prop (eqA x x_27_)) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + equ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + preo : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PreOrder A R) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.PartialOrder A eqA equ R preo) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hx : cupicef.Term cupicef.prop (eqA x x_27_) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hy : cupicef.Term cupicef.prop (eqA y y_27_) => + Hr : cupicef.Term cupicef.prop (R x y) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eqA x x_27_) + (__ : cupicef.Term cupicef.prop (eqA x x_27_) => + Coq__Init__Logic.and (R x x_27_) (R x_27_ x))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) => + eqA x x_27_))) => + H1 : + (__ : cupicef.Term cupicef.prop (eqA x x_27_) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x))) => + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) -> + cupicef.Term cupicef.prop (eqA x x_27_)) => + H1 Hx. + +def let__H12 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + equ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + preo : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PreOrder A R) -> + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.PartialOrder A eqA equ R preo) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hx : cupicef.Term cupicef.prop (eqA x x_27_) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hy : cupicef.Term cupicef.prop (eqA y y_27_) -> + Hr : cupicef.Term cupicef.prop (R x y) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eqA x x_27_) + (__ : cupicef.Term cupicef.prop (eqA x x_27_) => + Coq__Init__Logic.and (R x x_27_) (R x_27_ x))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) => + eqA x x_27_))) -> + H1 : + (__ : cupicef.Term cupicef.prop (eqA x x_27_) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.and (R x x_27_) (R x_27_ x))) -> + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) -> + cupicef.Term cupicef.prop (eqA x x_27_)) -> + H3 : cupicef.Term cupicef.prop (R x x_27_) -> + H4 : cupicef.Term cupicef.prop (R x_27_ x) -> + __ : cupicef.Term cupicef.prop (R x x_27_) -> + __0 : cupicef.Term cupicef.prop (R x_27_ x) -> + cupicef.Term cupicef.prop (eqA x x_27_) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + equ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + preo : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PreOrder A R) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.PartialOrder A eqA equ R preo) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hx : cupicef.Term cupicef.prop (eqA x x_27_) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hy : cupicef.Term cupicef.prop (eqA y y_27_) => + Hr : cupicef.Term cupicef.prop (R x y) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eqA x x_27_) + (__ : cupicef.Term cupicef.prop (eqA x x_27_) => + Coq__Init__Logic.and (R x x_27_) (R x_27_ x))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) => + eqA x x_27_))) => + H1 : + (__ : cupicef.Term cupicef.prop (eqA x x_27_) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x))) => + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) -> + cupicef.Term cupicef.prop (eqA x x_27_)) => + H3 : cupicef.Term cupicef.prop (R x x_27_) => + H4 : cupicef.Term cupicef.prop (R x_27_ x) => + H5 : cupicef.Term cupicef.prop (R x x_27_) => + H6 : cupicef.Term cupicef.prop (R x_27_ x) => + H2 (Coq__Init__Logic.conj (R x x_27_) (R x_27_ x) H5 H6). + +def let__H13 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + equ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + preo : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PreOrder A R) -> + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.PartialOrder A eqA equ R preo) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hx : cupicef.Term cupicef.prop (eqA x x_27_) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hy : cupicef.Term cupicef.prop (eqA y y_27_) -> + Hr : cupicef.Term cupicef.prop (R x y) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eqA x x_27_) + (__ : cupicef.Term cupicef.prop (eqA x x_27_) => + Coq__Init__Logic.and (R x x_27_) (R x_27_ x))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) => + eqA x x_27_))) -> + H1 : + (__ : cupicef.Term cupicef.prop (eqA x x_27_) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.and (R x x_27_) (R x_27_ x))) -> + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) -> + cupicef.Term cupicef.prop (eqA x x_27_)) -> + H3 : cupicef.Term cupicef.prop (R x x_27_) -> + H4 : cupicef.Term cupicef.prop (R x_27_ x) -> + __ : cupicef.Term cupicef.prop (R x_27_ x) -> + cupicef.Term cupicef.prop (eqA x x_27_) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + equ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + preo : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PreOrder A R) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.PartialOrder A eqA equ R preo) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hx : cupicef.Term cupicef.prop (eqA x x_27_) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hy : cupicef.Term cupicef.prop (eqA y y_27_) => + Hr : cupicef.Term cupicef.prop (R x y) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eqA x x_27_) + (__ : cupicef.Term cupicef.prop (eqA x x_27_) => + Coq__Init__Logic.and (R x x_27_) (R x_27_ x))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) => + eqA x x_27_))) => + H1 : + (__ : cupicef.Term cupicef.prop (eqA x x_27_) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x))) => + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) -> + cupicef.Term cupicef.prop (eqA x x_27_)) => + H3 : cupicef.Term cupicef.prop (R x x_27_) => + H4 : cupicef.Term cupicef.prop (R x_27_ x) => + let__H12 A eqA equ R preo H x x_27_ Hx y y_27_ Hy Hr H0 H1 H2 H3 H4 H3. + +def let__H14 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + equ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + preo : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PreOrder A R) -> + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.PartialOrder A eqA equ R preo) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hx : cupicef.Term cupicef.prop (eqA x x_27_) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hy : cupicef.Term cupicef.prop (eqA y y_27_) -> + Hr : cupicef.Term cupicef.prop (R x y) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eqA x x_27_) + (__ : cupicef.Term cupicef.prop (eqA x x_27_) => + Coq__Init__Logic.and (R x x_27_) (R x_27_ x))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) => + eqA x x_27_))) -> + H1 : + (__ : cupicef.Term cupicef.prop (eqA x x_27_) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.and (R x x_27_) (R x_27_ x))) -> + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) -> + cupicef.Term cupicef.prop (eqA x x_27_)) -> + H3 : cupicef.Term cupicef.prop (R x x_27_) -> + H4 : cupicef.Term cupicef.prop (R x_27_ x) -> + cupicef.Term cupicef.prop (eqA x x_27_) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + equ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + preo : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PreOrder A R) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.PartialOrder A eqA equ R preo) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hx : cupicef.Term cupicef.prop (eqA x x_27_) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hy : cupicef.Term cupicef.prop (eqA y y_27_) => + Hr : cupicef.Term cupicef.prop (R x y) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eqA x x_27_) + (__ : cupicef.Term cupicef.prop (eqA x x_27_) => + Coq__Init__Logic.and (R x x_27_) (R x_27_ x))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) => + eqA x x_27_))) => + H1 : + (__ : cupicef.Term cupicef.prop (eqA x x_27_) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x))) => + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) -> + cupicef.Term cupicef.prop (eqA x x_27_)) => + H3 : cupicef.Term cupicef.prop (R x x_27_) => + H4 : cupicef.Term cupicef.prop (R x_27_ x) => + let__H13 A eqA equ R preo H x x_27_ Hx y y_27_ Hy Hr H0 H1 H2 H3 H4 H4. + +def let__H15 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + equ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + preo : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PreOrder A R) -> + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.PartialOrder A eqA equ R preo) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hx : cupicef.Term cupicef.prop (eqA x x_27_) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hy : cupicef.Term cupicef.prop (eqA y y_27_) -> + Hr : cupicef.Term cupicef.prop (R x y) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eqA y y_27_) + (__ : cupicef.Term cupicef.prop (eqA y y_27_) => + Coq__Init__Logic.and (R y y_27_) (R y_27_ y))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) => + eqA y y_27_))) -> + H1 : + (__ : cupicef.Term cupicef.prop (eqA y y_27_) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.and (R y y_27_) (R y_27_ y))) -> + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) -> + cupicef.Term cupicef.prop (eqA y y_27_)) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + equ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + preo : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PreOrder A R) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.PartialOrder A eqA equ R preo) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hx : cupicef.Term cupicef.prop (eqA x x_27_) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hy : cupicef.Term cupicef.prop (eqA y y_27_) => + Hr : cupicef.Term cupicef.prop (R x y) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eqA y y_27_) + (__ : cupicef.Term cupicef.prop (eqA y y_27_) => + Coq__Init__Logic.and (R y y_27_) (R y_27_ y))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) => + eqA y y_27_))) => + H1 : + (__ : cupicef.Term cupicef.prop (eqA y y_27_) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y))) => + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) -> + cupicef.Term cupicef.prop (eqA y y_27_)) => + H1 Hy. + +def let__H16 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + equ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + preo : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PreOrder A R) -> + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.PartialOrder A eqA equ R preo) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hx : cupicef.Term cupicef.prop (eqA x x_27_) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hy : cupicef.Term cupicef.prop (eqA y y_27_) -> + Hr : cupicef.Term cupicef.prop (R x y) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eqA y y_27_) + (__ : cupicef.Term cupicef.prop (eqA y y_27_) => + Coq__Init__Logic.and (R y y_27_) (R y_27_ y))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) => + eqA y y_27_))) -> + H1 : + (__ : cupicef.Term cupicef.prop (eqA y y_27_) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.and (R y y_27_) (R y_27_ y))) -> + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) -> + cupicef.Term cupicef.prop (eqA y y_27_)) -> + H3 : cupicef.Term cupicef.prop (R y y_27_) -> + H4 : cupicef.Term cupicef.prop (R y_27_ y) -> + __ : cupicef.Term cupicef.prop (R y y_27_) -> + __0 : cupicef.Term cupicef.prop (R y_27_ y) -> + cupicef.Term cupicef.prop (eqA y y_27_) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + equ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + preo : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PreOrder A R) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.PartialOrder A eqA equ R preo) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hx : cupicef.Term cupicef.prop (eqA x x_27_) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hy : cupicef.Term cupicef.prop (eqA y y_27_) => + Hr : cupicef.Term cupicef.prop (R x y) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eqA y y_27_) + (__ : cupicef.Term cupicef.prop (eqA y y_27_) => + Coq__Init__Logic.and (R y y_27_) (R y_27_ y))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) => + eqA y y_27_))) => + H1 : + (__ : cupicef.Term cupicef.prop (eqA y y_27_) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y))) => + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) -> + cupicef.Term cupicef.prop (eqA y y_27_)) => + H3 : cupicef.Term cupicef.prop (R y y_27_) => + H4 : cupicef.Term cupicef.prop (R y_27_ y) => + H5 : cupicef.Term cupicef.prop (R y y_27_) => + H6 : cupicef.Term cupicef.prop (R y_27_ y) => + H2 (Coq__Init__Logic.conj (R y y_27_) (R y_27_ y) H5 H6). + +def let__H17 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + equ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + preo : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PreOrder A R) -> + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.PartialOrder A eqA equ R preo) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hx : cupicef.Term cupicef.prop (eqA x x_27_) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hy : cupicef.Term cupicef.prop (eqA y y_27_) -> + Hr : cupicef.Term cupicef.prop (R x y) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eqA y y_27_) + (__ : cupicef.Term cupicef.prop (eqA y y_27_) => + Coq__Init__Logic.and (R y y_27_) (R y_27_ y))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) => + eqA y y_27_))) -> + H1 : + (__ : cupicef.Term cupicef.prop (eqA y y_27_) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.and (R y y_27_) (R y_27_ y))) -> + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) -> + cupicef.Term cupicef.prop (eqA y y_27_)) -> + H3 : cupicef.Term cupicef.prop (R y y_27_) -> + H4 : cupicef.Term cupicef.prop (R y_27_ y) -> + __ : cupicef.Term cupicef.prop (R y_27_ y) -> + cupicef.Term cupicef.prop (eqA y y_27_) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + equ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + preo : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PreOrder A R) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.PartialOrder A eqA equ R preo) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hx : cupicef.Term cupicef.prop (eqA x x_27_) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hy : cupicef.Term cupicef.prop (eqA y y_27_) => + Hr : cupicef.Term cupicef.prop (R x y) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eqA y y_27_) + (__ : cupicef.Term cupicef.prop (eqA y y_27_) => + Coq__Init__Logic.and (R y y_27_) (R y_27_ y))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) => + eqA y y_27_))) => + H1 : + (__ : cupicef.Term cupicef.prop (eqA y y_27_) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y))) => + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) -> + cupicef.Term cupicef.prop (eqA y y_27_)) => + H3 : cupicef.Term cupicef.prop (R y y_27_) => + H4 : cupicef.Term cupicef.prop (R y_27_ y) => + let__H16 A eqA equ R preo H x x_27_ Hx y y_27_ Hy Hr H0 H1 H2 H3 H4 H3. + +def let__H18 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + equ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + preo : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PreOrder A R) -> + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.PartialOrder A eqA equ R preo) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hx : cupicef.Term cupicef.prop (eqA x x_27_) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hy : cupicef.Term cupicef.prop (eqA y y_27_) -> + Hr : cupicef.Term cupicef.prop (R x y) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eqA y y_27_) + (__ : cupicef.Term cupicef.prop (eqA y y_27_) => + Coq__Init__Logic.and (R y y_27_) (R y_27_ y))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) => + eqA y y_27_))) -> + H1 : + (__ : cupicef.Term cupicef.prop (eqA y y_27_) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.and (R y y_27_) (R y_27_ y))) -> + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) -> + cupicef.Term cupicef.prop (eqA y y_27_)) -> + H3 : cupicef.Term cupicef.prop (R y y_27_) -> + H4 : cupicef.Term cupicef.prop (R y_27_ y) -> + cupicef.Term cupicef.prop (eqA y y_27_) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + equ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + preo : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PreOrder A R) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.PartialOrder A eqA equ R preo) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hx : cupicef.Term cupicef.prop (eqA x x_27_) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hy : cupicef.Term cupicef.prop (eqA y y_27_) => + Hr : cupicef.Term cupicef.prop (R x y) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eqA y y_27_) + (__ : cupicef.Term cupicef.prop (eqA y y_27_) => + Coq__Init__Logic.and (R y y_27_) (R y_27_ y))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) => + eqA y y_27_))) => + H1 : + (__ : cupicef.Term cupicef.prop (eqA y y_27_) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y))) => + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) -> + cupicef.Term cupicef.prop (eqA y y_27_)) => + H3 : cupicef.Term cupicef.prop (R y y_27_) => + H4 : cupicef.Term cupicef.prop (R y_27_ y) => + let__H17 A eqA equ R preo H x x_27_ Hx y y_27_ Hy Hr H0 H1 H2 H3 H4 H4. + +thm PartialOrder__proper : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + equ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + preo : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PreOrder A R) -> + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.PartialOrder A eqA equ R preo) -> + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + eqA + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + eqA Coq__Init__Logic.iff)) + R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + equ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + preo : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PreOrder A R) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.PartialOrder A eqA equ R preo) => + proper__sym__impl__iff__2 A eqA + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + eqA x y) + (H0 : cupicef.Term cupicef.prop (eqA x y) => eqA y x)))) + (Coq__Classes__RelationClasses.Symmetric A eqA) cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (eqA x y) => + Coq__Classes__RelationClasses.symmetry A eqA + (Coq__Classes__RelationClasses.Equivalence__Symmetric A eqA equ) x + y H0)) + A eqA + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + eqA x y) + (H0 : cupicef.Term cupicef.prop (eqA x y) => eqA y x)))) + (Coq__Classes__RelationClasses.Symmetric A eqA) cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (eqA x y) => + Coq__Classes__RelationClasses.symmetry A eqA + (Coq__Classes__RelationClasses.Equivalence__Symmetric A eqA equ) x + y H0)) + R + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eqA x x_27_) + (Hx : cupicef.Term cupicef.prop (eqA x x_27_) => + respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + eqA Coq__Program__Basics.impl (R x) (R x_27_))))) + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + eqA + (respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + eqA Coq__Program__Basics.impl)) + R) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hx : cupicef.Term cupicef.prop (eqA x x_27_) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eqA y y_27_) + (Hy : cupicef.Term cupicef.prop (eqA y y_27_) => + Coq__Program__Basics.impl (R x y) (R x_27_ y_27_))))) + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + eqA Coq__Program__Basics.impl (R x) (R x_27_)) + cupicef.I + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hy : cupicef.Term cupicef.prop (eqA y y_27_) => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + R x y) + (Hr : cupicef.Term cupicef.prop (R x y) => R x_27_ y_27_)) + (Coq__Program__Basics.impl (R x y) (R x_27_ y_27_)) cupicef.I + (Hr : cupicef.Term cupicef.prop (R x y) => + Coq__Classes__RelationClasses.transitivity A R + (Coq__Classes__RelationClasses.PreOrder__Transitive A R preo) + x_27_ x y_27_ + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (eqA x x_27_) + (__ : cupicef.Term cupicef.prop (eqA x x_27_) => + Coq__Init__Logic.and (R x x_27_) (R x_27_ x))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x_27_) + (R x_27_ x)) => + eqA x x_27_))) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (eqA x x_27_) + (__ : + cupicef.Term cupicef.prop (eqA x x_27_) => + Coq__Init__Logic.and (R x x_27_) + (R x_27_ x))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I + (Coq__Init__Logic.and (R x x_27_) + (R x_27_ x)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x_27_) + (R x_27_ x)) => + eqA x x_27_))) => + R x_27_ x)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.iff (eqA x x_27_) + (Coq__Classes__RelationClasses.relation__conjunction + A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x x_27_)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (eqA x x_27_) + (Coq__Classes__RelationClasses.relation__conjunction + A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)) + R) + x x_27_)) => + R x_27_ x)) + cupicef.I + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (eqA x x_27_) + (__ : cupicef.Term cupicef.prop (eqA x x_27_) => + Coq__Init__Logic.and (R x x_27_) (R x_27_ x))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x_27_) + (R x_27_ x)) => + eqA x x_27_))) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eqA x x_27_) + (__ : cupicef.Term cupicef.prop (eqA x x_27_) => + Coq__Init__Logic.and (R x x_27_) (R x_27_ x))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) => + eqA x x_27_)) + (R x_27_ x) + (H1 : + (__ : cupicef.Term cupicef.prop (eqA x x_27_) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x))) => + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x_27_) (R x_27_ x)) -> + cupicef.Term cupicef.prop (eqA x x_27_)) => + Coq__Init__Logic.and__ind (R x x_27_) (R x_27_ x) + (R x_27_ x) + (H3 : cupicef.Term cupicef.prop (R x x_27_) => + H4 : cupicef.Term cupicef.prop (R x_27_ x) => + H4) + (let__H0 A eqA equ R preo H x x_27_ Hx y y_27_ Hy + Hr H0 H1 H2)) + H0) + (Coq__Classes__RelationClasses.partial__order__equivalence + A eqA equ R preo H x x_27_)) + (Coq__Classes__RelationClasses.transitivity A R + (Coq__Classes__RelationClasses.PreOrder__Transitive A R + preo) + x y y_27_ Hr + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (eqA y y_27_) + (__ : cupicef.Term cupicef.prop (eqA y y_27_) => + Coq__Init__Logic.and (R y y_27_) (R y_27_ y))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y y_27_) + (R y_27_ y)) => + eqA y y_27_))) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (eqA y y_27_) + (__ : + cupicef.Term cupicef.prop + (eqA y y_27_) => + Coq__Init__Logic.and (R y y_27_) + (R y_27_ y))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I + (Coq__Init__Logic.and (R y y_27_) + (R y_27_ y)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y y_27_) + (R y_27_ y)) => + eqA y y_27_))) => + R y y_27_)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.iff (eqA y y_27_) + (Coq__Classes__RelationClasses.relation__conjunction + A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + y y_27_)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (eqA y y_27_) + (Coq__Classes__RelationClasses.relation__conjunction + A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)) + R) + y y_27_)) => + R y y_27_)) + cupicef.I + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (eqA y y_27_) + (__ : + cupicef.Term cupicef.prop (eqA y y_27_) => + Coq__Init__Logic.and (R y y_27_) + (R y_27_ y))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I + (Coq__Init__Logic.and (R y y_27_) + (R y_27_ y)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y y_27_) + (R y_27_ y)) => + eqA y y_27_))) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (eqA y y_27_) + (__ : cupicef.Term cupicef.prop (eqA y y_27_) => + Coq__Init__Logic.and (R y y_27_) (R y_27_ y))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y y_27_) + (R y_27_ y)) => + eqA y y_27_)) + (R y y_27_) + (H1 : + (__ : cupicef.Term cupicef.prop (eqA y y_27_) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y y_27_) (R y_27_ y))) => + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y y_27_) + (R y_27_ y)) -> + cupicef.Term cupicef.prop (eqA y y_27_)) => + Coq__Init__Logic.and__ind (R y y_27_) (R y_27_ y) + (R y y_27_) + (H3 : cupicef.Term cupicef.prop (R y y_27_) => + H4 : cupicef.Term cupicef.prop (R y_27_ y) => + H3) + (let__H15 A eqA equ R preo H x x_27_ Hx y y_27_ + Hy Hr H0 H1 H2)) + H0) + (Coq__Classes__RelationClasses.partial__order__equivalence + A eqA equ R preo H y y_27_))))))). + +#TYPE_CLASS_INSTANCE PartialOrder__proper. + +thm PartialOrder__StrictOrder : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + equ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + preo : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PreOrder A R) -> + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.PartialOrder A eqA equ R preo) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.StrictOrder A + (Coq__Classes__RelationClasses.relation__conjunction A R + (Coq__Classes__RelationClasses.complement A eqA))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + equ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + preo : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.PreOrder A R) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.PartialOrder A eqA equ R preo) => + Coq__Classes__RelationClasses.Build__StrictOrder A + (Coq__Classes__RelationClasses.relation__conjunction A R + (Coq__Classes__RelationClasses.complement A eqA)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (R x x) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eqA x x) + (__ : cupicef.Term cupicef.prop (eqA x x) => + Coq__Init__Logic.False))) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eqA x x) + (__ : cupicef.Term cupicef.prop (eqA x x) => + Coq__Init__Logic.False))) => + Coq__Init__Logic.False))) + (Coq__Classes__RelationClasses.Irreflexive A + (Coq__Classes__RelationClasses.relation__conjunction A R + (Coq__Classes__RelationClasses.complement A eqA))) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eqA x x) + (__ : cupicef.Term cupicef.prop (eqA x x) => + Coq__Init__Logic.False))) => + Coq__Init__Logic.match____and cupicef.prop (R x x) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eqA x x) + (__ : cupicef.Term cupicef.prop (eqA x x) => + Coq__Init__Logic.False)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x x) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eqA x x) + (__ : cupicef.Term cupicef.prop (eqA x x) => + Coq__Init__Logic.False))) => + Coq__Init__Logic.False) + (__tmp : cupicef.Term cupicef.prop (R x x) => + Hx : + (__ : cupicef.Term cupicef.prop (eqA x x) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Hx + (Coq__Classes__RelationClasses.Equivalence__Reflexive A eqA equ + x)) + H0)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Init__Logic.and (R x y) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eqA x y) + (__ : cupicef.Term cupicef.prop (eqA x y) => + Coq__Init__Logic.False))) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x y) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (eqA x y) + (__ : cupicef.Term cupicef.prop (eqA x y) => + Coq__Init__Logic.False))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (R y z) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (eqA y z) + (__ : cupicef.Term cupicef.prop (eqA y z) => + Coq__Init__Logic.False))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y z) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (eqA y z) + (__ : cupicef.Term cupicef.prop (eqA y z) => + Coq__Init__Logic.False))) => + Coq__Init__Logic.and (R x z) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (eqA x z) + (__0 : cupicef.Term cupicef.prop (eqA x z) => + Coq__Init__Logic.False)))))))) + (Coq__Classes__RelationClasses.Transitive A + (Coq__Classes__RelationClasses.relation__conjunction A R + (Coq__Classes__RelationClasses.complement A eqA))) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x y) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eqA x y) + (__ : cupicef.Term cupicef.prop (eqA x y) => + Coq__Init__Logic.False))) => + Coq__Init__Logic.match____and cupicef.prop (R x y) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eqA x y) + (__ : cupicef.Term cupicef.prop (eqA x y) => + Coq__Init__Logic.False)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R x y) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eqA x y) + (__ : cupicef.Term cupicef.prop (eqA x y) => + Coq__Init__Logic.False))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and (R y z) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eqA y z) + (__0 : cupicef.Term cupicef.prop (eqA y z) => + Coq__Init__Logic.False))) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y z) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eqA y z) + (__0 : cupicef.Term cupicef.prop (eqA y z) => + Coq__Init__Logic.False))) => + Coq__Init__Logic.and (R x z) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eqA x z) + (__1 : cupicef.Term cupicef.prop (eqA x z) => + Coq__Init__Logic.False)))) + (Hxy : cupicef.Term cupicef.prop (R x y) => + Hxy_27_ : + (__ : cupicef.Term cupicef.prop (eqA x y) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y z) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eqA y z) + (__ : cupicef.Term cupicef.prop (eqA y z) => + Coq__Init__Logic.False))) => + Coq__Init__Logic.match____and cupicef.prop (R y z) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + eqA y z) + (__ : cupicef.Term cupicef.prop (eqA y z) => + Coq__Init__Logic.False)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (R y z) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eqA y z) + (__ : cupicef.Term cupicef.prop (eqA y z) => + Coq__Init__Logic.False))) => + Coq__Init__Logic.and (R x z) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eqA x z) + (__0 : cupicef.Term cupicef.prop (eqA x z) => + Coq__Init__Logic.False))) + (Hyz : cupicef.Term cupicef.prop (R y z) => + Hyz_27_ : + (__ : cupicef.Term cupicef.prop (eqA y z) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.conj (R x z) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eqA x z) + (__ : cupicef.Term cupicef.prop (eqA x z) => + Coq__Init__Logic.False)) + (Coq__Classes__RelationClasses.PreOrder__Transitive A R preo + x y z Hxy Hyz) + (Hxz : cupicef.Term cupicef.prop (eqA x z) => + Hxy_27_ + (Coq__Classes__RelationClasses.partial__order__antisym A + eqA equ R preo H x y Hxy + ((lemma : cupicef.Term cupicef.prop (eqA x z) => + Reflexive__partial__app__morphism A + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + eqA + (respectful A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + eqA + (Coq__Program__Basics.flip + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl)) + R + (subrelation__proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type + (cupicef.s cupicef.z)) + A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type + (cupicef.s cupicef.z)) + A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + eqA + (respectful A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + eqA Coq__Init__Logic.iff)) + R (PartialOrder__proper A eqA equ R preo H) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type + (cupicef.s cupicef.z)) + A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + eqA + (respectful A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + eqA + (Coq__Program__Basics.flip + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl))) + Coq__Init__Datatypes.tt + (subrelation__respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type + (cupicef.s cupicef.z)) + A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + eqA eqA (subrelation__refl A eqA) + (respectful A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + eqA Coq__Init__Logic.iff) + (respectful A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + eqA + (Coq__Program__Basics.flip + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl)) + (subrelation__respectful A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + eqA eqA (subrelation__refl A eqA) + Coq__Init__Logic.iff + (Coq__Program__Basics.flip + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl) + iff__flip__impl__subrelation))) + y + (reflexive__proper__proxy A eqA + (Coq__Classes__RelationClasses.Equivalence__Reflexive + A eqA equ) + y) + x z lemma) + Hxz Hyz)))) + H1) + H0)). + +thm StrictOrder__PreOrder : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H0 : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.StrictOrder A R) -> + H1 : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + eqA + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + eqA Coq__Init__Logic.iff)) + R) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.PreOrder A + (Coq__Classes__RelationClasses.relation__disjunction A R eqA)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H0 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.StrictOrder A R) => + H1 : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + eqA + (respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + eqA Coq__Init__Logic.iff)) + R) => + Coq__Classes__RelationClasses.Build__PreOrder A + (Coq__Classes__RelationClasses.relation__disjunction A R eqA) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.or (R x x) (eqA x x))) + (Coq__Classes__RelationClasses.Reflexive A + (Coq__Classes__RelationClasses.relation__disjunction A R eqA)) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.or__intror (R x x) (eqA x x) + (Coq__Classes__RelationClasses.reflexivity A eqA + (Coq__Classes__RelationClasses.Equivalence__Reflexive A eqA H) + x))) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Classes__RelationClasses.relation__disjunction A R + eqA x y) + (H2 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.relation__disjunction + A R eqA x y) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Classes__RelationClasses.relation__disjunction A + R eqA y z) + (__ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.relation__disjunction + A R eqA y z) => + Coq__Classes__RelationClasses.relation__disjunction A + R eqA x z)))))) + (Coq__Classes__RelationClasses.Transitive A + (Coq__Classes__RelationClasses.relation__disjunction A R eqA)) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H2 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.relation__disjunction A R eqA x y) => + Coq__Init__Logic.match____or cupicef.prop (R x y) (eqA x y) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (R x y) (eqA x y)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Classes__RelationClasses.relation__disjunction A R eqA y + z) + (__0 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.relation__disjunction A R + eqA y z) => + Coq__Classes__RelationClasses.relation__disjunction A R eqA x + z)) + (Hxy : cupicef.Term cupicef.prop (R x y) => + H3 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.relation__disjunction A R eqA + y z) => + Coq__Init__Logic.match____or cupicef.prop (R y z) (eqA y z) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (R y z) (eqA y z)) => + Coq__Classes__RelationClasses.relation__disjunction A R eqA x + z) + (Hyz : cupicef.Term cupicef.prop (R y z) => + Coq__Init__Logic.or__introl (R x z) (eqA x z) + (Coq__Classes__RelationClasses.transitivity A R + (Coq__Classes__RelationClasses.StrictOrder__Transitive A + R H0) + x y z Hxy Hyz)) + (Hyz : cupicef.Term cupicef.prop (eqA y z) => + Coq__Init__Logic.or__introl (R x z) (eqA x z) + ((lemma : cupicef.Term cupicef.prop (eqA y z) => + Reflexive__partial__app__morphism A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + eqA + (respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + eqA + (Coq__Program__Basics.flip + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl)) + R + (subrelation__proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + eqA + (respectful A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + eqA Coq__Init__Logic.iff)) + R H1 + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + eqA + (respectful A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + eqA + (Coq__Program__Basics.flip + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl))) + Coq__Init__Datatypes.tt + (subrelation__respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + 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cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + eqA Coq__Init__Logic.iff) + (respectful A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + eqA + (Coq__Program__Basics.flip + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl)) + (subrelation__respectful A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + eqA eqA (subrelation__refl A eqA) + Coq__Init__Logic.iff + (Coq__Program__Basics.flip + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl) + iff__flip__impl__subrelation))) + x + (reflexive__proper__proxy A eqA + (Coq__Classes__RelationClasses.Equivalence__Reflexive + A eqA H) + x) + z y + (Coq__Classes__RelationClasses.symmetry A eqA + (Coq__Classes__RelationClasses.Equivalence__Symmetric + A eqA H) + y z lemma)) + Hyz Hxy)) + H3) + (Hxy : cupicef.Term cupicef.prop (eqA x y) => + H3 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.relation__disjunction A R eqA + y z) => + Coq__Init__Logic.match____or cupicef.prop (R y z) (eqA y z) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (R y z) (eqA y z)) => + Coq__Classes__RelationClasses.relation__disjunction A R eqA x + z) + (Hyz : cupicef.Term cupicef.prop (R y z) => + Coq__Init__Logic.or__introl (R x z) (eqA x z) + ((lemma : cupicef.Term cupicef.prop (eqA x y) => + subrelation__proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + eqA + (respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + eqA Coq__Init__Logic.iff)) + R H1 + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + 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(cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + eqA + (Coq__Program__Basics.flip + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl))) + Coq__Init__Datatypes.tt + (subrelation__respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + eqA eqA (subrelation__refl A eqA) + (respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + eqA Coq__Init__Logic.iff) + (respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + eqA + (Coq__Program__Basics.flip + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl)) + (subrelation__respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + eqA eqA (subrelation__refl A eqA) + Coq__Init__Logic.iff + (Coq__Program__Basics.flip + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl) + iff__flip__impl__subrelation)) + x y lemma z z + (reflexive__proper__proxy A eqA + (Coq__Classes__RelationClasses.Equivalence__Reflexive + A eqA H) + z)) + Hxy Hyz)) + (Hyz : cupicef.Term cupicef.prop (eqA y z) => + Coq__Init__Logic.or__intror (R x z) (eqA x z) + (Coq__Classes__RelationClasses.transitivity A eqA + (Coq__Classes__RelationClasses.Equivalence__Transitive A + eqA H) + x y z Hxy Hyz)) + H3) + H2)). + +thm StrictOrder__PartialOrder : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H0 : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.StrictOrder A R) -> + H1 : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + eqA + (respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + eqA Coq__Init__Logic.iff)) + R) -> + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.PartialOrder A eqA H + (Coq__Classes__RelationClasses.relation__disjunction A R eqA) + (StrictOrder__PreOrder A eqA H R H0 H1)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H0 : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.StrictOrder A R) => + H1 : + cupicef.Term cupicef.prop + (Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + eqA + (respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + eqA Coq__Init__Logic.iff)) + R) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Classes__RelationClasses.pointwise__lifting + Coq__Init__Logic.iff + (Coq__Classes__RelationClasses.Tcons A + Coq__Classes__RelationClasses.Tnil) + (eqA x) + (Coq__Classes__RelationClasses.relation__conjunction A + (Coq__Classes__RelationClasses.relation__disjunction A R eqA) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Classes__RelationClasses.relation__disjunction A R eqA)) + x))) + (Coq__Classes__RelationClasses.PartialOrder A eqA H + (Coq__Classes__RelationClasses.relation__disjunction A R eqA) + (StrictOrder__PreOrder A eqA H R H0 H1)) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Classes__RelationClasses.pointwise__lifting + Coq__Init__Logic.iff Coq__Classes__RelationClasses.Tnil + (eqA x y) + (Coq__Classes__RelationClasses.relation__conjunction A + (Coq__Classes__RelationClasses.relation__disjunction A R eqA) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Classes__RelationClasses.relation__disjunction A R + eqA)) + x y))) + (Coq__Classes__RelationClasses.pointwise__lifting + Coq__Init__Logic.iff + (Coq__Classes__RelationClasses.Tcons A + Coq__Classes__RelationClasses.Tnil) + (eqA x) + (Coq__Classes__RelationClasses.relation__conjunction A + (Coq__Classes__RelationClasses.relation__disjunction A R eqA) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Classes__RelationClasses.relation__disjunction A R eqA)) + x)) + cupicef.I + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + eqA x y) + (__ : cupicef.Term cupicef.prop (eqA x y) => + Coq__Init__Logic.and + (Coq__Init__Logic.or (R x y) (eqA x y)) + (Coq__Init__Logic.or (R y x) (eqA y x)))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.or (R x y) (eqA x y)) + (Coq__Init__Logic.or (R y x) (eqA y x))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.or (R x y) (eqA x y)) + (Coq__Init__Logic.or (R y x) (eqA y x))) => + eqA x y))) + (Coq__Classes__RelationClasses.pointwise__lifting + Coq__Init__Logic.iff Coq__Classes__RelationClasses.Tnil + (eqA x y) + (Coq__Classes__RelationClasses.relation__conjunction A + (Coq__Classes__RelationClasses.relation__disjunction A R eqA) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Classes__RelationClasses.relation__disjunction A R + eqA)) + x y)) + cupicef.I + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + eqA x y) + (__ : cupicef.Term cupicef.prop (eqA x y) => + Coq__Init__Logic.and + (Coq__Init__Logic.or (R x y) (eqA x y)) + (Coq__Init__Logic.or (R y x) (eqA y x)))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.or (R x y) (eqA x y)) + (Coq__Init__Logic.or (R y x) (eqA y x))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.or (R x y) (eqA x y)) + (Coq__Init__Logic.or (R y x) (eqA y x))) => + eqA x y)) + (H2 : cupicef.Term cupicef.prop (eqA x y) => + Coq__Init__Logic.conj (Coq__Init__Logic.or (R x y) (eqA x y)) + (Coq__Init__Logic.or (R y x) (eqA y x)) + (Coq__Init__Logic.or__intror (R x y) (eqA x y) H2) + (Coq__Init__Logic.or__intror (R y x) (eqA y x) + (cupicef.cast cupicef.prop cupicef.prop (eqA y x) + (eqA y x) cupicef.I + (Coq__Classes__RelationClasses.symmetry A eqA + (Coq__Classes__RelationClasses.Equivalence__Symmetric + A eqA H) + x y H2)))) + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.or (R x y) (eqA x y)) + (Coq__Init__Logic.or (R y x) (eqA y x))) => + Coq__Init__Logic.and__ind + (Coq__Init__Logic.or (R x y) (eqA x y)) + (Coq__Init__Logic.or (R y x) (eqA y x)) (eqA x y) + (H3 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (R x y) (eqA x y)) => + H4 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (R y x) (eqA y x)) => + Coq__Init__Logic.or__ind (R x y) (eqA x y) (eqA x y) + (H5 : cupicef.Term cupicef.prop (R x y) => + Coq__Init__Logic.or__ind (R y x) (eqA y x) (eqA x y) + (H6 : cupicef.Term cupicef.prop (R y x) => + Coq__Init__Logic.False__ind (eqA x y) + (Coq__Classes__RelationClasses.StrictOrder__Irreflexive + A R H0 x + (Coq__Classes__RelationClasses.transitivity A R + (Coq__Classes__RelationClasses.StrictOrder__Transitive + A R H0) + x y x H5 H6))) + (H6 : cupicef.Term cupicef.prop (eqA y x) => + cupicef.cast cupicef.prop cupicef.prop (eqA x y) + (eqA x y) cupicef.I + (Coq__Classes__RelationClasses.symmetry A eqA + (Coq__Classes__RelationClasses.Equivalence__Symmetric + A eqA H) + y x H6)) + H4) + (H5 : cupicef.Term cupicef.prop (eqA x y) => + Coq__Init__Logic.or__ind (R y x) (eqA y x) (eqA x y) + (H6 : cupicef.Term cupicef.prop (R y x) => H5) + (H6 : cupicef.Term cupicef.prop (eqA y x) => H5) H4) + H3) + H2)))). + +(; End of translation. ;) diff --git a/out/Coq__Classes__Morphisms__Prop.dk b/out/Coq__Classes__Morphisms__Prop.dk new file mode 100644 index 000000000..23fe9ecfc --- /dev/null +++ b/out/Coq__Classes__Morphisms__Prop.dk @@ -0,0 +1,5654 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def not__impl__morphism__obligation__1 : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.respectful + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Program__Basics.flip + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl) + Coq__Program__Basics.impl Coq__Init__Logic.not Coq__Init__Logic.not) + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not x) => + H1 : cupicef.Term cupicef.prop y => + H0 (H H1). + +def not__impl__morphism : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.Proper + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (__ : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (Coq__Classes__Morphisms.respectful + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Program__Basics.flip + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Program__Basics.impl) + Coq__Program__Basics.impl) + Coq__Init__Logic.not) + +:= not__impl__morphism__obligation__1. + +#TYPE_CLASS_INSTANCE not__impl__morphism. + +def let__H3 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + H0 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H1 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not x) -> + H3 : cupicef.Term cupicef.prop y -> + cupicef.Term cupicef.prop x + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + H0 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H1 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not x) => + H3 : cupicef.Term cupicef.prop y => + H1 H3. + +def let__H1 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + H0 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H1 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not x) -> + H3 : cupicef.Term cupicef.prop y -> + cupicef.Term cupicef.prop y + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + H0 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H1 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not x) => + H3 : cupicef.Term cupicef.prop y => + H0 (let__H3 x y H H0 H1 H2 H3). + +def let__H0 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + H0 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H1 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not x) -> + H3 : cupicef.Term cupicef.prop y -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + H0 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H1 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not x) => + H3 : cupicef.Term cupicef.prop y => + H2 (let__H3 x y H H0 H1 H2 H3). + +def let__H4 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + H0 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H1 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not y) -> + H3 : cupicef.Term cupicef.prop x -> + cupicef.Term cupicef.prop y + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + H0 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H1 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not y) => + H3 : cupicef.Term cupicef.prop x => + H0 H3. + +def let__H2 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + H0 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H1 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not y) -> + H3 : cupicef.Term cupicef.prop x -> + cupicef.Term cupicef.prop x + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + H0 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H1 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not y) => + H3 : cupicef.Term cupicef.prop x => + H1 (let__H4 x y H H0 H1 H2 H3). + +def let__H5 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + H0 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H1 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not y) -> + H3 : cupicef.Term cupicef.prop x -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + H0 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H1 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not y) => + H3 : cupicef.Term cupicef.prop x => + H2 (let__H4 x y H H0 H1 H2 H3). + +def not__iff__morphism__obligation__1 : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.respectful + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff Coq__Init__Logic.iff Coq__Init__Logic.not + Coq__Init__Logic.not) + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I x + (__ : cupicef.Term cupicef.prop x => y)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I y + (__ : cupicef.Term cupicef.prop y => x)) + (Coq__Init__Logic.iff (Coq__Init__Logic.not x) (Coq__Init__Logic.not y)) + (H0 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H1 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not x) + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.not x) => + Coq__Init__Logic.not y)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not y) + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.not y) => + Coq__Init__Logic.not x)) + (H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not x) => + H3 : cupicef.Term cupicef.prop y => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H0 x y H H0 H1 H2 H3)) + (H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not y) => + H3 : cupicef.Term cupicef.prop x => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H5 x y H H0 H1 H2 H3))) + H. + +def not__iff__morphism : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.Proper + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (__ : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (Coq__Classes__Morphisms.respectful + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff Coq__Init__Logic.iff) + Coq__Init__Logic.not) + +:= not__iff__morphism__obligation__1. + +#TYPE_CLASS_INSTANCE not__iff__morphism. + +def and__impl__morphism__obligation__1 : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.respectful + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (__ : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + Coq__Program__Basics.impl + (Coq__Classes__Morphisms.respectful + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl Coq__Program__Basics.impl) + Coq__Init__Logic.and Coq__Init__Logic.and) + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.and x x0) => + Coq__Init__Logic.match____and cupicef.prop x x0 + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.and x x0) => + Coq__Init__Logic.and y y0) + (H2 : cupicef.Term cupicef.prop x => + H3 : cupicef.Term cupicef.prop x0 => + Coq__Init__Logic.conj y y0 (H H2) (H0 H3)) + H1. + +def and__impl__morphism : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.Proper + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (__ : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (__0 : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (Coq__Classes__Morphisms.respectful + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (__ : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + Coq__Program__Basics.impl + (Coq__Classes__Morphisms.respectful + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Program__Basics.impl Coq__Program__Basics.impl)) + Coq__Init__Logic.and) + +:= and__impl__morphism__obligation__1. + +#TYPE_CLASS_INSTANCE and__impl__morphism. + +def let__H6 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) -> + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) -> + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.and x x0) -> + H6 : cupicef.Term cupicef.prop x -> + H7 : cupicef.Term cupicef.prop x0 -> + cupicef.Term cupicef.prop y + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.and x x0) => + H6 : cupicef.Term cupicef.prop x => + H7 : cupicef.Term cupicef.prop x0 => + H1 H6. + +def let__H7 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) -> + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) -> + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.and x x0) -> + H6 : cupicef.Term cupicef.prop x -> + H7 : cupicef.Term cupicef.prop x0 -> + cupicef.Term cupicef.prop y0 + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.and x x0) => + H6 : cupicef.Term cupicef.prop x => + H7 : cupicef.Term cupicef.prop x0 => + H3 H7. + +def let__H : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) -> + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) -> + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.and x x0) -> + H6 : cupicef.Term cupicef.prop x -> + H7 : cupicef.Term cupicef.prop x0 -> + cupicef.Term cupicef.prop x + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.and x x0) => + H6 : cupicef.Term cupicef.prop x => + H7 : cupicef.Term cupicef.prop x0 => + H2 (let__H6 x y H x0 y0 H0 H1 H2 H3 H4 H5 H6 H7). + +def let__H8 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) -> + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) -> + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.and x x0) -> + H6 : cupicef.Term cupicef.prop x -> + H7 : cupicef.Term cupicef.prop x0 -> + cupicef.Term cupicef.prop x0 + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.and x x0) => + H6 : cupicef.Term cupicef.prop x => + H7 : cupicef.Term cupicef.prop x0 => + H4 (let__H7 x y H x0 y0 H0 H1 H2 H3 H4 H5 H6 H7). + +def let__H9 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) -> + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) -> + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.and y y0) -> + H6 : cupicef.Term cupicef.prop y -> + H7 : cupicef.Term cupicef.prop y0 -> + cupicef.Term cupicef.prop x + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.and y y0) => + H6 : cupicef.Term cupicef.prop y => + H7 : cupicef.Term cupicef.prop y0 => + H2 H6. + +def let__H10 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) -> + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) -> + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.and y y0) -> + H6 : cupicef.Term cupicef.prop y -> + H7 : cupicef.Term cupicef.prop y0 -> + cupicef.Term cupicef.prop x0 + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.and y y0) => + H6 : cupicef.Term cupicef.prop y => + H7 : cupicef.Term cupicef.prop y0 => + H4 H7. + +def let__H11 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) -> + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) -> + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.and y y0) -> + H6 : cupicef.Term cupicef.prop y -> + H7 : cupicef.Term cupicef.prop y0 -> + cupicef.Term cupicef.prop y + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.and y y0) => + H6 : cupicef.Term cupicef.prop y => + H7 : cupicef.Term cupicef.prop y0 => + H1 (let__H9 x y H x0 y0 H0 H1 H2 H3 H4 H5 H6 H7). + +def let__H12 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) -> + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) -> + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.and y y0) -> + H6 : cupicef.Term cupicef.prop y -> + H7 : cupicef.Term cupicef.prop y0 -> + cupicef.Term cupicef.prop y0 + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.and y y0) => + H6 : cupicef.Term cupicef.prop y => + H7 : cupicef.Term cupicef.prop y0 => + H3 (let__H10 x y H x0 y0 H0 H1 H2 H3 H4 H5 H6 H7). + +def and__iff__morphism__obligation__1 : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.respectful + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (__ : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + Coq__Init__Logic.iff + (Coq__Classes__Morphisms.respectful + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff Coq__Init__Logic.iff) + Coq__Init__Logic.and Coq__Init__Logic.and) + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I x + (__ : cupicef.Term cupicef.prop x => y)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I y + (__ : cupicef.Term cupicef.prop y => x)) + (Coq__Init__Logic.iff (Coq__Init__Logic.and x x0) + (Coq__Init__Logic.and y y0)) + (H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I x0 + (__ : cupicef.Term cupicef.prop x0 => y0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I y0 + (__ : cupicef.Term cupicef.prop y0 => x0)) + (Coq__Init__Logic.iff (Coq__Init__Logic.and x x0) + (Coq__Init__Logic.and y y0)) + (H3 : + (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : + (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and x x0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.and x x0) => + Coq__Init__Logic.and y y0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and y y0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.and y y0) => + Coq__Init__Logic.and x x0)) + (H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.and x x0) => + Coq__Init__Logic.and__ind x x0 (Coq__Init__Logic.and y y0) + (H6 : cupicef.Term cupicef.prop x => + H7 : cupicef.Term cupicef.prop x0 => + Coq__Init__Logic.conj y y0 + (let__H6 x y H x0 y0 H0 H1 H2 H3 H4 H5 H6 H7) + (let__H7 x y H x0 y0 H0 H1 H2 H3 H4 H5 H6 H7)) + H5) + (H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.and y y0) => + Coq__Init__Logic.and__ind y y0 (Coq__Init__Logic.and x x0) + (H6 : cupicef.Term cupicef.prop y => + H7 : cupicef.Term cupicef.prop y0 => + Coq__Init__Logic.conj x x0 + (let__H9 x y H x0 y0 H0 H1 H2 H3 H4 H5 H6 H7) + (let__H10 x y H x0 y0 H0 H1 H2 H3 H4 H5 H6 H7)) + H5)) + H0) + H. + +def and__iff__morphism : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.Proper + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (__ : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (__0 : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (Coq__Classes__Morphisms.respectful + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (__ : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + Coq__Init__Logic.iff + (Coq__Classes__Morphisms.respectful + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Init__Logic.iff Coq__Init__Logic.iff)) + Coq__Init__Logic.and) + +:= and__iff__morphism__obligation__1. + +#TYPE_CLASS_INSTANCE and__iff__morphism. + +def let__H13 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.or x x0) -> + H2 : cupicef.Term cupicef.prop x -> + cupicef.Term cupicef.prop y + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.or x x0) => + H2 : cupicef.Term cupicef.prop x => + H H2. + +def let__H14 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.or x x0) -> + H2 : cupicef.Term cupicef.prop x0 -> + cupicef.Term cupicef.prop y0 + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.or x x0) => + H2 : cupicef.Term cupicef.prop x0 => + H0 H2. + +def or__impl__morphism__obligation__1 : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.respectful + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (__ : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + Coq__Program__Basics.impl + (Coq__Classes__Morphisms.respectful + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl Coq__Program__Basics.impl) + Coq__Init__Logic.or Coq__Init__Logic.or) + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.or x x0) => + Coq__Init__Logic.or__ind x x0 (Coq__Init__Logic.or y y0) + (H2 : cupicef.Term cupicef.prop x => + Coq__Init__Logic.or__introl y y0 (let__H13 x y H x0 y0 H0 H1 H2)) + (H2 : cupicef.Term cupicef.prop x0 => + Coq__Init__Logic.or__intror y y0 (let__H14 x y H x0 y0 H0 H1 H2)) + H1. + +def or__impl__morphism : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.Proper + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (__ : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (__0 : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (Coq__Classes__Morphisms.respectful + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (__ : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + Coq__Program__Basics.impl + (Coq__Classes__Morphisms.respectful + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Program__Basics.impl Coq__Program__Basics.impl)) + Coq__Init__Logic.or) + +:= or__impl__morphism__obligation__1. + +#TYPE_CLASS_INSTANCE or__impl__morphism. + +def let__H15 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) -> + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) -> + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.or x x0) -> + H6 : cupicef.Term cupicef.prop x -> + cupicef.Term cupicef.prop y + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.or x x0) => + H6 : cupicef.Term cupicef.prop x => + H1 H6. + +def let__H16 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) -> + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) -> + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.or x x0) -> + H6 : cupicef.Term cupicef.prop x -> + cupicef.Term cupicef.prop x + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.or x x0) => + H6 : cupicef.Term cupicef.prop x => + H2 (let__H15 x y H x0 y0 H0 H1 H2 H3 H4 H5 H6). + +def let__H17 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) -> + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) -> + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.or x x0) -> + H6 : cupicef.Term cupicef.prop x0 -> + cupicef.Term cupicef.prop y0 + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.or x x0) => + H6 : cupicef.Term cupicef.prop x0 => + H3 H6. + +def let__H18 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) -> + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) -> + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.or x x0) -> + H6 : cupicef.Term cupicef.prop x0 -> + cupicef.Term cupicef.prop x0 + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.or x x0) => + H6 : cupicef.Term cupicef.prop x0 => + H4 (let__H17 x y H x0 y0 H0 H1 H2 H3 H4 H5 H6). + +def let__H19 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) -> + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) -> + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.or y y0) -> + H6 : cupicef.Term cupicef.prop y -> + cupicef.Term cupicef.prop x + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.or y y0) => + H6 : cupicef.Term cupicef.prop y => + H2 H6. + +def let__H20 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) -> + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) -> + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.or y y0) -> + H6 : cupicef.Term cupicef.prop y -> + cupicef.Term cupicef.prop y + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.or y y0) => + H6 : cupicef.Term cupicef.prop y => + H1 (let__H19 x y H x0 y0 H0 H1 H2 H3 H4 H5 H6). + +def let__H21 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) -> + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) -> + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.or y y0) -> + H6 : cupicef.Term cupicef.prop y0 -> + cupicef.Term cupicef.prop x0 + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.or y y0) => + H6 : cupicef.Term cupicef.prop y0 => + H4 H6. + +def let__H22 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) -> + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) -> + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.or y y0) -> + H6 : cupicef.Term cupicef.prop y0 -> + cupicef.Term cupicef.prop y0 + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.or y y0) => + H6 : cupicef.Term cupicef.prop y0 => + H3 (let__H21 x y H x0 y0 H0 H1 H2 H3 H4 H5 H6). + +def or__iff__morphism__obligation__1 : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.respectful + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (__ : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + Coq__Init__Logic.iff + (Coq__Classes__Morphisms.respectful + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff Coq__Init__Logic.iff) + Coq__Init__Logic.or Coq__Init__Logic.or) + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I x + (__ : cupicef.Term cupicef.prop x => y)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I y + (__ : cupicef.Term cupicef.prop y => x)) + (Coq__Init__Logic.iff (Coq__Init__Logic.or x x0) + (Coq__Init__Logic.or y y0)) + (H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I x0 + (__ : cupicef.Term cupicef.prop x0 => y0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I y0 + (__ : cupicef.Term cupicef.prop y0 => x0)) + (Coq__Init__Logic.iff (Coq__Init__Logic.or x x0) + (Coq__Init__Logic.or y y0)) + (H3 : + (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : + (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or x x0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.or x x0) => + Coq__Init__Logic.or y y0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or y y0) + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.or y y0) => + Coq__Init__Logic.or x x0)) + (H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.or x x0) => + Coq__Init__Logic.or__ind x x0 (Coq__Init__Logic.or y y0) + (H6 : cupicef.Term cupicef.prop x => + Coq__Init__Logic.or__introl y y0 + (let__H15 x y H x0 y0 H0 H1 H2 H3 H4 H5 H6)) + (H6 : cupicef.Term cupicef.prop x0 => + Coq__Init__Logic.or__intror y y0 + (let__H17 x y H x0 y0 H0 H1 H2 H3 H4 H5 H6)) + H5) + (H5 : cupicef.Term cupicef.prop (Coq__Init__Logic.or y y0) => + Coq__Init__Logic.or__ind y y0 (Coq__Init__Logic.or x x0) + (H6 : cupicef.Term cupicef.prop y => + Coq__Init__Logic.or__introl x x0 + (let__H19 x y H x0 y0 H0 H1 H2 H3 H4 H5 H6)) + (H6 : cupicef.Term cupicef.prop y0 => + Coq__Init__Logic.or__intror x x0 + (let__H21 x y H x0 y0 H0 H1 H2 H3 H4 H5 H6)) + H5)) + H0) + H. + +def or__iff__morphism : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.Proper + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (__ : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (__0 : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (Coq__Classes__Morphisms.respectful + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (__ : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + Coq__Init__Logic.iff + (Coq__Classes__Morphisms.respectful + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Init__Logic.iff Coq__Init__Logic.iff)) + Coq__Init__Logic.or) + +:= or__iff__morphism__obligation__1. + +#TYPE_CLASS_INSTANCE or__iff__morphism. + +def let__H23 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) -> + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) -> + H5 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop x0) -> + H6 : cupicef.Term cupicef.prop y -> + cupicef.Term cupicef.prop x + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + H5 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop x0) => + H6 : cupicef.Term cupicef.prop y => + H2 H6. + +def let__H24 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) -> + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) -> + H5 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop x0) -> + H6 : cupicef.Term cupicef.prop y -> + cupicef.Term cupicef.prop y + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + H5 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop x0) => + H6 : cupicef.Term cupicef.prop y => + H1 (let__H23 x y H x0 y0 H0 H1 H2 H3 H4 H5 H6). + +def let__H25 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) -> + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) -> + H5 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop x0) -> + H6 : cupicef.Term cupicef.prop y -> + cupicef.Term cupicef.prop x0 + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + H5 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop x0) => + H6 : cupicef.Term cupicef.prop y => + H5 (let__H23 x y H x0 y0 H0 H1 H2 H3 H4 H5 H6). + +def let__H26 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) -> + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) -> + H5 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop x0) -> + H6 : cupicef.Term cupicef.prop y -> + cupicef.Term cupicef.prop y0 + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + H5 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop x0) => + H6 : cupicef.Term cupicef.prop y => + H3 (let__H25 x y H x0 y0 H0 H1 H2 H3 H4 H5 H6). + +def let__H27 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) -> + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) -> + H5 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop x0) -> + H6 : cupicef.Term cupicef.prop y -> + cupicef.Term cupicef.prop x0 + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + H5 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop x0) => + H6 : cupicef.Term cupicef.prop y => + H4 (let__H26 x y H x0 y0 H0 H1 H2 H3 H4 H5 H6). + +def let__H28 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) -> + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) -> + H5 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop y0) -> + H6 : cupicef.Term cupicef.prop x -> + cupicef.Term cupicef.prop y + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + H5 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop y0) => + H6 : cupicef.Term cupicef.prop x => + H1 H6. + +def let__H29 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) -> + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) -> + H5 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop y0) -> + H6 : cupicef.Term cupicef.prop x -> + cupicef.Term cupicef.prop x + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + H5 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop y0) => + H6 : cupicef.Term cupicef.prop x => + H2 (let__H28 x y H x0 y0 H0 H1 H2 H3 H4 H5 H6). + +def let__H30 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) -> + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) -> + H5 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop y0) -> + H6 : cupicef.Term cupicef.prop x -> + cupicef.Term cupicef.prop y0 + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + H5 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop y0) => + H6 : cupicef.Term cupicef.prop x => + H5 (let__H28 x y H x0 y0 H0 H1 H2 H3 H4 H5 H6). + +def let__H31 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) -> + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) -> + H5 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop y0) -> + H6 : cupicef.Term cupicef.prop x -> + cupicef.Term cupicef.prop x0 + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + H5 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop y0) => + H6 : cupicef.Term cupicef.prop x => + H4 (let__H30 x y H x0 y0 H0 H1 H2 H3 H4 H5 H6). + +def let__H32 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) -> + x0 : cupicef.Univ cupicef.prop -> + y0 : cupicef.Univ cupicef.prop -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) -> + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) -> + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) -> + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) -> + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) -> + H5 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop y0) -> + H6 : cupicef.Term cupicef.prop x -> + cupicef.Term cupicef.prop y0 + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + H3 : (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + H5 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop y0) => + H6 : cupicef.Term cupicef.prop x => + H3 (let__H31 x y H x0 y0 H0 H1 H2 H3 H4 H5 H6). + +def iff__iff__iff__impl__morphism__obligation__1 : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.respectful + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (__ : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + Coq__Init__Logic.iff + (Coq__Classes__Morphisms.respectful + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff Coq__Init__Logic.iff) + Coq__Program__Basics.impl Coq__Program__Basics.impl) + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x y) => + x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff x0 y0) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I x + (__ : cupicef.Term cupicef.prop x => y)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I y + (__ : cupicef.Term cupicef.prop y => x)) + (Coq__Init__Logic.iff + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I x + (__ : cupicef.Term cupicef.prop x => x0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I y + (__ : cupicef.Term cupicef.prop y => y0))) + (H1 : (__ : cupicef.Term cupicef.prop x -> cupicef.Term cupicef.prop y) => + H2 : (__ : cupicef.Term cupicef.prop y -> cupicef.Term cupicef.prop x) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I x0 + (__ : cupicef.Term cupicef.prop x0 => y0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I y0 + (__ : cupicef.Term cupicef.prop y0 => x0)) + (Coq__Init__Logic.iff + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I x + (__ : cupicef.Term cupicef.prop x => x0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I y + (__ : cupicef.Term cupicef.prop y => y0))) + (H3 : + (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H4 : + (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + Coq__Init__Logic.conj + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I x + (__ : cupicef.Term cupicef.prop x => x0)) + (__ : + (__ : cupicef.Term cupicef.prop x -> + cupicef.Term cupicef.prop x0) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I y + (__0 : cupicef.Term cupicef.prop y => y0))) + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I y + (__ : cupicef.Term cupicef.prop y => y0)) + (__ : + (__ : cupicef.Term cupicef.prop y -> + cupicef.Term cupicef.prop y0) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I x + (__0 : cupicef.Term cupicef.prop x => x0))) + (H5 : + (__ : cupicef.Term cupicef.prop x -> + cupicef.Term cupicef.prop x0) => + H6 : cupicef.Term cupicef.prop y => + let__H26 x y H x0 y0 H0 H1 H2 H3 H4 H5 H6) + (H5 : + (__ : cupicef.Term cupicef.prop y -> + cupicef.Term cupicef.prop y0) => + H6 : cupicef.Term cupicef.prop x => + let__H31 x y H x0 y0 H0 H1 H2 H3 H4 H5 H6)) + H0) + H. + +def iff__iff__iff__impl__morphism : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.Proper + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (__ : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (__0 : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (Coq__Classes__Morphisms.respectful + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (__ : cupicef.Univ cupicef.prop => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + Coq__Init__Logic.iff + (Coq__Classes__Morphisms.respectful + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Init__Logic.iff Coq__Init__Logic.iff)) + Coq__Program__Basics.impl) + +:= iff__iff__iff__impl__morphism__obligation__1. + +#TYPE_CLASS_INSTANCE iff__iff__iff__impl__morphism. + +def ex__iff__morphism__obligation__1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Classes__Morphisms.pointwise__relation A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff) + Coq__Init__Logic.iff (Coq__Init__Logic.ex A) (Coq__Init__Logic.ex A)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + y : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.pointwise__relation A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff x y) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.ex A x) + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.ex A x) => + Coq__Init__Logic.ex A y)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.ex A y) + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.ex A y) => + Coq__Init__Logic.ex A x)) + (H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.ex A x) => + Coq__Init__Logic.ex__ind A x (Coq__Init__Logic.ex A y) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H1 : cupicef.Term cupicef.prop (x x0) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (x x0) + (__ : cupicef.Term cupicef.prop (x x0) => y x0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (y x0) + (__ : cupicef.Term cupicef.prop (y x0) => x x0)) + (Coq__Init__Logic.ex A y) + (H2 : + (__ : cupicef.Term cupicef.prop (x x0) -> + cupicef.Term cupicef.prop (y x0)) => + H3 : + (__ : cupicef.Term cupicef.prop (y x0) -> + cupicef.Term cupicef.prop (x x0)) => + Coq__Init__Logic.ex__intro A y x0 (H2 H1)) + (H x0)) + H0) + (H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.ex A y) => + Coq__Init__Logic.ex__ind A y (Coq__Init__Logic.ex A x) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H1 : cupicef.Term cupicef.prop (y x0) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (x x0) + (__ : cupicef.Term cupicef.prop (x x0) => y x0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (y x0) + (__ : cupicef.Term cupicef.prop (y x0) => x x0)) + (Coq__Init__Logic.ex A x) + (H2 : + (__ : cupicef.Term cupicef.prop (x x0) -> + cupicef.Term cupicef.prop (y x0)) => + H3 : + (__ : cupicef.Term cupicef.prop (y x0) -> + cupicef.Term cupicef.prop (x x0)) => + Coq__Init__Logic.ex__intro A x x0 (H3 H1)) + (H x0)) + H0). + +def ex__iff__morphism : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (__ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (Coq__Classes__Morphisms.respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Classes__Morphisms.pointwise__relation A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Init__Logic.iff) + Coq__Init__Logic.iff) + (Coq__Init__Logic.ex A)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + (A0 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + ex__iff__morphism__obligation__1 A0) A. + +#TYPE_CLASS_INSTANCE ex__iff__morphism. + +def ex__impl__morphism__obligation__1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Classes__Morphisms.pointwise__relation A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl) + Coq__Program__Basics.impl (Coq__Init__Logic.ex A) + (Coq__Init__Logic.ex A)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + y : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.pointwise__relation A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl x y) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.ex A x) => + Coq__Init__Logic.ex__ind A x (Coq__Init__Logic.ex A y) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H1 : cupicef.Term cupicef.prop (x x0) => + Coq__Init__Logic.ex__intro A y x0 (H x0 H1)) + H0. + +def ex__impl__morphism : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (__ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (Coq__Classes__Morphisms.respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Classes__Morphisms.pointwise__relation A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Program__Basics.impl) + Coq__Program__Basics.impl) + (Coq__Init__Logic.ex A)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + (A0 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + ex__impl__morphism__obligation__1 A0) A. + +#TYPE_CLASS_INSTANCE ex__impl__morphism. + +def ex__flip__impl__morphism__obligation__1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Classes__Morphisms.pointwise__relation A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Program__Basics.flip + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Program__Basics.impl)) + (Coq__Program__Basics.flip + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl) + (Coq__Init__Logic.ex A) (Coq__Init__Logic.ex A)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + y : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.pointwise__relation A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Program__Basics.flip + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Program__Basics.impl) + x y) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.ex A y) => + Coq__Init__Logic.ex__ind A y (Coq__Init__Logic.ex A x) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H1 : cupicef.Term cupicef.prop (y x0) => + Coq__Init__Logic.ex__intro A x x0 (H x0 H1)) + H0. + +def ex__flip__impl__morphism : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (__ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (Coq__Classes__Morphisms.respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Classes__Morphisms.pointwise__relation A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Program__Basics.flip + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Program__Basics.impl)) + (Coq__Program__Basics.flip + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Program__Basics.impl)) + (Coq__Init__Logic.ex A)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + (A0 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + ex__flip__impl__morphism__obligation__1 A0) A. + +#TYPE_CLASS_INSTANCE ex__flip__impl__morphism. + +def all__iff__morphism__obligation__1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Classes__Morphisms.pointwise__relation A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff) + Coq__Init__Logic.iff (Coq__Init__Logic.all A) (Coq__Init__Logic.all A)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + y : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.pointwise__relation A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff x y) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.all A x) + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.all A x) => + Coq__Init__Logic.all A y)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.all A y) + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.all A y) => + Coq__Init__Logic.all A x)) + (H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.all A x) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (x x0) + (__ : cupicef.Term cupicef.prop (x x0) => y x0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (y x0) + (__ : cupicef.Term cupicef.prop (y x0) => x x0)) + (y x0) + (H1 : + (__ : cupicef.Term cupicef.prop (x x0) -> + cupicef.Term cupicef.prop (y x0)) => + H2 : + (__ : cupicef.Term cupicef.prop (y x0) -> + cupicef.Term cupicef.prop (x x0)) => + H1 (H0 x0)) + (H x0)) + (H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.all A y) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (x x0) + (__ : cupicef.Term cupicef.prop (x x0) => y x0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (y x0) + (__ : cupicef.Term cupicef.prop (y x0) => x x0)) + (x x0) + (H1 : + (__ : cupicef.Term cupicef.prop (x x0) -> + cupicef.Term cupicef.prop (y x0)) => + H2 : + (__ : cupicef.Term cupicef.prop (y x0) -> + cupicef.Term cupicef.prop (x x0)) => + H2 (H0 x0)) + (H x0)). + +def all__iff__morphism : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (__ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (Coq__Classes__Morphisms.respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Classes__Morphisms.pointwise__relation A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Init__Logic.iff) + Coq__Init__Logic.iff) + (Coq__Init__Logic.all A)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + (A0 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + all__iff__morphism__obligation__1 A0) A. + +#TYPE_CLASS_INSTANCE all__iff__morphism. + +def all__impl__morphism__obligation__1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Classes__Morphisms.pointwise__relation A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl) + Coq__Program__Basics.impl (Coq__Init__Logic.all A) + (Coq__Init__Logic.all A)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + y : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.pointwise__relation A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (A0 : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A0 + (__ : cupicef.Term cupicef.prop A0 => B)) + x y) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.all A x) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H x0 (H0 x0). + +def all__impl__morphism : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (__ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (Coq__Classes__Morphisms.respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Classes__Morphisms.pointwise__relation A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Program__Basics.impl) + Coq__Program__Basics.impl) + (Coq__Init__Logic.all A)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + (A0 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + all__impl__morphism__obligation__1 A0) A. + +#TYPE_CLASS_INSTANCE all__impl__morphism. + +def all__flip__impl__morphism__obligation__1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Classes__Morphisms.pointwise__relation A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Program__Basics.flip + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Program__Basics.impl)) + (Coq__Program__Basics.flip + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl) + (Coq__Init__Logic.all A) (Coq__Init__Logic.all A)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + y : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.pointwise__relation A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (x0 : cupicef.Univ cupicef.prop => + y0 : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I y0 + (__ : cupicef.Term cupicef.prop y0 => x0)) + x y) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.all A y) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H x0 (H0 x0). + +def all__flip__impl__morphism : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (__ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (Coq__Classes__Morphisms.respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Classes__Morphisms.pointwise__relation A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Program__Basics.flip + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Program__Basics.impl)) + (Coq__Program__Basics.flip + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Program__Basics.impl)) + (Coq__Init__Logic.all A)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + (A0 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + all__flip__impl__morphism__obligation__1 A0) A. + +#TYPE_CLASS_INSTANCE all__flip__impl__morphism. + +thm Acc__pt__morphism : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + E : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + H : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Equivalence A E) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (Coq__Classes__Morphisms.respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + E + (Coq__Classes__Morphisms.respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + E Coq__Init__Logic.iff)) + R) -> + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (Coq__Classes__Morphisms.respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + E Coq__Init__Logic.iff) + (Coq__Init__Wf.Acc A R)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + E : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A E) => + H0 : + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (Coq__Classes__Morphisms.respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + E + (Coq__Classes__Morphisms.respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + E Coq__Init__Logic.iff)) + R) => + Coq__Classes__Morphisms.proper__sym__impl__iff A E + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + E x y) (H1 : cupicef.Term cupicef.prop (E x y) => E y x)))) + (Coq__Classes__RelationClasses.Symmetric A E) cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H1 : cupicef.Term cupicef.prop (E x y) => + Coq__Classes__RelationClasses.symmetry A E + (Coq__Classes__RelationClasses.Equivalence__Symmetric A E H) x y + H1)) + (Coq__Init__Wf.Acc A R) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + E x y) + (EQ : cupicef.Term cupicef.prop (E x y) => + Coq__Program__Basics.impl (Coq__Init__Wf.Acc A R x) + (Coq__Init__Wf.Acc A R y))))) + (Coq__Classes__Morphisms.Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__Morphisms.respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + E Coq__Program__Basics.impl) + (Coq__Init__Wf.Acc A R)) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + EQ : cupicef.Term cupicef.prop (E x y) => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Wf.Acc A R x) + (WF : cupicef.Term cupicef.prop (Coq__Init__Wf.Acc A R x) => + Coq__Init__Wf.Acc A R y)) + (Coq__Program__Basics.impl (Coq__Init__Wf.Acc A R x) + (Coq__Init__Wf.Acc A R y)) + cupicef.I + (WF : cupicef.Term cupicef.prop (Coq__Init__Wf.Acc A R x) => + Coq__Init__Wf.Acc__intro A R y + (z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hz : cupicef.Term cupicef.prop (R z y) => + Coq__Init__Wf.Acc__inv A R x WF z + ((lemma : cupicef.Term cupicef.prop (E x y) => + Coq__Classes__Morphisms.Reflexive__partial__app__morphism + A + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + E + (Coq__Classes__Morphisms.respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + E Coq__Program__Basics.impl) + R + (Coq__Classes__Morphisms.subrelation__proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) + (Coq__Classes__Morphisms.respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + E + (Coq__Classes__Morphisms.respectful A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + E Coq__Init__Logic.iff)) + R H0 + (Coq__Classes__Morphisms.respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + E + (Coq__Classes__Morphisms.respectful A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + E Coq__Program__Basics.impl)) + Coq__Init__Datatypes.tt + (Coq__Classes__Morphisms.subrelation__respectful A + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + E E + (Coq__Classes__Morphisms.subrelation__refl A E) + (Coq__Classes__Morphisms.respectful A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + E Coq__Init__Logic.iff) + (Coq__Classes__Morphisms.respectful A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + E Coq__Program__Basics.impl) + (Coq__Classes__Morphisms.subrelation__respectful A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + E E + (Coq__Classes__Morphisms.subrelation__refl A E) + Coq__Init__Logic.iff Coq__Program__Basics.impl + Coq__Classes__Morphisms.iff__impl__subrelation))) + z + (Coq__Classes__Morphisms.reflexive__proper__proxy A E + (Coq__Classes__RelationClasses.Equivalence__Reflexive + A E H) + z) + y x + (Coq__Classes__RelationClasses.symmetry A E + (Coq__Classes__RelationClasses.Equivalence__Symmetric + A E H) + x y lemma)) + EQ Hz))))). + +#TYPE_CLASS_INSTANCE Acc__pt__morphism. + +def let__H33 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + EQ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.relation__equivalence A R R_27_) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + a_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Ha : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a a_27_) -> + WF : cupicef.Term cupicef.prop (Coq__Init__Wf.Acc A R a) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __tmp : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (Coq__Init__Wf.Acc A R y)) -> + WF_27_ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (Coq__Init__Wf.Acc A R_27_ y)) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Ryx : cupicef.Term cupicef.prop (R_27_ y x) -> + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R_27_ x0 x1) -> + cupicef.Term cupicef.prop (R x0 x1) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + EQ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.relation__equivalence A R R_27_) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + a_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Ha : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a a_27_) => + WF : cupicef.Term cupicef.prop (Coq__Init__Wf.Acc A R a) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + __tmp : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (Coq__Init__Wf.Acc A R y)) => + WF_27_ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (Coq__Init__Wf.Acc A R_27_ y)) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Ryx : cupicef.Term cupicef.prop (R_27_ y x) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x0 x1) + (__ : cupicef.Term cupicef.prop (R x0 x1) => R_27_ x0 x1)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R_27_ x0 x1) + (__ : cupicef.Term cupicef.prop (R_27_ x0 x1) => R x0 x1)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x0 x1) + (__ : cupicef.Term cupicef.prop (R x0 x1) => R_27_ x0 x1)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (R_27_ x0 x1) + (__ : cupicef.Term cupicef.prop (R_27_ x0 x1) => R x0 x1))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R_27_ x0 x1) + (__0 : cupicef.Term cupicef.prop (R_27_ x0 x1) => R x0 x1)) + (__ : + (__ : cupicef.Term cupicef.prop (R x0 x1) -> + cupicef.Term cupicef.prop (R_27_ x0 x1)) => + __0 : + (__0 : cupicef.Term cupicef.prop (R_27_ x0 x1) -> + cupicef.Term cupicef.prop (R x0 x1)) => + __0) + (EQ x0 x1). + +thm Acc__rel__morphism : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (Coq__Classes__Morphisms.respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__RelationClasses.relation__equivalence A) + (Coq__Classes__Morphisms.respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + Coq__Init__Logic.iff)) + (Coq__Init__Wf.Acc A)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + Coq__Classes__Morphisms.proper__sym__impl__iff__2 + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (Coq__Classes__RelationClasses.relation__equivalence A) + (cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (x : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (y : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Classes__RelationClasses.relation__equivalence A x y) + (H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.relation__equivalence A + x y) => + Coq__Classes__RelationClasses.relation__equivalence A y x)))) + (Coq__Classes__RelationClasses.Symmetric + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (Coq__Classes__RelationClasses.relation__equivalence A)) + cupicef.I + (x : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + y : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.relation__equivalence A x y) => + Coq__Classes__RelationClasses.symmetry + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (Coq__Classes__RelationClasses.relation__equivalence A) + (Coq__Classes__RelationClasses.Equivalence__Symmetric + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (Coq__Classes__RelationClasses.relation__equivalence A) + (Coq__Classes__RelationClasses.relation__equivalence__equivalence + A)) + x y H)) + A + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y x)))) + (Coq__Classes__RelationClasses.Symmetric A + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A))) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y H)) + (Coq__Init__Wf.Acc A) + (cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Classes__RelationClasses.relation__equivalence A R + R_27_) + (EQ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.relation__equivalence A + R R_27_) => + Coq__Classes__Morphisms.respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + Coq__Program__Basics.impl (Coq__Init__Wf.Acc A R) + (Coq__Init__Wf.Acc A R_27_))))) + (Coq__Classes__Morphisms.Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (Coq__Classes__Morphisms.respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__RelationClasses.relation__equivalence A) + (Coq__Classes__Morphisms.respectful A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + Coq__Program__Basics.impl)) + (Coq__Init__Wf.Acc A)) + cupicef.I + (R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + EQ : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.relation__equivalence A R R_27_) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (a_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a a_27_) + (Ha : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a a_27_) => + Coq__Program__Basics.impl (Coq__Init__Wf.Acc A R a) + (Coq__Init__Wf.Acc A R_27_ a_27_))))) + (Coq__Classes__Morphisms.respectful A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + Coq__Program__Basics.impl (Coq__Init__Wf.Acc A R) + (Coq__Init__Wf.Acc A R_27_)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + a_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Ha : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a a_27_) => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Wf.Acc A R a) + (WF : cupicef.Term cupicef.prop (Coq__Init__Wf.Acc A R a) => + (a_27_0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Wf.Acc A R_27_ a_27_0) a_27_)) + (Coq__Program__Basics.impl (Coq__Init__Wf.Acc A R a) + (Coq__Init__Wf.Acc A R_27_ a_27_)) + cupicef.I + (WF : cupicef.Term cupicef.prop (Coq__Init__Wf.Acc A R a) => + Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a + (a_27_0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Wf.Acc A R_27_ a_27_0) + (Coq__Init__Wf.Acc__ind A R + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Wf.Acc A R_27_ a0) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + __tmp : + (y : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (Coq__Init__Wf.Acc A R y)) => + WF_27_ : + (y : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop + (Coq__Init__Wf.Acc A R_27_ y)) => + Coq__Init__Wf.Acc__intro A R_27_ x + (y : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Ryx : cupicef.Term cupicef.prop (R_27_ y x) => + WF_27_ y + (let__H33 A R R_27_ EQ a a_27_ Ha WF x __tmp WF_27_ + y Ryx y x Ryx))) + a WF) + a_27_ Ha)))). + +#TYPE_CLASS_INSTANCE Acc__rel__morphism. + +thm well__founded__morphism : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term cupicef.prop + (Coq__Classes__Morphisms.Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (Coq__Classes__Morphisms.respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Classes__RelationClasses.relation__equivalence A) + Coq__Init__Logic.iff) + (Coq__Init__Wf.well__founded A)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Classes__Morphisms.Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (Coq__Classes__Morphisms.respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Classes__RelationClasses.relation__equivalence A) + Coq__Init__Logic.iff) + (R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Wf.Acc A R a))) + (Coq__Classes__Morphisms.Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (Coq__Classes__Morphisms.respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Classes__RelationClasses.relation__equivalence A) + Coq__Init__Logic.iff) + (Coq__Init__Wf.well__founded A)) + cupicef.I + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Classes__Morphisms.respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Classes__RelationClasses.relation__equivalence A) + Coq__Init__Logic.iff + (R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Wf.Acc A R a)) + (R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Wf.Acc A R a))) + (Coq__Classes__Morphisms.Proper + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (Coq__Classes__Morphisms.respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Classes__RelationClasses.relation__equivalence A) + Coq__Init__Logic.iff) + (R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Wf.Acc A R a))) + cupicef.I + (cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (x : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (Coq__Relations__Relation__Definitions.relation A) + (y : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Classes__RelationClasses.relation__equivalence A x + y) + (H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.relation__equivalence + A x y) => + Coq__Init__Logic.iff + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Wf.Acc A x a)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Wf.Acc A y a)))))) + (Coq__Classes__Morphisms.respectful + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Classes__RelationClasses.relation__equivalence A) + Coq__Init__Logic.iff + (R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Wf.Acc A R a)) + (R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Wf.Acc A R a))) + cupicef.I + (x : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + y : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.relation__equivalence A x y) => + Coq__Classes__Morphisms.trans__co__eq__inv__impl__morphism + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Init__Logic.iff + Coq__Classes__RelationClasses.iff__Transitive + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Wf.Acc A x a)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Wf.Acc A y a)) + (all__iff__morphism A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Wf.Acc A x a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Wf.Acc A y a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Acc__rel__morphism A x y H a a + (Coq__Classes__Morphisms.eq__proper__proxy A a))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Wf.Acc A y a)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Wf.Acc A y a)) + (Coq__Classes__Morphisms.eq__proper__proxy + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Wf.Acc A y a))) + (Coq__Classes__RelationClasses.reflexivity + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + Coq__Init__Logic.iff + Coq__Classes__RelationClasses.iff__Reflexive + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Wf.Acc A y a)))))). + +#TYPE_CLASS_INSTANCE well__founded__morphism. + +(; End of translation. ;) diff --git a/out/Coq__Classes__RelationClasses.dk b/out/Coq__Classes__RelationClasses.dk new file mode 100644 index 000000000..d15be60fa --- /dev/null +++ b/out/Coq__Classes__RelationClasses.dk @@ -0,0 +1,17089 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def Reflexive : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => R x x). + +#TYPE_CLASS Reflexive. + +def reflexivity : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + Reflexive0 : cupicef.Term cupicef.prop (Reflexive A R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (R x x) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + Reflexive0 : cupicef.Term cupicef.prop (Reflexive A R) => + Reflexive0. + +def complement : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x y) + (__ : cupicef.Term cupicef.prop (R x y) => Coq__Init__Logic.False). + +thm complement__inverse : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (complement A + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R)) + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (complement A R))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (complement A R)). + +def Irreflexive : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + Reflexive A (complement A R). + +#TYPE_CLASS Irreflexive. + +def irreflexivity : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + Irreflexive0 : cupicef.Term cupicef.prop (Irreflexive A R) -> + cupicef.Term cupicef.prop (Reflexive A (complement A R)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + Irreflexive0 : cupicef.Term cupicef.prop (Irreflexive A R) => + Irreflexive0. + +def Symmetric : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x y) + (__ : cupicef.Term cupicef.prop (R x y) => R y x))). + +#TYPE_CLASS Symmetric. + +def symmetry : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + Symmetric0 : cupicef.Term cupicef.prop (Symmetric A R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (R y x) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + Symmetric0 : cupicef.Term cupicef.prop (Symmetric A R) => + Symmetric0. + +def Asymmetric : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (R x y) + (__ : cupicef.Term cupicef.prop (R x y) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R y x) + (__0 : cupicef.Term cupicef.prop (R y x) => + Coq__Init__Logic.False)))). + +#TYPE_CLASS Asymmetric. + +def asymmetry : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + Asymmetric0 : cupicef.Term cupicef.prop (Asymmetric A R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + __0 : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + Asymmetric0 : cupicef.Term cupicef.prop (Asymmetric A R) => + Asymmetric0. + +def Transitive : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (R x y) + (__ : cupicef.Term cupicef.prop (R x y) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + R y z) (__0 : cupicef.Term cupicef.prop (R y z) => R x z))))). + +#TYPE_CLASS Transitive. + +def transitivity : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + Transitive0 : cupicef.Term cupicef.prop (Transitive A R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + __0 : cupicef.Term cupicef.prop (R y z) -> + cupicef.Term cupicef.prop (R x z) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + Transitive0 : cupicef.Term cupicef.prop (Transitive A R) => + Transitive0. + +def PreOrder : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Univ cupicef.prop. + +#TYPE_CLASS PreOrder. + +def Build__PreOrder : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + PreOrder__Reflexive0 : cupicef.Term cupicef.prop (Reflexive A R) -> + PreOrder__Transitive0 : cupicef.Term cupicef.prop (Transitive A R) -> + cupicef.Term cupicef.prop (PreOrder A R). + +[ ] cupicef.code_guarded? _ (Build__PreOrder _ _ _ _) --> cupicef.guarded. + +def match____PreOrder : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + P : (cupicef.Term cupicef.prop (PreOrder A R) -> cupicef.Univ s) -> + case__Build__PreOrder : + (PreOrder__Reflexive0 : cupicef.Term cupicef.prop (Reflexive A R) -> + PreOrder__Transitive0 : cupicef.Term cupicef.prop (Transitive A R) -> + cupicef.Term s + (P (Build__PreOrder A R PreOrder__Reflexive0 PreOrder__Transitive0))) -> + x : cupicef.Term cupicef.prop (PreOrder A R) -> + cupicef.Term s (P x). + +[ s, A, R, P, case__Build__PreOrder, PreOrder__Reflexive0, PreOrder__Transitive0] + match____PreOrder s A R P case__Build__PreOrder + (Build__PreOrder A R PreOrder__Reflexive0 PreOrder__Transitive0) + --> + case__Build__PreOrder PreOrder__Reflexive0 PreOrder__Transitive0. + +[ s, A, R, P, s'] + match____PreOrder s A R + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____PreOrder s' A R (x => P x). + +def PreOrder__Reflexive : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + PreOrder0 : cupicef.Term cupicef.prop (PreOrder A R) -> + cupicef.Term cupicef.prop (Reflexive A R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + PreOrder0 : cupicef.Term cupicef.prop (PreOrder A R) => + match____PreOrder cupicef.prop A R + (PreOrder1 : cupicef.Term cupicef.prop (PreOrder A R) => Reflexive A R) + (PreOrder__Reflexive0 : cupicef.Term cupicef.prop (Reflexive A R) => + PreOrder__Transitive0 : cupicef.Term cupicef.prop (Transitive A R) => + PreOrder__Reflexive0) + PreOrder0. + +#TYPE_CLASS_INSTANCE PreOrder__Reflexive. + +def PreOrder__Transitive : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + PreOrder0 : cupicef.Term cupicef.prop (PreOrder A R) -> + cupicef.Term cupicef.prop (Transitive A R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + PreOrder0 : cupicef.Term cupicef.prop (PreOrder A R) => + match____PreOrder cupicef.prop A R + (PreOrder1 : cupicef.Term cupicef.prop (PreOrder A R) => Transitive A R) + (PreOrder__Reflexive0 : cupicef.Term cupicef.prop (Reflexive A R) => + PreOrder__Transitive0 : cupicef.Term cupicef.prop (Transitive A R) => + PreOrder__Transitive0) + PreOrder0. + +#TYPE_CLASS_INSTANCE PreOrder__Transitive. + +def StrictOrder : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Univ cupicef.prop. + +#TYPE_CLASS StrictOrder. + +def Build__StrictOrder : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + StrictOrder__Irreflexive0 : cupicef.Term cupicef.prop (Irreflexive A R) -> + StrictOrder__Transitive0 : cupicef.Term cupicef.prop (Transitive A R) -> + cupicef.Term cupicef.prop (StrictOrder A R). + +[ ] cupicef.code_guarded? _ (Build__StrictOrder _ _ _ _) --> cupicef.guarded. + +def match____StrictOrder : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + P : (cupicef.Term cupicef.prop (StrictOrder A R) -> cupicef.Univ s) -> + case__Build__StrictOrder : + (StrictOrder__Irreflexive0 : cupicef.Term cupicef.prop (Irreflexive A R) -> + StrictOrder__Transitive0 : cupicef.Term cupicef.prop (Transitive A R) -> + cupicef.Term s + (P + (Build__StrictOrder A R StrictOrder__Irreflexive0 + StrictOrder__Transitive0))) -> + x : cupicef.Term cupicef.prop (StrictOrder A R) -> + cupicef.Term s (P x). + +[ s, A, R, P, case__Build__StrictOrder, StrictOrder__Irreflexive0, StrictOrder__Transitive0] + match____StrictOrder s A R P case__Build__StrictOrder + (Build__StrictOrder A R StrictOrder__Irreflexive0 + StrictOrder__Transitive0) + --> + case__Build__StrictOrder StrictOrder__Irreflexive0 StrictOrder__Transitive0. + +[ s, A, R, P, s'] + match____StrictOrder s A R + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____StrictOrder s' A R (x => P x). + +def StrictOrder__Irreflexive : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + StrictOrder0 : cupicef.Term cupicef.prop (StrictOrder A R) -> + cupicef.Term cupicef.prop (Irreflexive A R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + StrictOrder0 : cupicef.Term cupicef.prop (StrictOrder A R) => + match____StrictOrder cupicef.prop A R + (StrictOrder1 : cupicef.Term cupicef.prop (StrictOrder A R) => + Irreflexive A R) + (StrictOrder__Irreflexive0 : cupicef.Term cupicef.prop (Irreflexive A R) => + StrictOrder__Transitive0 : cupicef.Term cupicef.prop (Transitive A R) => + StrictOrder__Irreflexive0) + StrictOrder0. + +#TYPE_CLASS_INSTANCE StrictOrder__Irreflexive. + +def StrictOrder__Transitive : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + StrictOrder0 : cupicef.Term cupicef.prop (StrictOrder A R) -> + cupicef.Term cupicef.prop (Transitive A R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + StrictOrder0 : cupicef.Term cupicef.prop (StrictOrder A R) => + match____StrictOrder cupicef.prop A R + (StrictOrder1 : cupicef.Term cupicef.prop (StrictOrder A R) => + Transitive A R) + (StrictOrder__Irreflexive0 : cupicef.Term cupicef.prop (Irreflexive A R) => + StrictOrder__Transitive0 : cupicef.Term cupicef.prop (Transitive A R) => + StrictOrder__Transitive0) + StrictOrder0. + +#TYPE_CLASS_INSTANCE StrictOrder__Transitive. + +thm StrictOrder__Asymmetric : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (StrictOrder A R) -> + cupicef.Term cupicef.prop (Asymmetric A R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (StrictOrder A R) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (R x y) + (H0 : cupicef.Term cupicef.prop (R x y) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + R y x) + (H1 : cupicef.Term cupicef.prop (R y x) => + Coq__Init__Logic.False))))) + (Asymmetric A R) cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x y) => + H1 : cupicef.Term cupicef.prop (R y x) => + match____StrictOrder cupicef.prop A R + (__ : cupicef.Term cupicef.prop (StrictOrder A R) => + Coq__Init__Logic.False) + (StrictOrder__Irreflexive0 : + cupicef.Term cupicef.prop (Irreflexive A R) => + StrictOrder__Transitive0 : + cupicef.Term cupicef.prop (Transitive A R) => + (H2 : + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y y0) -> + __0 : cupicef.Term cupicef.prop (R y0 z) -> + cupicef.Term cupicef.prop (R y z)) => + (H3 : cupicef.Term cupicef.prop (complement A R y y) => + (H4 : + (z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + __0 : cupicef.Term cupicef.prop (R x z) -> + cupicef.Term cupicef.prop (R y z)) => + H3 (H4 y H1 H0)) (H2 x)) (StrictOrder__Irreflexive0 y)) + (StrictOrder__Transitive0 y)) + H). + +#TYPE_CLASS_INSTANCE StrictOrder__Asymmetric. + +def PER : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Univ cupicef.prop. + +#TYPE_CLASS PER. + +def Build__PER : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + PER__Symmetric0 : cupicef.Term cupicef.prop (Symmetric A R) -> + PER__Transitive0 : cupicef.Term cupicef.prop (Transitive A R) -> + cupicef.Term cupicef.prop (PER A R). + +[ ] cupicef.code_guarded? _ (Build__PER _ _ _ _) --> cupicef.guarded. + +def match____PER : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + P : (cupicef.Term cupicef.prop (PER A R) -> cupicef.Univ s) -> + case__Build__PER : + (PER__Symmetric0 : cupicef.Term cupicef.prop (Symmetric A R) -> + PER__Transitive0 : cupicef.Term cupicef.prop (Transitive A R) -> + cupicef.Term s (P (Build__PER A R PER__Symmetric0 PER__Transitive0))) -> + x : cupicef.Term cupicef.prop (PER A R) -> + cupicef.Term s (P x). + +[ s, A, R, P, case__Build__PER, PER__Symmetric0, PER__Transitive0] + match____PER s A R P case__Build__PER + (Build__PER A R PER__Symmetric0 PER__Transitive0) + --> + case__Build__PER PER__Symmetric0 PER__Transitive0. + +[ s, A, R, P, s'] + match____PER s A R + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____PER s' A R (x => P x). + +def PER__Symmetric : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + PER0 : cupicef.Term cupicef.prop (PER A R) -> + cupicef.Term cupicef.prop (Symmetric A R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + PER0 : cupicef.Term cupicef.prop (PER A R) => + match____PER cupicef.prop A R + (PER1 : cupicef.Term cupicef.prop (PER A R) => Symmetric A R) + (PER__Symmetric0 : cupicef.Term cupicef.prop (Symmetric A R) => + PER__Transitive0 : cupicef.Term cupicef.prop (Transitive A R) => + PER__Symmetric0) + PER0. + +#TYPE_CLASS_INSTANCE PER__Symmetric. + +def PER__Transitive : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + PER0 : cupicef.Term cupicef.prop (PER A R) -> + cupicef.Term cupicef.prop (Transitive A R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + PER0 : cupicef.Term cupicef.prop (PER A R) => + match____PER cupicef.prop A R + (PER1 : cupicef.Term cupicef.prop (PER A R) => Transitive A R) + (PER__Symmetric0 : cupicef.Term cupicef.prop (Symmetric A R) => + PER__Transitive0 : cupicef.Term cupicef.prop (Transitive A R) => + PER__Transitive0) + PER0. + +#TYPE_CLASS_INSTANCE PER__Transitive. + +def Equivalence : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Univ cupicef.prop. + +#TYPE_CLASS Equivalence. + +def Build__Equivalence : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + Equivalence__Reflexive0 : cupicef.Term cupicef.prop (Reflexive A R) -> + Equivalence__Symmetric0 : cupicef.Term cupicef.prop (Symmetric A R) -> + Equivalence__Transitive0 : cupicef.Term cupicef.prop (Transitive A R) -> + cupicef.Term cupicef.prop (Equivalence A R). + +[ ] cupicef.code_guarded? _ (Build__Equivalence _ _ _ _ _) --> cupicef.guarded. + +def match____Equivalence : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + P : (cupicef.Term cupicef.prop (Equivalence A R) -> cupicef.Univ s) -> + case__Build__Equivalence : + (Equivalence__Reflexive0 : cupicef.Term cupicef.prop (Reflexive A R) -> + Equivalence__Symmetric0 : cupicef.Term cupicef.prop (Symmetric A R) -> + Equivalence__Transitive0 : cupicef.Term cupicef.prop (Transitive A R) -> + cupicef.Term s + (P + (Build__Equivalence A R Equivalence__Reflexive0 + Equivalence__Symmetric0 Equivalence__Transitive0))) -> + x : cupicef.Term cupicef.prop (Equivalence A R) -> + cupicef.Term s (P x). + +[ s, A, R, P, case__Build__Equivalence, Equivalence__Reflexive0, Equivalence__Symmetric0, Equivalence__Transitive0] + match____Equivalence s A R P case__Build__Equivalence + (Build__Equivalence A R Equivalence__Reflexive0 Equivalence__Symmetric0 + Equivalence__Transitive0) + --> + case__Build__Equivalence Equivalence__Reflexive0 Equivalence__Symmetric0 + Equivalence__Transitive0. + +[ s, A, R, P, s'] + match____Equivalence s A R + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____Equivalence s' A R (x => P x). + +def Equivalence__Reflexive : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + Equivalence0 : cupicef.Term cupicef.prop (Equivalence A R) -> + cupicef.Term cupicef.prop (Reflexive A R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + Equivalence0 : cupicef.Term cupicef.prop (Equivalence A R) => + match____Equivalence cupicef.prop A R + (Equivalence1 : cupicef.Term cupicef.prop (Equivalence A R) => + Reflexive A R) + (Equivalence__Reflexive0 : cupicef.Term cupicef.prop (Reflexive A R) => + Equivalence__Symmetric0 : cupicef.Term cupicef.prop (Symmetric A R) => + Equivalence__Transitive0 : cupicef.Term cupicef.prop (Transitive A R) => + Equivalence__Reflexive0) + Equivalence0. + +#TYPE_CLASS_INSTANCE Equivalence__Reflexive. + +def Equivalence__Symmetric : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + Equivalence0 : cupicef.Term cupicef.prop (Equivalence A R) -> + cupicef.Term cupicef.prop (Symmetric A R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + Equivalence0 : cupicef.Term cupicef.prop (Equivalence A R) => + match____Equivalence cupicef.prop A R + (Equivalence1 : cupicef.Term cupicef.prop (Equivalence A R) => + Symmetric A R) + (Equivalence__Reflexive0 : cupicef.Term cupicef.prop (Reflexive A R) => + Equivalence__Symmetric0 : cupicef.Term cupicef.prop (Symmetric A R) => + Equivalence__Transitive0 : cupicef.Term cupicef.prop (Transitive A R) => + Equivalence__Symmetric0) + Equivalence0. + +#TYPE_CLASS_INSTANCE Equivalence__Symmetric. + +def Equivalence__Transitive : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + Equivalence0 : cupicef.Term cupicef.prop (Equivalence A R) -> + cupicef.Term cupicef.prop (Transitive A R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + Equivalence0 : cupicef.Term cupicef.prop (Equivalence A R) => + match____Equivalence cupicef.prop A R + (Equivalence1 : cupicef.Term cupicef.prop (Equivalence A R) => + Transitive A R) + (Equivalence__Reflexive0 : cupicef.Term cupicef.prop (Reflexive A R) => + Equivalence__Symmetric0 : cupicef.Term cupicef.prop (Symmetric A R) => + Equivalence__Transitive0 : cupicef.Term cupicef.prop (Transitive A R) => + Equivalence__Transitive0) + Equivalence0. + +#TYPE_CLASS_INSTANCE Equivalence__Transitive. + +def Equivalence__PER : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + E : cupicef.Term cupicef.prop (Equivalence A R) -> + cupicef.Term cupicef.prop (PER A R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + E : cupicef.Term cupicef.prop (Equivalence A R) => + Build__PER A R (Equivalence__Symmetric A R E) + (Equivalence__Transitive A R E). + +#TYPE_CLASS_INSTANCE Equivalence__PER. + +def Equivalence__PreOrder : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + E : cupicef.Term cupicef.prop (Equivalence A R) -> + cupicef.Term cupicef.prop (PreOrder A R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + E : cupicef.Term cupicef.prop (Equivalence A R) => + Build__PreOrder A R (Equivalence__Reflexive A R E) + (Equivalence__Transitive A R E). + +#TYPE_CLASS_INSTANCE Equivalence__PreOrder. + +def Antisymmetric : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + equ : cupicef.Term cupicef.prop (Equivalence A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + equ : cupicef.Term cupicef.prop (Equivalence A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (R x y) + (__ : cupicef.Term cupicef.prop (R x y) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R y x) + (__0 : cupicef.Term cupicef.prop (R y x) => eqA x y)))). + +#TYPE_CLASS Antisymmetric. + +def antisymmetry : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + equ : cupicef.Term cupicef.prop (Equivalence A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + Antisymmetric0 : cupicef.Term cupicef.prop (Antisymmetric A eqA equ R) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + __0 : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (eqA x y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + equ : cupicef.Term cupicef.prop (Equivalence A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + Antisymmetric0 : cupicef.Term cupicef.prop (Antisymmetric A eqA equ R) => + Antisymmetric0. + +def subrelation : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x y) + (__ : cupicef.Term cupicef.prop (R x y) => R_27_ x y))). + +#TYPE_CLASS subrelation. + +def is__subrelation : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + subrelation0 : cupicef.Term cupicef.prop (subrelation A R R_27_) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (R_27_ x y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + subrelation0 : cupicef.Term cupicef.prop (subrelation A R R_27_) => + subrelation0. + +thm subrelation__symmetric : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Symmetric A R) -> + cupicef.Term cupicef.prop + (subrelation A + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (Symmetric A R) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x y) + (H0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x y) => + R x y)))) + (subrelation A + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + R) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x y) => + symmetry A R H y x H0). + +thm flip__Reflexive : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Reflexive A R) -> + cupicef.Term cupicef.prop + (Reflexive A + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (Reflexive A R) => + H. + +def flip__Irreflexive : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Irreflexive A R) -> + cupicef.Term cupicef.prop + (Irreflexive A + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (Irreflexive A R) => + irreflexivity A R H. + +def flip__Symmetric : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Symmetric A R) -> + cupicef.Term cupicef.prop + (Symmetric A + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (Symmetric A R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x y) => + symmetry A R H y x H0. + +def flip__Asymmetric : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Asymmetric A R) -> + cupicef.Term cupicef.prop + (Asymmetric A + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (Asymmetric A R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x y) => + H_27_ : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R y x) => + asymmetry A R H y x H0 H_27_. + +def flip__Transitive : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Transitive A R) -> + cupicef.Term cupicef.prop + (Transitive A + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (Transitive A R) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x y) => + H_27_ : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R y z) => + transitivity A R H z y x H_27_ H0. + +thm flip__Antisymmetric : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + equ : cupicef.Term cupicef.prop (Equivalence A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Antisymmetric A eqA equ R) -> + cupicef.Term cupicef.prop + (Antisymmetric A eqA equ + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + equ : cupicef.Term cupicef.prop (Equivalence A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (Antisymmetric A eqA equ R) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x y) + (H0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x y) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R y x) + (H1 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R y x) => + eqA x y))))) + (Antisymmetric A eqA equ + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R)) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x y) => + H1 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R y x) => + (H2 : + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y y0) -> + __0 : cupicef.Term cupicef.prop (R y0 y) -> + cupicef.Term cupicef.prop (eqA y y0)) => + (H3 : + (__ : cupicef.Term cupicef.prop (R y x) -> + __0 : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (eqA y x)) => + (H4 : + (__ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (eqA y x)) => + (H5 : cupicef.Term cupicef.prop (eqA y x) => + (H6 : + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y0) -> + __0 : cupicef.Term cupicef.prop (R y0 x) -> + cupicef.Term cupicef.prop (eqA x y0)) => + H x y H1 H0) (H x)) (H4 H1)) + (H3 H0)) + (H2 x)) + (H y)). + +thm flip__PreOrder : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (PreOrder A R) -> + cupicef.Term cupicef.prop + (PreOrder A + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (PreOrder A R) => + match____PreOrder cupicef.prop A R + (__ : cupicef.Term cupicef.prop (PreOrder A R) => + PreOrder A + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R)) + (PreOrder__Reflexive0 : cupicef.Term cupicef.prop (Reflexive A R) => + PreOrder__Transitive0 : cupicef.Term cupicef.prop (Transitive A R) => + Build__PreOrder A + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + PreOrder__Reflexive0 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x y) + (H0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x y) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R y z) + (H1 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R y z) => + R z x)))))) + (Transitive A + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R)) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x y) => + H1 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R y z) => + (H2 : + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R z y0) -> + __0 : cupicef.Term cupicef.prop (R y0 z0) -> + cupicef.Term cupicef.prop (R z z0)) => + (H3 : + (z0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R z y) -> + __0 : cupicef.Term cupicef.prop (R y z0) -> + cupicef.Term cupicef.prop (R z z0)) => + (H4 : + (__ : cupicef.Term cupicef.prop (R z y) -> + __0 : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (R z x)) => + (H5 : + (__ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (R z x)) => + (H6 : cupicef.Term cupicef.prop (R z x) => H6) (H5 H0)) + (H4 H1)) + (H3 x)) + (H2 y)) + (PreOrder__Transitive0 z)))) + H. + +thm flip__StrictOrder : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (StrictOrder A R) -> + cupicef.Term cupicef.prop + (StrictOrder A + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (StrictOrder A R) => + match____StrictOrder cupicef.prop A R + (__ : cupicef.Term cupicef.prop (StrictOrder A R) => + StrictOrder A + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R)) + (StrictOrder__Irreflexive0 : cupicef.Term cupicef.prop (Irreflexive A R) => + StrictOrder__Transitive0 : cupicef.Term cupicef.prop (Transitive A R) => + Build__StrictOrder A + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + StrictOrder__Irreflexive0 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x y) + (H0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x y) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R y z) + (H1 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R y z) => + R z x)))))) + (Transitive A + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R)) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x y) => + H1 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R y z) => + (H2 : + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R z y0) -> + __0 : cupicef.Term cupicef.prop (R y0 z0) -> + cupicef.Term cupicef.prop (R z z0)) => + (H3 : + (z0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R z y) -> + __0 : cupicef.Term cupicef.prop (R y z0) -> + cupicef.Term cupicef.prop (R z z0)) => + (H4 : + (__ : cupicef.Term cupicef.prop (R z y) -> + __0 : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (R z x)) => + (H5 : + (__ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (R z x)) => + (H6 : cupicef.Term cupicef.prop (R z x) => H6) (H5 H0)) + (H4 H1)) + (H3 x)) + (H2 y)) + (StrictOrder__Transitive0 z)))) + H. + +thm flip__PER : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (PER A R) -> + cupicef.Term cupicef.prop + (PER A + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (PER A R) => + match____PER cupicef.prop A R + (__ : cupicef.Term cupicef.prop (PER A R) => + PER A + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R)) + (PER__Symmetric0 : cupicef.Term cupicef.prop (Symmetric A R) => + PER__Transitive0 : cupicef.Term cupicef.prop (Transitive A R) => + Build__PER A + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x y) + (H0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x y) => + R x y)))) + (Symmetric A + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R)) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x y) => + (H1 : + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y y0) -> + __0 : cupicef.Term cupicef.prop (R y0 z) -> + cupicef.Term cupicef.prop (R y z)) => + (H2 : + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop (R y0 y)) => + (H3 : + (__ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (R x y)) => + (H4 : cupicef.Term cupicef.prop (R x y) => H4) (H3 H0)) ( + H2 x)) + (PER__Symmetric0 y)) + (PER__Transitive0 y))) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x y) + (H0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x y) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R y z) + (H1 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R y z) => + R z x)))))) + (Transitive A + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R)) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x y) => + H1 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R y z) => + (H2 : + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R z y0) -> + __0 : cupicef.Term cupicef.prop (R y0 z0) -> + cupicef.Term cupicef.prop (R z z0)) => + (H3 : + (z0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R z y) -> + __0 : cupicef.Term cupicef.prop (R y z0) -> + cupicef.Term cupicef.prop (R z z0)) => + (H4 : + (__ : cupicef.Term cupicef.prop (R z y) -> + __0 : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (R z x)) => + (H5 : + (__ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (R z x)) => + (H6 : cupicef.Term cupicef.prop (R z x) => H6) (H5 H0)) + (H4 H1)) + (H3 x)) + (H2 y)) + (PER__Transitive0 z)))) + H. + +thm flip__Equivalence : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Equivalence A R) -> + cupicef.Term cupicef.prop + (Equivalence A + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (Equivalence A R) => + match____Equivalence cupicef.prop A R + (__ : cupicef.Term cupicef.prop (Equivalence A R) => + Equivalence A + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R)) + (Equivalence__Reflexive0 : cupicef.Term cupicef.prop (Reflexive A R) => + Equivalence__Symmetric0 : cupicef.Term cupicef.prop (Symmetric A R) => + Equivalence__Transitive0 : cupicef.Term cupicef.prop (Transitive A R) => + Build__Equivalence A + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + Equivalence__Reflexive0 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x y) + (H0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x y) => + R x y)))) + (Symmetric A + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R)) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x y) => + (H1 : + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y y0) -> + __0 : cupicef.Term cupicef.prop (R y0 z) -> + cupicef.Term cupicef.prop (R y z)) => + (H2 : + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop (R y0 y)) => + (H3 : + (__ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (R x y)) => + (H4 : cupicef.Term cupicef.prop (R x y) => H4) (H3 H0)) ( + H2 x)) + (Equivalence__Symmetric0 y)) + (Equivalence__Transitive0 y))) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x y) + (H0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x y) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R y z) + (H1 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R y z) => + R z x)))))) + (Transitive A + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R)) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x y) => + H1 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R y z) => + (H2 : + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R z y0) -> + __0 : cupicef.Term cupicef.prop (R y0 z0) -> + cupicef.Term cupicef.prop (R z z0)) => + (H3 : + (z0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R z y) -> + __0 : cupicef.Term cupicef.prop (R y z0) -> + cupicef.Term cupicef.prop (R z z0)) => + (H4 : + (__ : cupicef.Term cupicef.prop (R z y) -> + __0 : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (R z x)) => + (H5 : + (__ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (R z x)) => + (H6 : cupicef.Term cupicef.prop (R z x) => H6) (H5 H0)) + (H4 H1)) + (H3 x)) + (H2 y)) + (Equivalence__Transitive0 z)))) + H. + +thm complement__Irreflexive : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Reflexive A R) -> + cupicef.Term cupicef.prop (Irreflexive A (complement A R)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (Reflexive A R) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + complement A (complement A R) x x)) + (Irreflexive A (complement A R)) cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (complement A R x x) + (H0 : cupicef.Term cupicef.prop (complement A R x x) => + Coq__Init__Logic.False)) + (complement A (complement A R) x x) cupicef.I + (H0 : cupicef.Term cupicef.prop (complement A R x x) => + (H1 : cupicef.Term cupicef.prop (R x x) => + (H2 : cupicef.Term cupicef.prop Coq__Init__Logic.False => H2) + (H0 H1)) + (H x))). + +thm complement__Symmetric : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Symmetric A R) -> + cupicef.Term cupicef.prop (Symmetric A (complement A R)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (Symmetric A R) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (complement A R x y) + (H0 : cupicef.Term cupicef.prop (complement A R x y) => + complement A R y x)))) + (Symmetric A (complement A R)) cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (complement A R x y) => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R y x) + (H1 : cupicef.Term cupicef.prop (R y x) => Coq__Init__Logic.False)) + (complement A R y x) cupicef.I + (H1 : cupicef.Term cupicef.prop (R y x) => + (H2 : + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y y0) -> + cupicef.Term cupicef.prop (R y0 y)) => + (H3 : + (__ : cupicef.Term cupicef.prop (R y y) -> + cupicef.Term cupicef.prop (R y y)) => + (H4 : + (__ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (R x y)) => + (H5 : cupicef.Term cupicef.prop (R x y) => + (H6 : cupicef.Term cupicef.prop Coq__Init__Logic.False => H6) + (H0 H5)) + (H4 H1)) + (H2 x)) + (H2 y)) + (H y))). + +def RewriteRelation : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Univ cupicef.prop. + +#TYPE_CLASS RewriteRelation. + +def Build__RewriteRelation : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Term cupicef.prop (RewriteRelation A RA). + +[ ] cupicef.code_guarded? _ (Build__RewriteRelation _ _) --> cupicef.guarded. + +def match____RewriteRelation : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + RA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + P : (cupicef.Term cupicef.prop (RewriteRelation A RA) -> cupicef.Univ s) -> + case__Build__RewriteRelation : + cupicef.Term s (P (Build__RewriteRelation A RA)) -> + x : cupicef.Term cupicef.prop (RewriteRelation A RA) -> + cupicef.Term s (P x). + +[ s, A, RA, P, case__Build__RewriteRelation] + match____RewriteRelation s A RA P case__Build__RewriteRelation + (Build__RewriteRelation A RA) + --> + case__Build__RewriteRelation. + +[ s, A, RA, P, s'] + match____RewriteRelation s A RA + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____RewriteRelation s' A RA (x => P x). + +def equivalence__rewrite__relation : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (Equivalence A eqA) -> + cupicef.Term cupicef.prop (RewriteRelation A eqA) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (Equivalence A eqA) => + Build__RewriteRelation A eqA. + +#TYPE_CLASS_INSTANCE equivalence__rewrite__relation. + +def eq__Reflexive : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term cupicef.prop + (Reflexive A + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A). + +#TYPE_CLASS_INSTANCE eq__Reflexive. + +def eq__Symmetric : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term cupicef.prop + (Symmetric A + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A). + +#TYPE_CLASS_INSTANCE eq__Symmetric. + +def eq__Transitive : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term cupicef.prop + (Transitive A + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + Coq__Init__Logic.eq__trans + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A). + +#TYPE_CLASS_INSTANCE eq__Transitive. + +def eq__equivalence : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term cupicef.prop + (Equivalence A + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + Build__Equivalence A + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + (eq__Reflexive A) (eq__Symmetric A) (eq__Transitive A). + +#TYPE_CLASS_INSTANCE eq__equivalence. + +def RewriteRelation__instance__0 : + cupicef.Term cupicef.prop + (RewriteRelation + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl) + +:= Build__RewriteRelation + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl. + +#TYPE_CLASS_INSTANCE RewriteRelation__instance__0. + +def RewriteRelation__instance__1 : + cupicef.Term cupicef.prop + (RewriteRelation + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff) + +:= Build__RewriteRelation + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff. + +#TYPE_CLASS_INSTANCE RewriteRelation__instance__1. + +def impl__Reflexive__obligation__1 : + x : cupicef.Univ cupicef.prop -> + cupicef.Term cupicef.prop (Coq__Program__Basics.impl x x) + +:= x : cupicef.Univ cupicef.prop => H : cupicef.Term cupicef.prop x => H. + +def impl__Reflexive : + cupicef.Term cupicef.prop + (Reflexive + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl) + +:= x : cupicef.Univ cupicef.prop => impl__Reflexive__obligation__1 x. + +#TYPE_CLASS_INSTANCE impl__Reflexive. + +def impl__Transitive__obligation__1 : + x : cupicef.Univ cupicef.prop -> + y : cupicef.Univ cupicef.prop -> + z : cupicef.Univ cupicef.prop -> + H : cupicef.Term cupicef.prop (Coq__Program__Basics.impl x y) -> + H0 : cupicef.Term cupicef.prop (Coq__Program__Basics.impl y z) -> + cupicef.Term cupicef.prop (Coq__Program__Basics.impl x z) + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + z : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Program__Basics.impl x y) => + H0 : cupicef.Term cupicef.prop (Coq__Program__Basics.impl y z) => + H1 : cupicef.Term cupicef.prop x => + H0 (H H1). + +def impl__Transitive : + cupicef.Term cupicef.prop + (Transitive + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Program__Basics.impl) + +:= x : cupicef.Univ cupicef.prop => + y : cupicef.Univ cupicef.prop => + z : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (Coq__Program__Basics.impl x y) => + H0 : cupicef.Term cupicef.prop (Coq__Program__Basics.impl y z) => + impl__Transitive__obligation__1 x y z H H0. + +#TYPE_CLASS_INSTANCE impl__Transitive. + +def iff__Reflexive : + cupicef.Term cupicef.prop + (Reflexive + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff) + +:= Coq__Init__Logic.iff__refl. + +#TYPE_CLASS_INSTANCE iff__Reflexive. + +def iff__Symmetric : + cupicef.Term cupicef.prop + (Symmetric + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff) + +:= Coq__Init__Logic.iff__sym. + +#TYPE_CLASS_INSTANCE iff__Symmetric. + +def iff__Transitive : + cupicef.Term cupicef.prop + (Transitive + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff) + +:= Coq__Init__Logic.iff__trans. + +#TYPE_CLASS_INSTANCE iff__Transitive. + +def iff__equivalence : + cupicef.Term cupicef.prop + (Equivalence + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff) + +:= Build__Equivalence + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.iff iff__Reflexive iff__Symmetric iff__Transitive. + +#TYPE_CLASS_INSTANCE iff__equivalence. + +def Tlist : + cupicef.Univ + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set). + +Tlist' : + cupicef.Univ cupicef.sinf. + +[ ] + cupicef.code (cupicef.cU _) Tlist + --> + Tlist'. + +def Tnil : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist. + +def Tcons : + __ : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + __0 : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist. + +[ ] cupicef.code_guarded? _ Tnil --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (Tcons _ _) --> cupicef.guarded. + +def match____Tlist : + s : cupicef.Sort -> + P : + (cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Univ s) -> + case__Tnil : cupicef.Term s (P Tnil) -> + case__Tcons : + (__ : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + __0 : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Term s (P (Tcons __ __0))) -> + x : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Term s (P x). + +[ s, P, case__Tnil, case__Tcons] + match____Tlist s P case__Tnil case__Tcons Tnil + --> + case__Tnil. + +[ s, P, case__Tnil, case__Tcons, __, __0] + match____Tlist s P case__Tnil case__Tcons (Tcons __ __0) + --> + case__Tcons __ __0. + +[ s, P, s'] + match____Tlist s + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____Tlist s' (x => P x). + +def Tlist__rect : + P : + (t : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P Tnil) -> + f0 : + (T : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + t : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P t) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (Tcons T t))) -> + t : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P t) + +:= P : + (t : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P Tnil) => + f0 : + (T : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + t : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P t) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (Tcons T t))) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) cupicef.set)) + (cupicef.prod + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (t : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + P t)))) + (c => + c + (F => + t : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist (cupicef.type (cupicef.s cupicef.z)) + (t0 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + P t0) + f + (T : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + t0 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + f0 T t0 (F t0)) + t)) + cupicef.0. + +def Tlist__ind : + P : + (t : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Univ cupicef.prop) -> + f : cupicef.Term cupicef.prop (P Tnil) -> + f0 : + (T : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + t : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + __ : cupicef.Term cupicef.prop (P t) -> + cupicef.Term cupicef.prop (P (Tcons T t))) -> + t : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Term cupicef.prop (P t) + +:= P : + (t : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Univ cupicef.prop) => + f : cupicef.Term cupicef.prop (P Tnil) => + f0 : + (T : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + t : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + __ : cupicef.Term cupicef.prop (P t) -> + cupicef.Term cupicef.prop (P (Tcons T t))) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.prop + (cupicef.prod + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + cupicef.prop + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + cupicef.prop) + cupicef.I Tlist + (t : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + P t)))) + (c => + c + (F => + t : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist cupicef.prop + (t0 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + P t0) + f + (T : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + t0 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + f0 T t0 (F t0)) + t)) + cupicef.0. + +def Tlist__rec : + P : + (t : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Univ cupicef.set) -> + f : cupicef.Term cupicef.set (P Tnil) -> + f0 : + (T : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + t : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + __ : cupicef.Term cupicef.set (P t) -> + cupicef.Term cupicef.set (P (Tcons T t))) -> + t : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Term cupicef.set (P t) + +:= P : + (t : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Univ cupicef.set) => + Tlist__rect + (cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (t : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I Tlist + (t : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P). + +def arrows : + l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + r : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set))) + (cupicef.prod + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.rule + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.rule + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I Tlist + (l : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + cupicef.prod + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (r : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I))))) + (c => + c + (arrows0 => + l : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + r : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + match____Tlist (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (l0 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + r + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + l_27_ : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + arrows0 l_27_ r)) + l)) + cupicef.0. + +def unary__operation : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + arrows (Tcons A Tnil) A. + +def binary__operation : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + arrows (Tcons A (Tcons A Tnil)) A. + +def ternary__operation : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + arrows (Tcons A (Tcons A (Tcons A Tnil))) A. + +def unary__predicate : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + arrows (Tcons A Tnil) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)). + +def binary__relation : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + arrows (Tcons A (Tcons A Tnil)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)). + +def predicate__all : + l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + __ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) -> + cupicef.Univ cupicef.prop + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I Tlist + (l : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (arrows l + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))))) + (c => + c + (predicate__all0 => + l : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (l0 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (f : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + f) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + f : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons A tl) + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + predicate__all0 tl (f x))) + l)) + cupicef.0. + +def predicate__exists : + l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + __ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) -> + cupicef.Univ cupicef.prop + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I Tlist + (l : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (arrows l + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))))) + (c => + c + (predicate__exists0 => + l : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (l0 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (f : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + f) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + f : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons A tl) + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + Coq__Init__Logic.ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + predicate__exists0 tl (f x))) + l)) + cupicef.0. + +def pointwise__extension : + T : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (binary__operation T) -> + l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__operation (arrows l T)) + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.2 + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set))) + (cupicef.prod (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (T : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (binary__operation T) + (op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__operation T) => + cupicef.prod + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__operation (arrows l T))))))) + (c => + c + (pointwise__extension0 => + T : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__operation T) => + l : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist (cupicef.type (cupicef.s cupicef.z)) + (l0 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__operation (arrows l0 T)) + (R : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil T) => + R_27_ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil T) => + op R R_27_) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons A tl) T) => + R_27_ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons A tl) T) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + pointwise__extension0 T op tl (R x) (R_27_ x)) + l)) + cupicef.0. + +def pointwise__lifting : + op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) -> + l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) cupicef.set)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + cupicef.prod + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))))))) + (c => + c + (pointwise__lifting0 => + op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + l : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist (cupicef.type (cupicef.s cupicef.z)) + (l0 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (R : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + R_27_ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + op R R_27_) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons A tl) + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + R_27_ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons A tl) + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + pointwise__lifting0 op tl (R x) (R_27_ x))) + l)) + cupicef.0. + +def predicate__equivalence : + l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) + +:= l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + pointwise__lifting Coq__Init__Logic.iff l. + +def predicate__implication : + l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) + +:= l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + pointwise__lifting Coq__Program__Basics.impl l. + +def predicate__intersection : + l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__operation + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) + +:= pointwise__extension + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.and. + +def predicate__union : + l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__operation + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) + +:= pointwise__extension + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + Coq__Init__Logic.or. + +def true__predicate : + l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) cupicef.set)) + (cupicef.prod + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + arrows l + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))))) + (c => + c + (true__predicate0 => + l : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist (cupicef.type (cupicef.s cupicef.z)) + (l0 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + arrows l0 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + Coq__Init__Logic.True + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + true__predicate0 tl) + l)) + cupicef.0. + +def false__predicate : + l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) cupicef.set)) + (cupicef.prod + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + arrows l + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))))) + (c => + c + (false__predicate0 => + l : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist (cupicef.type (cupicef.s cupicef.z)) + (l0 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + arrows l0 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + Coq__Init__Logic.False + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + false__predicate0 tl) + l)) + cupicef.0. + +thm predicate__equivalence__equivalence__obligation__1 : + l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Term cupicef.prop + (Reflexive + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (predicate__equivalence l)) + +:= l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + x : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + Tlist__ind + (l0 : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I + (arrows l0 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + cupicef.set)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l1 + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))))))) + (c => + c + (pointwise__lifting0 => + op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist (cupicef.type (cupicef.s cupicef.z)) + (l2 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l2 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (R : cupicef.Univ cupicef.prop => + R_27_ : cupicef.Univ cupicef.prop => + op R R_27_) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) => + R_27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + pointwise__lifting0 op tl (R x1) (R_27_ x1))) + l1)) + cupicef.0 Coq__Init__Logic.iff l0 x0 x0)) + (x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I x0 + (__ : cupicef.Term cupicef.prop x0 => x0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I x0 + (__ : cupicef.Term cupicef.prop x0 => x0)) + (H : cupicef.Term cupicef.prop x0 => H) + (H : cupicef.Term cupicef.prop x0 => H)) + (T : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + l0 : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + IHl : + (x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) -> + cupicef.Term cupicef.prop + (cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + cupicef.set)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l1 + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))))))) + (c => + c + (pointwise__lifting0 => + op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist (cupicef.type (cupicef.s cupicef.z)) + (l2 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l2 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) + (R : cupicef.Univ cupicef.prop => + R_27_ : cupicef.Univ cupicef.prop => + op R R_27_) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) => + R_27_ : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (x1 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + pointwise__lifting0 op tl (R x1) (R_27_ x1))) + l1)) + cupicef.0 Coq__Init__Logic.iff l0 x0 x0)) => + x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons T l0) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) T => + IHl (x0 x1)) + l x. + +thm predicate__equivalence__equivalence__obligation__2 : + l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Term cupicef.prop + (Symmetric + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (predicate__equivalence l)) + +:= l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + x : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + y : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + H : cupicef.Term cupicef.prop (predicate__equivalence l x y) => + Tlist__ind + (l0 : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (arrows l0 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (arrows l0 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (y0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (predicate__equivalence l0 x0 y0) + (H0 : + cupicef.Term cupicef.prop (predicate__equivalence l0 x0 y0) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + cupicef.set)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (op : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l1 + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))))))) + (c => + c + (pointwise__lifting0 => + op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist (cupicef.type (cupicef.s cupicef.z)) + (l2 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l2 + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) + (R : cupicef.Univ cupicef.prop => + R_27_ : cupicef.Univ cupicef.prop => + op R R_27_) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R : + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) => + R_27_ : + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + pointwise__lifting0 op tl (R x1) (R_27_ x1))) + l1)) + cupicef.0 Coq__Init__Logic.iff l0 y0 x0)))) + (x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + y0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + H0 : cupicef.Term cupicef.prop (predicate__equivalence Tnil x0 y0) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I x0 + (__ : cupicef.Term cupicef.prop x0 => y0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I y0 + (__ : cupicef.Term cupicef.prop y0 => x0)) + (Coq__Init__Logic.iff y0 x0) + (H1 : + (__ : cupicef.Term cupicef.prop x0 -> cupicef.Term cupicef.prop y0) => + H2 : + (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop x0) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I y0 + (__ : cupicef.Term cupicef.prop y0 => x0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I x0 + (__ : cupicef.Term cupicef.prop x0 => y0)) + H2 H1) + H0) + (T : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + l0 : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + IHl : + (x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) -> + y0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) -> + H0 : cupicef.Term cupicef.prop (predicate__equivalence l0 x0 y0) -> + cupicef.Term cupicef.prop + (cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + cupicef.set)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l1 + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))))))) + (c => + c + (pointwise__lifting0 => + op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist (cupicef.type (cupicef.s cupicef.z)) + (l2 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l2 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) + (R : cupicef.Univ cupicef.prop => + R_27_ : cupicef.Univ cupicef.prop => + op R R_27_) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) => + R_27_ : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (x1 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + pointwise__lifting0 op tl (R x1) (R_27_ x1))) + l1)) + cupicef.0 Coq__Init__Logic.iff l0 y0 x0)) => + x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons T l0) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + y0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons T l0) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + H0 : + cupicef.Term cupicef.prop (predicate__equivalence (Tcons T l0) x0 y0) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) T => + IHl (x0 x1) (y0 x1) (H0 x1)) + l x y H. + +def let__p : + l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + x : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) -> + y : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) -> + z : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) -> + H : cupicef.Term cupicef.prop (predicate__equivalence l x y) -> + H0 : cupicef.Term cupicef.prop (predicate__equivalence l y z) -> + T : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + l0 : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + IHl : + (x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) -> + y0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) -> + z0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) -> + H1 : cupicef.Term cupicef.prop (predicate__equivalence l0 x0 y0) -> + H2 : cupicef.Term cupicef.prop (predicate__equivalence l0 y0 z0) -> + cupicef.Term cupicef.prop + (cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + cupicef.set)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l1 + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))))))) + (c => + c + (pointwise__lifting0 => + op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist (cupicef.type (cupicef.s cupicef.z)) + (l2 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l2 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (R : cupicef.Univ cupicef.prop => + R_27_ : cupicef.Univ cupicef.prop => + op R R_27_) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) => + R_27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + pointwise__lifting0 op tl (R x1) (R_27_ x1))) + l1)) + cupicef.0 Coq__Init__Logic.iff l0 x0 z0)) -> + x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons T l0) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) -> + y0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons T l0) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) -> + z0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons T l0) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) -> + H1 : cupicef.Term cupicef.prop (predicate__equivalence (Tcons T l0) x0 y0) -> + H2 : cupicef.Term cupicef.prop (predicate__equivalence (Tcons T l0) y0 z0) -> + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) T -> + H3 : cupicef.Term cupicef.prop (predicate__equivalence l0 (x0 x1) (y0 x1)) -> + H4 : cupicef.Term cupicef.prop (predicate__equivalence l0 (y0 x1) (z0 x1)) -> + cupicef.Term cupicef.prop + (cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) cupicef.set)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + cupicef.prod + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l1 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))))))) + (c => + c + (pointwise__lifting0 => + op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist (cupicef.type (cupicef.s cupicef.z)) + (l2 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l2 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (R : cupicef.Univ cupicef.prop => + R_27_ : cupicef.Univ cupicef.prop => + op R R_27_) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) => + R_27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + pointwise__lifting0 op tl (R x2) (R_27_ x2))) + l1)) + cupicef.0 Coq__Init__Logic.iff l0 (x0 x1) (z0 x1)) + +:= l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + x : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + y : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + z : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + H : cupicef.Term cupicef.prop (predicate__equivalence l x y) => + H0 : cupicef.Term cupicef.prop (predicate__equivalence l y z) => + T : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + l0 : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + IHl : + (x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) -> + y0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) -> + z0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) -> + H1 : cupicef.Term cupicef.prop (predicate__equivalence l0 x0 y0) -> + H2 : cupicef.Term cupicef.prop (predicate__equivalence l0 y0 z0) -> + cupicef.Term cupicef.prop + (cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + cupicef.set)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l1 + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))))))) + (c => + c + (pointwise__lifting0 => + op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist (cupicef.type (cupicef.s cupicef.z)) + (l2 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l2 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (R : cupicef.Univ cupicef.prop => + R_27_ : cupicef.Univ cupicef.prop => + op R R_27_) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) => + R_27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + pointwise__lifting0 op tl (R x1) (R_27_ x1))) + l1)) + cupicef.0 Coq__Init__Logic.iff l0 x0 z0)) => + x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons T l0) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + y0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons T l0) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + z0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons T l0) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + H1 : cupicef.Term cupicef.prop (predicate__equivalence (Tcons T l0) x0 y0) => + H2 : cupicef.Term cupicef.prop (predicate__equivalence (Tcons T l0) y0 z0) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) T => + IHl (x0 x1) (y0 x1) (z0 x1). + +thm predicate__equivalence__equivalence__obligation__3 : + l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Term cupicef.prop + (Transitive + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (predicate__equivalence l)) + +:= l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + x : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + y : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + z : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + H : cupicef.Term cupicef.prop (predicate__equivalence l x y) => + H0 : cupicef.Term cupicef.prop (predicate__equivalence l y z) => + Tlist__ind + (l0 : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.I + (arrows l0 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (arrows l0 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (y0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (z0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (predicate__equivalence l0 x0 y0) + (H1 : + cupicef.Term cupicef.prop + (predicate__equivalence l0 x0 y0) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (predicate__equivalence l0 y0 z0) + (H2 : + cupicef.Term cupicef.prop + (predicate__equivalence l0 y0 z0) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + cupicef.set)) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (op : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l1 + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)))))))) + (c => + c + (pointwise__lifting0 => + op : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist + (cupicef.type (cupicef.s cupicef.z)) + (l2 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l2 + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) + (R : cupicef.Univ cupicef.prop => + R_27_ : cupicef.Univ cupicef.prop => + op R R_27_) + (A : + cupicef.Univ + (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R : + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)))) => + R_27_ : + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)))) => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + pointwise__lifting0 op tl (R x1) (R_27_ x1))) + l1)) + cupicef.0 Coq__Init__Logic.iff l0 x0 z0)))))) + (x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + y0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + z0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + H1 : cupicef.Term cupicef.prop (predicate__equivalence Tnil x0 y0) => + H2 : cupicef.Term cupicef.prop (predicate__equivalence Tnil y0 z0) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I y0 + (__ : cupicef.Term cupicef.prop y0 => z0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I z0 + (__ : cupicef.Term cupicef.prop z0 => y0)) + (Coq__Init__Logic.iff x0 z0) + (H3 : + (__ : cupicef.Term cupicef.prop y0 -> cupicef.Term cupicef.prop z0) => + H4 : + (__ : cupicef.Term cupicef.prop z0 -> cupicef.Term cupicef.prop y0) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I x0 + (__ : cupicef.Term cupicef.prop x0 => y0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I y0 + (__ : cupicef.Term cupicef.prop y0 => x0)) + (Coq__Init__Logic.iff x0 z0) + (H5 : + (__ : cupicef.Term cupicef.prop x0 -> + cupicef.Term cupicef.prop y0) => + H6 : + (__ : cupicef.Term cupicef.prop y0 -> + cupicef.Term cupicef.prop x0) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I x0 + (__ : cupicef.Term cupicef.prop x0 => z0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I z0 + (__ : cupicef.Term cupicef.prop z0 => x0)) + (H7 : cupicef.Term cupicef.prop x0 => H3 (H5 H7)) + (H7 : cupicef.Term cupicef.prop z0 => H6 (H4 H7))) + H1) + H2) + (T : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + l0 : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + IHl : + (x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) -> + y0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) -> + z0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) -> + H1 : cupicef.Term cupicef.prop (predicate__equivalence l0 x0 y0) -> + H2 : cupicef.Term cupicef.prop (predicate__equivalence l0 y0 z0) -> + cupicef.Term cupicef.prop + (cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + cupicef.set)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l1 + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))))))) + (c => + c + (pointwise__lifting0 => + op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist (cupicef.type (cupicef.s cupicef.z)) + (l2 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l2 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) + (R : cupicef.Univ cupicef.prop => + R_27_ : cupicef.Univ cupicef.prop => + op R R_27_) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) => + R_27_ : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (x1 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + pointwise__lifting0 op tl (R x1) (R_27_ x1))) + l1)) + cupicef.0 Coq__Init__Logic.iff l0 x0 z0)) => + x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons T l0) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + y0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons T l0) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + z0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons T l0) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + H1 : + cupicef.Term cupicef.prop (predicate__equivalence (Tcons T l0) x0 y0) => + H2 : + cupicef.Term cupicef.prop (predicate__equivalence (Tcons T l0) y0 z0) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) T => + let__p l x y z H H0 T l0 IHl x0 y0 z0 H1 H2 x1 (H1 x1) (H2 x1)) + l x y z H H0. + +def predicate__equivalence__equivalence : + l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Term cupicef.prop + (Equivalence + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (predicate__equivalence l)) + +:= l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + Build__Equivalence + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (predicate__equivalence l) + ((l0 : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + predicate__equivalence__equivalence__obligation__1 l0) l) + ((l0 : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + predicate__equivalence__equivalence__obligation__2 l0) l) + ((l0 : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + predicate__equivalence__equivalence__obligation__3 l0) l). + +#TYPE_CLASS_INSTANCE predicate__equivalence__equivalence. + +thm predicate__implication__preorder__obligation__1 : + l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Term cupicef.prop + (Reflexive + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (predicate__implication l)) + +:= l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + x : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + Tlist__ind + (l0 : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I + (arrows l0 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + cupicef.set)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l1 + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))))))) + (c => + c + (pointwise__lifting0 => + op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist (cupicef.type (cupicef.s cupicef.z)) + (l2 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l2 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (R : cupicef.Univ cupicef.prop => + R_27_ : cupicef.Univ cupicef.prop => + op R R_27_) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) => + R_27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + pointwise__lifting0 op tl (R x1) (R_27_ x1))) + l1)) + cupicef.0 Coq__Program__Basics.impl l0 x0 x0)) + (x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + H : cupicef.Term cupicef.prop x0 => + H) + (T : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + l0 : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + IHl : + (x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) -> + cupicef.Term cupicef.prop + (cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + cupicef.set)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l1 + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))))))) + (c => + c + (pointwise__lifting0 => + op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist (cupicef.type (cupicef.s cupicef.z)) + (l2 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l2 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) + (R : cupicef.Univ cupicef.prop => + R_27_ : cupicef.Univ cupicef.prop => + op R R_27_) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) => + R_27_ : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (x1 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + pointwise__lifting0 op tl (R x1) (R_27_ x1))) + l1)) + cupicef.0 Coq__Program__Basics.impl l0 x0 x0)) => + x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons T l0) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) T => + IHl (x0 x1)) + l x. + +def let__p0 : + l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + x : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) -> + y : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) -> + z : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) -> + H : cupicef.Term cupicef.prop (predicate__implication l x y) -> + H0 : cupicef.Term cupicef.prop (predicate__implication l y z) -> + T : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + l0 : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + IHl : + (x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) -> + y0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) -> + z0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) -> + H1 : cupicef.Term cupicef.prop (predicate__implication l0 x0 y0) -> + H2 : cupicef.Term cupicef.prop (predicate__implication l0 y0 z0) -> + cupicef.Term cupicef.prop + (cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + cupicef.set)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l1 + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))))))) + (c => + c + (pointwise__lifting0 => + op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist (cupicef.type (cupicef.s cupicef.z)) + (l2 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l2 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (R : cupicef.Univ cupicef.prop => + R_27_ : cupicef.Univ cupicef.prop => + op R R_27_) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) => + R_27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + pointwise__lifting0 op tl (R x1) (R_27_ x1))) + l1)) + cupicef.0 Coq__Program__Basics.impl l0 x0 z0)) -> + x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons T l0) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) -> + y0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons T l0) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) -> + z0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons T l0) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) -> + H1 : cupicef.Term cupicef.prop (predicate__implication (Tcons T l0) x0 y0) -> + H2 : cupicef.Term cupicef.prop (predicate__implication (Tcons T l0) y0 z0) -> + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) T -> + H3 : + cupicef.Term cupicef.prop + (pointwise__lifting Coq__Program__Basics.impl l0 (x0 x1) (y0 x1)) -> + H4 : + cupicef.Term cupicef.prop + (pointwise__lifting Coq__Program__Basics.impl l0 (y0 x1) (z0 x1)) -> + cupicef.Term cupicef.prop + (cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) cupicef.set)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + cupicef.prod + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l1 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))))))) + (c => + c + (pointwise__lifting0 => + op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist (cupicef.type (cupicef.s cupicef.z)) + (l2 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l2 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (R : cupicef.Univ cupicef.prop => + R_27_ : cupicef.Univ cupicef.prop => + op R R_27_) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) => + R_27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + pointwise__lifting0 op tl (R x2) (R_27_ x2))) + l1)) + cupicef.0 Coq__Program__Basics.impl l0 (x0 x1) (z0 x1)) + +:= l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + x : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + y : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + z : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + H : cupicef.Term cupicef.prop (predicate__implication l x y) => + H0 : cupicef.Term cupicef.prop (predicate__implication l y z) => + T : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + l0 : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + IHl : + (x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) -> + y0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) -> + z0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) -> + H1 : cupicef.Term cupicef.prop (predicate__implication l0 x0 y0) -> + H2 : cupicef.Term cupicef.prop (predicate__implication l0 y0 z0) -> + cupicef.Term cupicef.prop + (cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + cupicef.set)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l1 + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))))))) + (c => + c + (pointwise__lifting0 => + op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist (cupicef.type (cupicef.s cupicef.z)) + (l2 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l2 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)))) + (R : cupicef.Univ cupicef.prop => + R_27_ : cupicef.Univ cupicef.prop => + op R R_27_) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) => + R_27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + pointwise__lifting0 op tl (R x1) (R_27_ x1))) + l1)) + cupicef.0 Coq__Program__Basics.impl l0 x0 z0)) => + x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons T l0) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + y0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons T l0) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + z0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons T l0) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + H1 : cupicef.Term cupicef.prop (predicate__implication (Tcons T l0) x0 y0) => + H2 : cupicef.Term cupicef.prop (predicate__implication (Tcons T l0) y0 z0) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) T => + IHl (x0 x1) (y0 x1) (z0 x1). + +thm predicate__implication__preorder__obligation__2 : + l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Term cupicef.prop + (Transitive + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (predicate__implication l)) + +:= l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + x : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + y : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + z : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + H : cupicef.Term cupicef.prop (predicate__implication l x y) => + H0 : cupicef.Term cupicef.prop (predicate__implication l y z) => + Tlist__ind + (l0 : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.I + (arrows l0 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (arrows l0 + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (y0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (z0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (predicate__implication l0 x0 y0) + (H1 : + cupicef.Term cupicef.prop + (predicate__implication l0 x0 y0) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (predicate__implication l0 y0 z0) + (H2 : + cupicef.Term cupicef.prop + (predicate__implication l0 y0 z0) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + cupicef.set)) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (op : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l1 + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)))))))) + (c => + c + (pointwise__lifting0 => + op : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist + (cupicef.type (cupicef.s cupicef.z)) + (l2 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l2 + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) + (R : cupicef.Univ cupicef.prop => + R_27_ : cupicef.Univ cupicef.prop => + op R R_27_) + (A : + cupicef.Univ + (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R : + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)))) => + R_27_ : + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)))) => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + pointwise__lifting0 op tl (R x1) (R_27_ x1))) + l1)) + cupicef.0 Coq__Program__Basics.impl l0 x0 z0)))))) + (x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + y0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + z0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + H1 : cupicef.Term cupicef.prop (predicate__implication Tnil x0 y0) => + H2 : cupicef.Term cupicef.prop (predicate__implication Tnil y0 z0) => + H3 : cupicef.Term cupicef.prop x0 => + H2 (H1 H3)) + (T : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + l0 : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + IHl : + (x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) -> + y0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) -> + z0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows l0 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) -> + H1 : cupicef.Term cupicef.prop (predicate__implication l0 x0 y0) -> + H2 : cupicef.Term cupicef.prop (predicate__implication l0 y0 z0) -> + cupicef.Term cupicef.prop + (cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + cupicef.set)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l1 + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))))))) + (c => + c + (pointwise__lifting0 => + op : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + l1 : + cupicef.Term + (cupicef.sup + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist (cupicef.type (cupicef.s cupicef.z)) + (l2 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l2 + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) + (R : cupicef.Univ cupicef.prop => + R_27_ : cupicef.Univ cupicef.prop => + op R R_27_) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) => + R_27_ : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows tl + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (x1 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + pointwise__lifting0 op tl (R x1) (R_27_ x1))) + l1)) + cupicef.0 Coq__Program__Basics.impl l0 x0 z0)) => + x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons T l0) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + y0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons T l0) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + z0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons T l0) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) => + H1 : + cupicef.Term cupicef.prop (predicate__implication (Tcons T l0) x0 y0) => + H2 : + cupicef.Term cupicef.prop (predicate__implication (Tcons T l0) y0 z0) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) T => + let__p0 l x y z H H0 T l0 IHl x0 y0 z0 H1 H2 x1 (H1 x1) (H2 x1)) + l x y z H H0. + +def predicate__implication__preorder : + l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist -> + cupicef.Term cupicef.prop + (PreOrder + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (predicate__implication l)) + +:= l : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + Build__PreOrder + (arrows l + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (predicate__implication l) + ((l0 : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + predicate__implication__preorder__obligation__1 l0) l) + ((l0 : + cupicef.Term + (cupicef.sup (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + predicate__implication__preorder__obligation__2 l0) l). + +#TYPE_CLASS_INSTANCE predicate__implication__preorder. + +def relation__equivalence : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + predicate__equivalence (Tcons A (Tcons A Tnil)). + +def RewriteRelation__instance__2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term cupicef.prop + (RewriteRelation + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (relation__equivalence A)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + Build__RewriteRelation + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (relation__equivalence A). + +#TYPE_CLASS_INSTANCE RewriteRelation__instance__2. + +def relation__conjunction : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + predicate__intersection (Tcons A (Tcons A Tnil)) R R_27_. + +def relation__disjunction : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R_27_ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + predicate__union (Tcons A (Tcons A Tnil)) R R_27_. + +thm relation__equivalence__equivalence : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term cupicef.prop + (Equivalence + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (relation__equivalence A)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + predicate__equivalence__equivalence (Tcons A (Tcons A Tnil)). + +#TYPE_CLASS_INSTANCE relation__equivalence__equivalence. + +thm relation__implication__preorder : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term cupicef.prop + (PreOrder + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (subrelation A)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + predicate__implication__preorder (Tcons A (Tcons A Tnil)). + +#TYPE_CLASS_INSTANCE relation__implication__preorder. + +def PartialOrder : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + equ : cupicef.Term cupicef.prop (Equivalence A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + preo : cupicef.Term cupicef.prop (PreOrder A R) -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + equ : cupicef.Term cupicef.prop (Equivalence A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + preo : cupicef.Term cupicef.prop (PreOrder A R) => + relation__equivalence A eqA + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R)). + +#TYPE_CLASS PartialOrder. + +def partial__order__equivalence : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + equ : cupicef.Term cupicef.prop (Equivalence A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + preo : cupicef.Term cupicef.prop (PreOrder A R) -> + PartialOrder0 : cupicef.Term cupicef.prop (PartialOrder A eqA equ R preo) -> + cupicef.Term cupicef.prop + (relation__equivalence A eqA + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + equ : cupicef.Term cupicef.prop (Equivalence A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + preo : cupicef.Term cupicef.prop (PreOrder A R) => + PartialOrder0 : cupicef.Term cupicef.prop (PartialOrder A eqA equ R preo) => + PartialOrder0. + +def let__poe : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + equ : cupicef.Term cupicef.prop (Equivalence A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + preo : cupicef.Term cupicef.prop (PreOrder A R) -> + H : cupicef.Term cupicef.prop (PartialOrder A eqA equ R preo) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H0 : cupicef.Term cupicef.prop (R x y) -> + H1 : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop + (relation__equivalence A eqA + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + equ : cupicef.Term cupicef.prop (Equivalence A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + preo : cupicef.Term cupicef.prop (PreOrder A R) => + H : cupicef.Term cupicef.prop (PartialOrder A eqA equ R preo) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x y) => + H1 : cupicef.Term cupicef.prop (R y x) => + partial__order__equivalence A eqA equ R preo H. + +def let__H2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + equ : cupicef.Term cupicef.prop (Equivalence A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + preo : cupicef.Term cupicef.prop (PreOrder A R) -> + H : cupicef.Term cupicef.prop (PartialOrder A eqA equ R preo) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H0 : cupicef.Term cupicef.prop (R x y) -> + H1 : cupicef.Term cupicef.prop (R y x) -> + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (eqA x0 x1) + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + x0 x1)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + equ : cupicef.Term cupicef.prop (Equivalence A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + preo : cupicef.Term cupicef.prop (PreOrder A R) => + H : cupicef.Term cupicef.prop (PartialOrder A eqA equ R preo) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x y) => + H1 : cupicef.Term cupicef.prop (R y x) => + let__poe A eqA equ R preo H x y H0 H1. + +def let__a : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + equ : cupicef.Term cupicef.prop (Equivalence A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + preo : cupicef.Term cupicef.prop (PreOrder A R) -> + H : cupicef.Term cupicef.prop (PartialOrder A eqA equ R preo) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H0 : cupicef.Term cupicef.prop (R x y) -> + H1 : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + equ : cupicef.Term cupicef.prop (Equivalence A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + preo : cupicef.Term cupicef.prop (PreOrder A R) => + H : cupicef.Term cupicef.prop (PartialOrder A eqA equ R preo) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x y) => + H1 : cupicef.Term cupicef.prop (R y x) => + x. + +def let__H3 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + equ : cupicef.Term cupicef.prop (Equivalence A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + preo : cupicef.Term cupicef.prop (PreOrder A R) -> + H : cupicef.Term cupicef.prop (PartialOrder A eqA equ R preo) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H0 : cupicef.Term cupicef.prop (R x y) -> + H1 : cupicef.Term cupicef.prop (R y x) -> + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (eqA (let__a A eqA equ R preo H x y H0 H1) x0) + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + (let__a A eqA equ R preo H x y H0 H1) x0)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + equ : cupicef.Term cupicef.prop (Equivalence A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + preo : cupicef.Term cupicef.prop (PreOrder A R) => + H : cupicef.Term cupicef.prop (PartialOrder A eqA equ R preo) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x y) => + H1 : cupicef.Term cupicef.prop (R y x) => + let__H2 A eqA equ R preo H x y H0 H1 (let__a A eqA equ R preo H x y H0 H1). + +def let__a0 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + equ : cupicef.Term cupicef.prop (Equivalence A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + preo : cupicef.Term cupicef.prop (PreOrder A R) -> + H : cupicef.Term cupicef.prop (PartialOrder A eqA equ R preo) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H0 : cupicef.Term cupicef.prop (R x y) -> + H1 : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + equ : cupicef.Term cupicef.prop (Equivalence A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + preo : cupicef.Term cupicef.prop (PreOrder A R) => + H : cupicef.Term cupicef.prop (PartialOrder A eqA equ R preo) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x y) => + H1 : cupicef.Term cupicef.prop (R y x) => + y. + +def let__H4 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + equ : cupicef.Term cupicef.prop (Equivalence A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + preo : cupicef.Term cupicef.prop (PreOrder A R) -> + H : cupicef.Term cupicef.prop (PartialOrder A eqA equ R preo) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H0 : cupicef.Term cupicef.prop (R x y) -> + H1 : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff (eqA x (let__a0 A eqA equ R preo H x y H0 H1)) + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + x (let__a0 A eqA equ R preo H x y H0 H1))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + equ : cupicef.Term cupicef.prop (Equivalence A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + preo : cupicef.Term cupicef.prop (PreOrder A R) => + H : cupicef.Term cupicef.prop (PartialOrder A eqA equ R preo) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x y) => + H1 : cupicef.Term cupicef.prop (R y x) => + let__H3 A eqA equ R preo H x y H0 H1 + (let__a0 A eqA equ R preo H x y H0 H1). + +thm partial__order__antisym : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + equ : cupicef.Term cupicef.prop (Equivalence A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + preo : cupicef.Term cupicef.prop (PreOrder A R) -> + H : cupicef.Term cupicef.prop (PartialOrder A eqA equ R preo) -> + cupicef.Term cupicef.prop (Antisymmetric A eqA equ R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + equ : cupicef.Term cupicef.prop (Equivalence A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + preo : cupicef.Term cupicef.prop (PreOrder A R) => + H : cupicef.Term cupicef.prop (PartialOrder A eqA equ R preo) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (R x y) + (H0 : cupicef.Term cupicef.prop (R x y) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + R y x) (H1 : cupicef.Term cupicef.prop (R y x) => eqA x y))))) + (Antisymmetric A eqA equ R) cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x y) => + H1 : cupicef.Term cupicef.prop (R y x) => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eqA x (let__a0 A eqA equ R preo H x y H0 H1)) + (__ : + cupicef.Term cupicef.prop + (eqA x (let__a0 A eqA equ R preo H x y H0 H1)) => + relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + x (let__a0 A eqA equ R preo H x y H0 H1))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + x (let__a0 A eqA equ R preo H x y H0 H1)) + (__ : + cupicef.Term cupicef.prop + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + x (let__a0 A eqA equ R preo H x y H0 H1)) => + eqA x (let__a0 A eqA equ R preo H x y H0 H1))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eqA x y) + (__ : cupicef.Term cupicef.prop (eqA x y) => + relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + x y)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + x y) + (__ : + cupicef.Term cupicef.prop + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x y) => + eqA x y))) => + eqA x y) + (__tmp : + (__ : cupicef.Term cupicef.prop (eqA x y) -> + cupicef.Term cupicef.prop + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + x y)) => + H4 : + (__ : + cupicef.Term cupicef.prop + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + x y) -> + cupicef.Term cupicef.prop (eqA x y)) => + H4 + (Coq__Init__Logic.conj (R x y) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x y) + H0 H1)) + (let__H4 A eqA equ R preo H x y H0 H1)). + +#TYPE_CLASS_INSTANCE partial__order__antisym. + +thm PartialOrder__inverse : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + equ : cupicef.Term cupicef.prop (Equivalence A eqA) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + preo : cupicef.Term cupicef.prop (PreOrder A R) -> + H : cupicef.Term cupicef.prop (PartialOrder A eqA equ R preo) -> + cupicef.Term cupicef.prop + (PartialOrder A eqA equ + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + (flip__PreOrder A R preo)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eqA : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + equ : cupicef.Term cupicef.prop (Equivalence A eqA) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + preo : cupicef.Term cupicef.prop (PreOrder A R) => + H : cupicef.Term cupicef.prop (PartialOrder A eqA equ R preo) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + pointwise__lifting Coq__Init__Logic.iff (Tcons A Tnil) (eqA x) + (relation__conjunction A + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R)) + x))) + (PartialOrder A eqA equ + (Coq__Program__Basics.flip A A + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + (flip__PreOrder A R preo)) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eqA x x0) + (__ : cupicef.Term cupicef.prop (eqA x x0) => + relation__conjunction A + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R)) + x x0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (relation__conjunction A + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R)) + x x0) + (__ : + cupicef.Term cupicef.prop + (relation__conjunction A + (Coq__Program__Basics.flip A A + (cupicef.cast + 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+ (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R)) + x x0) => + eqA x x0)))) + (pointwise__lifting Coq__Init__Logic.iff (Tcons A Tnil) (eqA x) + (relation__conjunction A + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R)) + x)) + cupicef.I + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eqA x x0) + (__ : cupicef.Term cupicef.prop (eqA x x0) => + relation__conjunction A + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ 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(cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R)) + x x0) + (__ : + cupicef.Term cupicef.prop + (relation__conjunction A + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R)) + x x0) => + eqA x x0)) + (H0 : cupicef.Term cupicef.prop (eqA x x0) => + Coq__Init__Logic.conj + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x x0) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + x x0) + ((H1 : + cupicef.Term cupicef.prop + (cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + cupicef.set)) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (op : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)))))))) + (c => + c + (pointwise__lifting0 => + op : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + l : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist + (cupicef.type (cupicef.s cupicef.z)) + (l0 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l0 + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) + (R0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + R_27_ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + op R0 R_27_) + (A0 : + cupicef.Univ + (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons A0 tl) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + R_27_ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons A0 tl) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A0 + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A0 => + pointwise__lifting0 op tl (R0 x1) (R_27_ x1))) + l)) + cupicef.0 Coq__Init__Logic.iff (Tcons A Tnil) ( + eqA x0) + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x0)) => + (H2 : + cupicef.Term cupicef.prop + (cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + cupicef.set)) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (op : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)))))))) + (c => + c + (pointwise__lifting0 => + op : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + l : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist + (cupicef.type (cupicef.s cupicef.z)) + (l0 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l0 + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) + (R0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + R_27_ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + op R0 R_27_) + (A0 : + cupicef.Univ + (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons A0 tl) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + R_27_ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons A0 tl) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A0 + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A0 => + pointwise__lifting0 op tl (R0 x1) + (R_27_ x1))) + l)) + cupicef.0 Coq__Init__Logic.iff (Tcons A Tnil) ( + eqA x) + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x)) => + (H3 : + cupicef.Term cupicef.prop + (cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + cupicef.set)) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) + (op : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)))))))) + (c => + c + (pointwise__lifting0 => + op : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + l : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist + (cupicef.type (cupicef.s cupicef.z)) + (l0 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l0 + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)))) + (R0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + R_27_ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + op R0 R_27_) + (A0 : + cupicef.Univ + (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons A0 tl) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + R_27_ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons A0 tl) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A0 + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A0 => + pointwise__lifting0 op tl (R0 x1) + (R_27_ x1))) + l)) + cupicef.0 Coq__Init__Logic.iff Tnil (eqA x x0) + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x x0)) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eqA x x0) + (__ : cupicef.Term cupicef.prop (eqA x x0) => + relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x x0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x x0) + (__ : + cupicef.Term cupicef.prop + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x x0) => + eqA x x0)) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x x0) + (H4 : + (__ : cupicef.Term cupicef.prop (eqA x x0) -> + cupicef.Term cupicef.prop + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x x0)) => + H5 : + (__ : + cupicef.Term cupicef.prop + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x x0) -> + cupicef.Term cupicef.prop (eqA x x0)) => + (H6 : + (__ : cupicef.Term cupicef.prop (R x x0) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x x0) -> + cupicef.Term cupicef.prop (eqA x x0)) => + (H7 : + cupicef.Term cupicef.prop + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x x0) => + Coq__Init__Logic.and__ind (R x x0) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x x0) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x x0) + (H8 : cupicef.Term cupicef.prop (R x x0) => + H9 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x x0) => + H9) + H7) + (H4 H0)) + (__ : cupicef.Term cupicef.prop (R x x0) => + __0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x x0) => + H5 + (Coq__Init__Logic.conj (R x x0) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x x0) + __ __0))) + H3) + (H2 x0)) + (H x)) + (H x0)) + ((H1 : + cupicef.Term cupicef.prop + (cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + cupicef.set)) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (op : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)))))))) + (c => + c + (pointwise__lifting0 => + op : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + l : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist + (cupicef.type (cupicef.s cupicef.z)) + (l0 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l0 + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) + (R0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + R_27_ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + op R0 R_27_) + (A0 : + cupicef.Univ + (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons A0 tl) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + R_27_ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons A0 tl) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A0 + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A0 => + pointwise__lifting0 op tl (R0 x1) (R_27_ x1))) + l)) + cupicef.0 Coq__Init__Logic.iff (Tcons A Tnil) ( + eqA x0) + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x0)) => + (H2 : + cupicef.Term cupicef.prop + (cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + cupicef.set)) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (op : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)))))))) + (c => + c + (pointwise__lifting0 => + op : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + l : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist + (cupicef.type (cupicef.s cupicef.z)) + (l0 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l0 + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) + (R0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + R_27_ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + op R0 R_27_) + (A0 : + cupicef.Univ + (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons A0 tl) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + R_27_ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons A0 tl) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A0 + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A0 => + pointwise__lifting0 op tl (R0 x1) + (R_27_ x1))) + l)) + cupicef.0 Coq__Init__Logic.iff (Tcons A Tnil) ( + eqA x) + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x)) => + (H3 : + cupicef.Term cupicef.prop + (cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + cupicef.set)) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) + (op : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)))))))) + (c => + c + (pointwise__lifting0 => + op : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + l : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist + (cupicef.type (cupicef.s cupicef.z)) + (l0 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l0 + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)))) + (R0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + R_27_ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + op R0 R_27_) + (A0 : + cupicef.Univ + (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons A0 tl) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + R_27_ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons A0 tl) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A0 + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A0 => + pointwise__lifting0 op tl (R0 x1) + (R_27_ x1))) + l)) + cupicef.0 Coq__Init__Logic.iff Tnil (eqA x x0) + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x x0)) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eqA x x0) + (__ : cupicef.Term cupicef.prop (eqA x x0) => + relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x x0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x x0) + (__ : + cupicef.Term cupicef.prop + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x x0) => + eqA x x0)) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x x0) + (H4 : + (__ : cupicef.Term cupicef.prop (eqA x x0) -> + cupicef.Term cupicef.prop + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x x0)) => + H5 : + (__ : + cupicef.Term cupicef.prop + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x x0) -> + cupicef.Term cupicef.prop (eqA x x0)) => + (H6 : + (__ : cupicef.Term cupicef.prop (R x x0) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x x0) -> + cupicef.Term cupicef.prop (eqA x x0)) => + (H7 : + cupicef.Term cupicef.prop + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x x0) => + Coq__Init__Logic.and__ind (R x x0) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x x0) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x x0) + (H8 : cupicef.Term cupicef.prop (R x x0) => + H9 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x x0) => + H8) + H7) + (H4 H0)) + (__ : cupicef.Term cupicef.prop (R x x0) => + __0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x x0) => + H5 + (Coq__Init__Logic.conj (R x x0) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x x0) + __ __0))) + H3) + (H2 x0)) + (H x)) + (H x0))) + (H0 : + cupicef.Term cupicef.prop + (relation__conjunction A + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R)) + x x0) => + Coq__Init__Logic.and__ind + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x x0) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + x x0) + (eqA x x0) + (H1 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R x x0) => + H2 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + R) + x x0) => + (H3 : + cupicef.Term cupicef.prop + (cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + cupicef.set)) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (op : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)))))))) + (c => + c + (pointwise__lifting0 => + op : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + l : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist + (cupicef.type (cupicef.s cupicef.z)) + (l0 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l0 + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) + (R0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + R_27_ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + op R0 R_27_) + (A0 : + cupicef.Univ + (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons A0 tl) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + R_27_ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons A0 tl) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A0 + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A0 => + pointwise__lifting0 op tl (R0 x1) (R_27_ x1))) + l)) + cupicef.0 Coq__Init__Logic.iff (Tcons A Tnil) ( + eqA x0) + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x0)) => + (H4 : + cupicef.Term cupicef.prop + (cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + cupicef.set)) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (op : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Tlist + (l : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)))))))) + (c => + c + (pointwise__lifting0 => + op : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) => + l : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist + (cupicef.type (cupicef.s cupicef.z)) + (l0 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l0 + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)))) + (R0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + R_27_ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + op R0 R_27_) + (A0 : + cupicef.Univ + (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons A0 tl) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + R_27_ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons A0 tl) + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A0 + (x1 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A0 => + pointwise__lifting0 op tl (R0 x1) + (R_27_ x1))) + l)) + cupicef.0 Coq__Init__Logic.iff Tnil (eqA x0 x0) + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x0 x0)) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eqA x0 x0) + (__ : cupicef.Term cupicef.prop (eqA x0 x0) => + relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x0 x0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x0 x0) + (__ : + cupicef.Term cupicef.prop + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x0 x0) => + eqA x0 x0)) + (eqA x x0) + (H5 : + (__ : cupicef.Term cupicef.prop (eqA x0 x0) -> + cupicef.Term cupicef.prop + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x0 x0)) => + H6 : + (__ : + cupicef.Term cupicef.prop + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x0 x0) -> + cupicef.Term cupicef.prop (eqA x0 x0)) => + (H7 : + (__ : cupicef.Term cupicef.prop (R x0 x0) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x0 x0) -> + cupicef.Term cupicef.prop (eqA x0 x0)) => + (H8 : + cupicef.Term cupicef.prop + (cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + cupicef.set)) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type + (cupicef.s cupicef.z)))) + cupicef.I + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) + (op : + cupicef.Term + (cupicef.type + (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s + cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I Tlist + (l : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s + cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)))))))) + (c => + c + (pointwise__lifting0 => + op : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (binary__relation + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + l : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + match____Tlist + (cupicef.type (cupicef.s cupicef.z)) + (l0 : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + binary__relation + (arrows l0 + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)))) + (R0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + R_27_ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows Tnil + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + op R0 R_27_) + (A0 : + cupicef.Univ + (cupicef.type (cupicef.s cupicef.z)) => + tl : + cupicef.Term + (cupicef.sup + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.set) + Tlist => + R0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons A0 tl) + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + R_27_ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (arrows (Tcons A0 tl) + (cupicef.cast + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I))) => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A0 + (x1 : + cupicef.Term + (cupicef.type + (cupicef.s cupicef.z)) + A0 => + pointwise__lifting0 op tl (R0 x1) + (R_27_ x1))) + l)) + cupicef.0 Coq__Init__Logic.iff Tnil (eqA x0 x) + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x0 x)) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eqA x0 x) + (__ : cupicef.Term cupicef.prop (eqA x0 x) => + relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x0 x)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x0 x) + (__ : + cupicef.Term cupicef.prop + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)) + R) + x0 x) => + eqA x0 x)) + (eqA x x0) + (H9 : + (__ : cupicef.Term cupicef.prop (eqA x0 x) -> + cupicef.Term cupicef.prop + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R) + x0 x)) => + H10 : + (__ : + cupicef.Term cupicef.prop + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)) + R) + x0 x) -> + cupicef.Term cupicef.prop (eqA x0 x)) => + (H11 : + (__ : cupicef.Term cupicef.prop (R x0 x) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x0 x) -> + cupicef.Term cupicef.prop (eqA x0 x)) => + (H12 : + (__ : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)) + R x0 x) -> + cupicef.Term cupicef.prop (eqA x0 x)) => + (H13 : cupicef.Term cupicef.prop (eqA x0 x) => + (H14 : + cupicef.Term cupicef.prop + (relation__conjunction A R + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)) + R) + x0 x) => + Coq__Init__Logic.and__ind (R x0 x) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x0 x) + (eqA x x0) + (H15 : cupicef.Term cupicef.prop (R x0 x) => + H16 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I)) + R x0 x) => + symmetry A eqA + (Equivalence__Symmetric A eqA equ) x0 x H13) + H14) + (H9 H13)) + (H12 H2)) + (H11 H1)) + (__ : cupicef.Term cupicef.prop (R x0 x) => + __0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x0 x) => + H10 + (Coq__Init__Logic.conj (R x0 x) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x0 x) + __ __0))) + H8) + (H3 x)) + (__ : cupicef.Term cupicef.prop (R x0 x0) => + __0 : + cupicef.Term cupicef.prop + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x0 x0) => + H6 + (Coq__Init__Logic.conj (R x0 x0) + (Coq__Program__Basics.flip A A + (cupicef.cast + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + R x0 x0) + __ __0))) + H4) + (H3 x0)) + (H x0)) + H0))). + +thm subrelation__partial__order__obligation__1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term cupicef.prop + (relation__equivalence + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (relation__equivalence A) + (relation__conjunction + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (subrelation A) + (Coq__Program__Basics.flip + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (subrelation A)))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + Coq__Init__Logic.conj + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop)) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop)) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop)) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (x x1 x2) + (__ : cupicef.Term cupicef.prop (x x1 x2) => x0 x1 x2)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (x0 x1 x2) + (__ : cupicef.Term cupicef.prop (x0 x1 x2) => x x1 x2))))) + (__ : + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (x x1 x2) + (__ : cupicef.Term cupicef.prop (x x1 x2) => x0 x1 x2)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (x0 x1 x2) + (__ : cupicef.Term cupicef.prop (x0 x1 x2) => x x1 x2)))) => + Coq__Init__Logic.and + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (x x1 y) + (__0 : cupicef.Term cupicef.prop (x x1 y) => x0 x1 y)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (x0 x1 y) + (__0 : cupicef.Term cupicef.prop (x0 x1 y) => x x1 y)))))) + (cupicef.prod cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop))) + cupicef.I + (Coq__Init__Logic.and + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (x x1 y) + (__ : cupicef.Term cupicef.prop (x x1 y) => x0 x1 y)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (x0 x1 y) + (__ : cupicef.Term cupicef.prop (x0 x1 y) => x x1 y))))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (x x1 y) + (__ : cupicef.Term cupicef.prop (x x1 y) => x0 x1 y)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (x0 x1 y) + (__ : cupicef.Term cupicef.prop (x0 x1 y) => x x1 y))))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop)) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (x x1 x2) + (__0 : cupicef.Term cupicef.prop (x x1 x2) => x0 x1 x2)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (x0 x1 x2) + (__0 : cupicef.Term cupicef.prop (x0 x1 x2) => x x1 x2)))))) + (H : + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + x x1 x2) + (__ : cupicef.Term cupicef.prop (x x1 x2) => x0 x1 x2)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (x0 x1 x2) + (__ : cupicef.Term cupicef.prop (x0 x1 x2) => x x1 x2)))) => + Coq__Init__Logic.conj + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + x x1 y) (__ : cupicef.Term cupicef.prop (x x1 y) => x0 x1 y)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + x0 x1 y) + (__ : cupicef.Term cupicef.prop (x0 x1 y) => x x1 y)))) + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (x x1 y) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (x x1 y) + (__ : cupicef.Term cupicef.prop (x x1 y) => x0 x1 y)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (x0 x1 y) + (__ : cupicef.Term cupicef.prop (x0 x1 y) => x x1 y)) + (x0 x1 y) + (H2 : + (__ : cupicef.Term cupicef.prop (x x1 y) -> + cupicef.Term cupicef.prop (x0 x1 y)) => + H3 : + (__ : cupicef.Term cupicef.prop (x0 x1 y) -> + cupicef.Term cupicef.prop (x x1 y)) => + H2 H0) + (H x1 y)) + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (x0 x1 y) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (x x1 y) + (__ : cupicef.Term cupicef.prop (x x1 y) => x0 x1 y)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (x0 x1 y) + (__ : cupicef.Term cupicef.prop (x0 x1 y) => x x1 y)) + (x x1 y) + (H2 : + (__ : cupicef.Term cupicef.prop (x x1 y) -> + cupicef.Term cupicef.prop (x0 x1 y)) => + H3 : + (__ : cupicef.Term cupicef.prop (x0 x1 y) -> + cupicef.Term cupicef.prop (x x1 y)) => + H3 H0) + (H x1 y))) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (x x1 y) + (__ : cupicef.Term cupicef.prop (x x1 y) => x0 x1 y)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (x0 x1 y) + (__ : cupicef.Term cupicef.prop (x0 x1 y) => x x1 y))))) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.and__ind + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + x x3 y) (__ : cupicef.Term cupicef.prop (x x3 y) => x0 x3 y)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + x0 x3 y) + (__ : cupicef.Term cupicef.prop (x0 x3 y) => x x3 y)))) + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (x x1 x2) + (__ : cupicef.Term cupicef.prop (x x1 x2) => x0 x1 x2)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (x0 x1 x2) + (__ : cupicef.Term cupicef.prop (x0 x1 x2) => x x1 x2))) + (H0 : + (x3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (x x3 y) -> + cupicef.Term cupicef.prop (x0 x3 y)) => + H1 : + (x3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (x0 x3 y) -> + cupicef.Term cupicef.prop (x x3 y)) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (x x1 x2) + (__ : cupicef.Term cupicef.prop (x x1 x2) => x0 x1 x2)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (x0 x1 x2) + (__ : cupicef.Term cupicef.prop (x0 x1 x2) => x x1 x2)) + (H2 : cupicef.Term cupicef.prop (x x1 x2) => H0 x1 x2 H2) + (H2 : cupicef.Term cupicef.prop (x0 x1 x2) => H1 x1 x2 H2)) + H). + +def subrelation__partial__order : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term cupicef.prop + (PartialOrder + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I (Coq__Relations__Relation__Definitions.relation A)) + (relation__equivalence A) (relation__equivalence__equivalence A) + (subrelation A) (relation__implication__preorder A)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + (A0 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + subrelation__partial__order__obligation__1 A0) A. + +#TYPE_CLASS_INSTANCE subrelation__partial__order. + +(; End of translation. ;) diff --git a/out/Coq__Classes__SetoidTactics.dk b/out/Coq__Classes__SetoidTactics.dk new file mode 100644 index 000000000..d37f97cb9 --- /dev/null +++ b/out/Coq__Classes__SetoidTactics.dk @@ -0,0 +1,98 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def DefaultRelation : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Univ cupicef.prop. + +#TYPE_CLASS DefaultRelation. + +def Build__DefaultRelation : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Term cupicef.prop (DefaultRelation A R). + +[ ] cupicef.code_guarded? _ (Build__DefaultRelation _ _) --> cupicef.guarded. + +def match____DefaultRelation : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + P : (cupicef.Term cupicef.prop (DefaultRelation A R) -> cupicef.Univ s) -> + case__Build__DefaultRelation : + cupicef.Term s (P (Build__DefaultRelation A R)) -> + x : cupicef.Term cupicef.prop (DefaultRelation A R) -> + cupicef.Term s (P x). + +[ s, A, R, P, case__Build__DefaultRelation] + match____DefaultRelation s A R P case__Build__DefaultRelation + (Build__DefaultRelation A R) + --> + case__Build__DefaultRelation. + +[ s, A, R, P, s'] + match____DefaultRelation s A R + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____DefaultRelation s' A R (x => P x). + +def default__relation : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : cupicef.Term cupicef.prop (DefaultRelation A R) -> + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : cupicef.Term cupicef.prop (DefaultRelation A R) => + R. + +def equivalence__default : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + H : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Equivalence A R) -> + cupicef.Term cupicef.prop (DefaultRelation A R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + H : + cupicef.Term cupicef.prop + (Coq__Classes__RelationClasses.Equivalence A R) => + Build__DefaultRelation A R. + +#TYPE_CLASS_INSTANCE equivalence__default. + +(; End of translation. ;) diff --git a/out/Coq__Init__Datatypes.dk b/out/Coq__Init__Datatypes.dk new file mode 100644 index 000000000..386b9d26f --- /dev/null +++ b/out/Coq__Init__Datatypes.dk @@ -0,0 +1,9568 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +#REQUIRE cupicef. +#REQUIRE Coq__Init__Logic. + +def Empty__set : + cupicef.Univ cupicef.set. + +def match____Empty__set : + s : cupicef.Sort -> + P : (cupicef.Term cupicef.set Empty__set -> cupicef.Univ s) -> + x : cupicef.Term cupicef.set Empty__set -> + cupicef.Term s (P x). + +[ s, P, s'] + match____Empty__set s + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____Empty__set s' (x => P x). + +def Empty__set__rect : + P : + (e : cupicef.Term cupicef.set Empty__set -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + e : cupicef.Term cupicef.set Empty__set -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P e) + +:= P : + (e : cupicef.Term cupicef.set Empty__set -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + e : cupicef.Term cupicef.set Empty__set => + match____Empty__set (cupicef.type (cupicef.s cupicef.z)) + (e0 : cupicef.Term cupicef.set Empty__set => P e0) e. + +def Empty__set__ind : + P : (e : cupicef.Term cupicef.set Empty__set -> cupicef.Univ cupicef.prop) -> + e : cupicef.Term cupicef.set Empty__set -> + cupicef.Term cupicef.prop (P e) + +:= P : (e : cupicef.Term cupicef.set Empty__set -> cupicef.Univ cupicef.prop) => + e : cupicef.Term cupicef.set Empty__set => + match____Empty__set cupicef.prop + (e0 : cupicef.Term cupicef.set Empty__set => P e0) e. + +def Empty__set__rec : + P : (e : cupicef.Term cupicef.set Empty__set -> cupicef.Univ cupicef.set) -> + e : cupicef.Term cupicef.set Empty__set -> + cupicef.Term cupicef.set (P e) + +:= P : (e : cupicef.Term cupicef.set Empty__set -> cupicef.Univ cupicef.set) => + Empty__set__rect + (cupicef.cast + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.set (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Empty__set + (e : cupicef.Term cupicef.set Empty__set => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I Empty__set + (e : cupicef.Term cupicef.set Empty__set => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P). + +def unit : + cupicef.Univ cupicef.set. + +def tt : + cupicef.Term cupicef.set unit. + +[ ] cupicef.code_guarded? _ tt --> cupicef.guarded. + +def match____unit : + s : cupicef.Sort -> + P : (cupicef.Term cupicef.set unit -> cupicef.Univ s) -> + case__tt : cupicef.Term s (P tt) -> + x : cupicef.Term cupicef.set unit -> + cupicef.Term s (P x). + +[ s, P, case__tt] + match____unit s P case__tt tt + --> + case__tt. + +[ s, P, s'] + match____unit s + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____unit s' (x => P x). + +def unit__rect : + P : + (u : cupicef.Term cupicef.set unit -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P tt) -> + u : cupicef.Term cupicef.set unit -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P u) + +:= P : + (u : cupicef.Term cupicef.set unit -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P tt) => + u : cupicef.Term cupicef.set unit => + match____unit (cupicef.type (cupicef.s cupicef.z)) + (u0 : cupicef.Term cupicef.set unit => P u0) f u. + +def unit__ind : + P : (u : cupicef.Term cupicef.set unit -> cupicef.Univ cupicef.prop) -> + f : cupicef.Term cupicef.prop (P tt) -> + u : cupicef.Term cupicef.set unit -> + cupicef.Term cupicef.prop (P u) + +:= P : (u : cupicef.Term cupicef.set unit -> cupicef.Univ cupicef.prop) => + f : cupicef.Term cupicef.prop (P tt) => + u : cupicef.Term cupicef.set unit => + match____unit cupicef.prop (u0 : cupicef.Term cupicef.set unit => P u0) f + u. + +def unit__rec : + P : (u : cupicef.Term cupicef.set unit -> cupicef.Univ cupicef.set) -> + f : cupicef.Term cupicef.set (P tt) -> + u : cupicef.Term cupicef.set unit -> + cupicef.Term cupicef.set (P u) + +:= P : (u : cupicef.Term cupicef.set unit -> cupicef.Univ cupicef.set) => + unit__rect + (cupicef.cast + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.set (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + cupicef.I unit + (u : cupicef.Term cupicef.set unit => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I unit + (u : cupicef.Term cupicef.set unit => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P). + +def bool : + cupicef.Univ cupicef.set. + +def true : + cupicef.Term cupicef.set bool. + +def false : + cupicef.Term cupicef.set bool. + +[ ] cupicef.code_guarded? _ true --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ false --> cupicef.guarded. + +def match____bool : + s : cupicef.Sort -> + P : (cupicef.Term cupicef.set bool -> cupicef.Univ s) -> + case__true : cupicef.Term s (P true) -> + case__false : cupicef.Term s (P false) -> + x : cupicef.Term cupicef.set bool -> + cupicef.Term s (P x). + +[ s, P, case__true, case__false] + match____bool s P case__true case__false true + --> + case__true. + +[ s, P, case__true, case__false] + match____bool s P case__true case__false false + --> + case__false. + +[ s, P, s'] + match____bool s + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____bool s' (x => P x). + +def bool__rect : + P : + (b : cupicef.Term cupicef.set bool -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P true) -> + f0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P false) -> + b : cupicef.Term cupicef.set bool -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P b) + +:= P : + (b : cupicef.Term cupicef.set bool -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P true) => + f0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P false) => + b : cupicef.Term cupicef.set bool => + match____bool (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set bool => P b0) f f0 b. + +def bool__ind : + P : (b : cupicef.Term cupicef.set bool -> cupicef.Univ cupicef.prop) -> + f : cupicef.Term cupicef.prop (P true) -> + f0 : cupicef.Term cupicef.prop (P false) -> + b : cupicef.Term cupicef.set bool -> + cupicef.Term cupicef.prop (P b) + +:= P : (b : cupicef.Term cupicef.set bool -> cupicef.Univ cupicef.prop) => + f : cupicef.Term cupicef.prop (P true) => + f0 : cupicef.Term cupicef.prop (P false) => + b : cupicef.Term cupicef.set bool => + match____bool cupicef.prop (b0 : cupicef.Term cupicef.set bool => P b0) f + f0 b. + +def bool__rec : + P : (b : cupicef.Term cupicef.set bool -> cupicef.Univ cupicef.set) -> + f : cupicef.Term cupicef.set (P true) -> + f0 : cupicef.Term cupicef.set (P false) -> + b : cupicef.Term cupicef.set bool -> + cupicef.Term cupicef.set (P b) + +:= P : (b : cupicef.Term cupicef.set bool -> cupicef.Univ cupicef.set) => + bool__rect + (cupicef.cast + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.set (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + cupicef.I bool + (b : cupicef.Term cupicef.set bool => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I bool + (b : cupicef.Term cupicef.set bool => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P). + +def andb : + b1 : cupicef.Term cupicef.set bool -> + b2 : cupicef.Term cupicef.set bool -> + cupicef.Term cupicef.set bool + +:= b1 : cupicef.Term cupicef.set bool => + b2 : cupicef.Term cupicef.set bool => + match____bool cupicef.set (b3 : cupicef.Term cupicef.set bool => bool) b2 + false b1. + +def orb : + b1 : cupicef.Term cupicef.set bool -> + b2 : cupicef.Term cupicef.set bool -> + cupicef.Term cupicef.set bool + +:= b1 : cupicef.Term cupicef.set bool => + b2 : cupicef.Term cupicef.set bool => + match____bool cupicef.set (b3 : cupicef.Term cupicef.set bool => bool) + true b2 b1. + +def implb : + b1 : cupicef.Term cupicef.set bool -> + b2 : cupicef.Term cupicef.set bool -> + cupicef.Term cupicef.set bool + +:= b1 : cupicef.Term cupicef.set bool => + b2 : cupicef.Term cupicef.set bool => + match____bool cupicef.set (b3 : cupicef.Term cupicef.set bool => bool) b2 + true b1. + +def xorb : + b1 : cupicef.Term cupicef.set bool -> + b2 : cupicef.Term cupicef.set bool -> + cupicef.Term cupicef.set bool + +:= b1 : cupicef.Term cupicef.set bool => + b2 : cupicef.Term cupicef.set bool => + match____bool cupicef.set (b3 : cupicef.Term cupicef.set bool => bool) + (match____bool cupicef.set (b3 : cupicef.Term cupicef.set bool => bool) + false true b2) + (match____bool cupicef.set (b3 : cupicef.Term cupicef.set bool => bool) + true false b2) + b1. + +def negb : + b : cupicef.Term cupicef.set bool -> cupicef.Term cupicef.set bool + +:= b : cupicef.Term cupicef.set bool => + match____bool cupicef.set (b0 : cupicef.Term cupicef.set bool => bool) + false true b. + +thm andb__prop : + a : cupicef.Term cupicef.set bool -> + b : cupicef.Term cupicef.set bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + (andb a b) true) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + a true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + b true)) + +:= a : cupicef.Term cupicef.set bool => + match____bool cupicef.prop + (b : cupicef.Term cupicef.set bool => + cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I bool + (b0 : cupicef.Term cupicef.set bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + (andb b b0) true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + (andb b b0) true) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + b true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + b0 true)))) + (b : cupicef.Term cupicef.set bool => + match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + (andb true b0) true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + (andb true b0) true) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + b0 true))) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + (andb true true) true) => + Coq__Init__Logic.conj + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + (andb true false) true) => + Coq__Init__Logic.conj + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true) + H) + b) + (b : cupicef.Term cupicef.set bool => + match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + (andb false b0) true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + (andb false b0) true) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + b0 true))) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + (andb false true) true) => + Coq__Init__Logic.conj + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) + H + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + (andb false false) true) => + Coq__Init__Logic.conj + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) + H H) + b) + a. + +thm andb__true__intro : + b1 : cupicef.Term cupicef.set bool -> + b2 : cupicef.Term cupicef.set bool -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + b1 true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + b2 true)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + (andb b1 b2) true) + +:= b1 : cupicef.Term cupicef.set bool => + match____bool cupicef.prop + (b : cupicef.Term cupicef.set bool => + cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I bool + (b2 : cupicef.Term cupicef.set bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + b true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + b2 true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + b true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + b2 true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + (andb b b2) true))) + (b2 : cupicef.Term cupicef.set bool => + match____bool cupicef.prop + (b : cupicef.Term cupicef.set bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + b true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + b true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + (andb true b) true)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + (andb true true) true)) + cupicef.I + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true)) => + Coq__Init__Logic.match____and cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) => + H1) + H)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + (andb true false) true)) + cupicef.I + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true)) => + Coq__Init__Logic.match____and cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) => + H1) + H)) + b2) + (b2 : cupicef.Term cupicef.set bool => + match____bool cupicef.prop + (b : cupicef.Term cupicef.set bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + b true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + b true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + (andb false b) true)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + (andb false true) true)) + cupicef.I + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true)) => + Coq__Init__Logic.match____and cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + true true) => + H0) + H)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + (andb false false) true)) + cupicef.I + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true)) => + Coq__Init__Logic.match____and cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + false true) => + H1) + H)) + b2) + b1. + +def eq__true : + __ : cupicef.Term cupicef.set bool -> cupicef.Univ cupicef.prop. + +def is__eq__true : + cupicef.Term cupicef.prop (eq__true true). + +[ ] cupicef.code_guarded? _ is__eq__true --> cupicef.guarded. + +def match____eq__true : + s : cupicef.Sort -> + P : + (__ : cupicef.Term cupicef.set bool -> + cupicef.Term cupicef.prop (eq__true __) -> + cupicef.Univ s) -> + case__is__eq__true : cupicef.Term s (P true is__eq__true) -> + __ : cupicef.Term cupicef.set bool -> + x : cupicef.Term cupicef.prop (eq__true __) -> + cupicef.Term s (P __ x). + +[ s, P, case__is__eq__true] + match____eq__true s P case__is__eq__true {true} is__eq__true + --> + case__is__eq__true. + +[ s, P, s'] + match____eq__true s + (__ => x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P __ x))) + --> + match____eq__true s' (__ => x => P __ x). + +def eq__true__rect : + P : + (__ : cupicef.Term cupicef.set bool -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P true) -> + b : cupicef.Term cupicef.set bool -> + e : cupicef.Term cupicef.prop (eq__true b) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P b) + +:= P : + (__ : cupicef.Term cupicef.set bool -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P true) => + b : cupicef.Term cupicef.set bool => + e : cupicef.Term cupicef.prop (eq__true b) => + match____eq__true (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set bool => + __ : cupicef.Term cupicef.prop (eq__true b0) => + P b0) + f b e. + +def eq__true__ind : + P : (__ : cupicef.Term cupicef.set bool -> cupicef.Univ cupicef.prop) -> + f : cupicef.Term cupicef.prop (P true) -> + b : cupicef.Term cupicef.set bool -> + e : cupicef.Term cupicef.prop (eq__true b) -> + cupicef.Term cupicef.prop (P b) + +:= P : (__ : cupicef.Term cupicef.set bool -> cupicef.Univ cupicef.prop) => + f : cupicef.Term cupicef.prop (P true) => + b : cupicef.Term cupicef.set bool => + e : cupicef.Term cupicef.prop (eq__true b) => + match____eq__true cupicef.prop + (b0 : cupicef.Term cupicef.set bool => + __ : cupicef.Term cupicef.prop (eq__true b0) => + P b0) + f b e. + +def eq__true__rec : + P : (__ : cupicef.Term cupicef.set bool -> cupicef.Univ cupicef.set) -> + f : cupicef.Term cupicef.set (P true) -> + b : cupicef.Term cupicef.set bool -> + e : cupicef.Term cupicef.prop (eq__true b) -> + cupicef.Term cupicef.set (P b) + +:= P : (__ : cupicef.Term cupicef.set bool -> cupicef.Univ cupicef.set) => + eq__true__rect + (cupicef.cast + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.set (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + cupicef.I bool + (__ : cupicef.Term cupicef.set bool => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I bool + (__ : cupicef.Term cupicef.set bool => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P). + +def is__true : + b : cupicef.Term cupicef.set bool -> cupicef.Univ cupicef.prop + +:= b : cupicef.Term cupicef.set bool => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I bool) + b true. + +def eq__true__ind__r : + P : (__ : cupicef.Term cupicef.set bool -> cupicef.Univ cupicef.prop) -> + b : cupicef.Term cupicef.set bool -> + __ : cupicef.Term cupicef.prop (P b) -> + __0 : cupicef.Term cupicef.prop (eq__true b) -> + cupicef.Term cupicef.prop (P true) + +:= P : (__ : cupicef.Term cupicef.set bool -> cupicef.Univ cupicef.prop) => + b : cupicef.Term cupicef.set bool => + H : cupicef.Term cupicef.prop (P b) => + H0 : cupicef.Term cupicef.prop (eq__true b) => + match____eq__true cupicef.prop + (b0 : cupicef.Term cupicef.set bool => + __ : cupicef.Term cupicef.prop (eq__true b0) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (P b0) + (H1 : cupicef.Term cupicef.prop (P b0) => P true)) + (H1 : cupicef.Term cupicef.prop (P true) => H1) b H0 H. + +def eq__true__rec__r : + P : (__ : cupicef.Term cupicef.set bool -> cupicef.Univ cupicef.set) -> + b : cupicef.Term cupicef.set bool -> + __ : cupicef.Term cupicef.set (P b) -> + __0 : cupicef.Term cupicef.prop (eq__true b) -> + cupicef.Term cupicef.set (P true) + +:= P : (__ : cupicef.Term cupicef.set bool -> cupicef.Univ cupicef.set) => + b : cupicef.Term cupicef.set bool => + H : cupicef.Term cupicef.set (P b) => + H0 : cupicef.Term cupicef.prop (eq__true b) => + match____eq__true cupicef.set + (b0 : cupicef.Term cupicef.set bool => + __ : cupicef.Term cupicef.prop (eq__true b0) => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I (P b0) + (H1 : cupicef.Term cupicef.set (P b0) => P true)) + (H1 : cupicef.Term cupicef.set (P true) => H1) b H0 H. + +def eq__true__rect__r : + P : + (__ : cupicef.Term cupicef.set bool -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + b : cupicef.Term cupicef.set bool -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P b) -> + __0 : cupicef.Term cupicef.prop (eq__true b) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P true) + +:= P : + (__ : cupicef.Term cupicef.set bool -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + b : cupicef.Term cupicef.set bool => + H : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P b) => + H0 : cupicef.Term cupicef.prop (eq__true b) => + match____eq__true (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set bool => + __ : cupicef.Term cupicef.prop (eq__true b0) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P b0) + (H1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P b0) => + P true)) + (H1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P true) => H1) + b H0 H. + +def BoolSpec : + P : cupicef.Univ cupicef.prop -> + Q : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.set bool -> + cupicef.Univ cupicef.prop. + +def BoolSpecT : + P : cupicef.Univ cupicef.prop -> + Q : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop P -> + cupicef.Term cupicef.prop (BoolSpec P Q true). + +def BoolSpecF : + P : cupicef.Univ cupicef.prop -> + Q : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop Q -> + cupicef.Term cupicef.prop (BoolSpec P Q false). + +[ ] cupicef.code_guarded? _ (BoolSpecT _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (BoolSpecF _ _ _) --> cupicef.guarded. + +def match____BoolSpec : + s : cupicef.Sort -> + P : cupicef.Univ cupicef.prop -> + Q : cupicef.Univ cupicef.prop -> + P0 : + (__ : cupicef.Term cupicef.set bool -> + cupicef.Term cupicef.prop (BoolSpec P Q __) -> + cupicef.Univ s) -> + case__BoolSpecT : + (__ : cupicef.Term cupicef.prop P -> + cupicef.Term s (P0 true (BoolSpecT P Q __))) -> + case__BoolSpecF : + (__ : cupicef.Term cupicef.prop Q -> + cupicef.Term s (P0 false (BoolSpecF P Q __))) -> + __ : cupicef.Term cupicef.set bool -> + x : cupicef.Term cupicef.prop (BoolSpec P Q __) -> + cupicef.Term s (P0 __ x). + +[ s, P, Q, P0, case__BoolSpecT, case__BoolSpecF, __] + match____BoolSpec s P Q P0 case__BoolSpecT case__BoolSpecF {true} + (BoolSpecT P Q __) + --> + case__BoolSpecT __. + +[ s, P, Q, P0, case__BoolSpecT, case__BoolSpecF, __] + match____BoolSpec s P Q P0 case__BoolSpecT case__BoolSpecF {false} + (BoolSpecF P Q __) + --> + case__BoolSpecF __. + +[ s, P, Q, P0, s'] + match____BoolSpec s P Q + (__ => x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P0 __ x))) + --> + match____BoolSpec s' P Q (__ => x => P0 __ x). + +def BoolSpec__ind : + P : cupicef.Univ cupicef.prop -> + Q : cupicef.Univ cupicef.prop -> + P0 : (__ : cupicef.Term cupicef.set bool -> cupicef.Univ cupicef.prop) -> + f : + (__ : cupicef.Term cupicef.prop P -> cupicef.Term cupicef.prop (P0 true)) -> + f0 : + (__ : cupicef.Term cupicef.prop Q -> cupicef.Term cupicef.prop (P0 false)) -> + b : cupicef.Term cupicef.set bool -> + b0 : cupicef.Term cupicef.prop (BoolSpec P Q b) -> + cupicef.Term cupicef.prop (P0 b) + +:= P : cupicef.Univ cupicef.prop => + Q : cupicef.Univ cupicef.prop => + P0 : (__ : cupicef.Term cupicef.set bool -> cupicef.Univ cupicef.prop) => + f : + (__ : cupicef.Term cupicef.prop P -> cupicef.Term cupicef.prop (P0 true)) => + f0 : + (__ : cupicef.Term cupicef.prop Q -> + cupicef.Term cupicef.prop (P0 false)) => + b : cupicef.Term cupicef.set bool => + b0 : cupicef.Term cupicef.prop (BoolSpec P Q b) => + match____BoolSpec cupicef.prop P Q + (b1 : cupicef.Term cupicef.set bool => + __ : cupicef.Term cupicef.prop (BoolSpec P Q b1) => + P0 b1) + f f0 b b0. + +def nat : + cupicef.Univ cupicef.set. + +def O : + cupicef.Term cupicef.set nat. + +def S : + __ : cupicef.Term cupicef.set nat -> cupicef.Term cupicef.set nat. + +[ ] cupicef.code_guarded? _ O --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (S _) --> cupicef.guarded. + +def match____nat : + s : cupicef.Sort -> + P : (cupicef.Term cupicef.set nat -> cupicef.Univ s) -> + case__O : cupicef.Term s (P O) -> + case__S : (__ : cupicef.Term cupicef.set nat -> cupicef.Term s (P (S __))) -> + x : cupicef.Term cupicef.set nat -> + cupicef.Term s (P x). + +[ s, P, case__O, case__S] + match____nat s P case__O case__S O + --> + case__O. + +[ s, P, case__O, case__S, __] + match____nat s P case__O case__S (S __) + --> + case__S __. + +[ s, P, s'] + match____nat s (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____nat s' (x => P x). + +def nat__rect : + P : + (n : cupicef.Term cupicef.set nat -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P O) -> + f0 : + (n : cupicef.Term cupicef.set nat -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P n) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (S n))) -> + n : cupicef.Term cupicef.set nat -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P n) + +:= P : + (n : cupicef.Term cupicef.set nat -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P O) => + f0 : + (n : cupicef.Term cupicef.set nat -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P n) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (S n))) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.prod cupicef.set (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + cupicef.I nat (n : cupicef.Term cupicef.set nat => P n)))) + (c => + c + (F => + n : cupicef.Term cupicef.set nat => + match____nat (cupicef.type (cupicef.s cupicef.z)) + (n0 : cupicef.Term cupicef.set nat => P n0) f + (n0 : cupicef.Term cupicef.set nat => f0 n0 (F n0)) n)) + cupicef.0. + +def nat__ind : + P : (n : cupicef.Term cupicef.set nat -> cupicef.Univ cupicef.prop) -> + f : cupicef.Term cupicef.prop (P O) -> + f0 : + (n : cupicef.Term cupicef.set nat -> + __ : cupicef.Term cupicef.prop (P n) -> + cupicef.Term cupicef.prop (P (S n))) -> + n : cupicef.Term cupicef.set nat -> + cupicef.Term cupicef.prop (P n) + +:= P : (n : cupicef.Term cupicef.set nat -> cupicef.Univ cupicef.prop) => + f : cupicef.Term cupicef.prop (P O) => + f0 : + (n : cupicef.Term cupicef.set nat -> + __ : cupicef.Term cupicef.prop (P n) -> + cupicef.Term cupicef.prop (P (S n))) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.prop + (cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I nat + (n : cupicef.Term cupicef.set nat => P n)))) + (c => + c + (F => + n : cupicef.Term cupicef.set nat => + match____nat cupicef.prop + (n0 : cupicef.Term cupicef.set nat => P n0) f + (n0 : cupicef.Term cupicef.set nat => f0 n0 (F n0)) n)) + cupicef.0. + +def nat__rec : + P : (n : cupicef.Term cupicef.set nat -> cupicef.Univ cupicef.set) -> + f : cupicef.Term cupicef.set (P O) -> + f0 : + (n : cupicef.Term cupicef.set nat -> + __ : cupicef.Term cupicef.set (P n) -> + cupicef.Term cupicef.set (P (S n))) -> + n : cupicef.Term cupicef.set nat -> + cupicef.Term cupicef.set (P n) + +:= P : (n : cupicef.Term cupicef.set nat -> cupicef.Univ cupicef.set) => + nat__rect + (cupicef.cast + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.set (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + cupicef.I nat + (n : cupicef.Term cupicef.set nat => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I nat + (n : cupicef.Term cupicef.set nat => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P). + +def option : + Coq__Init__Datatypes__13 : cupicef.Sort -> + A : cupicef.Univ Coq__Init__Datatypes__13 -> + cupicef.Univ (cupicef.sup cupicef.set Coq__Init__Datatypes__13). + +option' : + A : cupicef.Univ cupicef.sinf -> cupicef.Univ cupicef.sinf. + +[ Coq__Init__Datatypes__13, A] + cupicef.code (cupicef.cU _) (option Coq__Init__Datatypes__13 A) + --> + option' (cupicef.code (cupicef.cU Coq__Init__Datatypes__13) A). + +def Coq__Init__Datatypes__13 : + cupicef.Nat + +:= cupicef.s cupicef.z. + +def Some : + A : cupicef.Univ (cupicef.type Coq__Init__Datatypes__13) -> + __ : cupicef.Term (cupicef.type Coq__Init__Datatypes__13) A -> + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type Coq__Init__Datatypes__13)) + (option (cupicef.type Coq__Init__Datatypes__13) A). + +def None : + A : cupicef.Univ (cupicef.type Coq__Init__Datatypes__13) -> + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type Coq__Init__Datatypes__13)) + (option (cupicef.type Coq__Init__Datatypes__13) A). + +[ ] cupicef.code_guarded? _ (Some _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (None _) --> cupicef.guarded. + +def match____option : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type Coq__Init__Datatypes__13) -> + P : + (cupicef.Term + (cupicef.sup cupicef.set (cupicef.type Coq__Init__Datatypes__13)) + (option (cupicef.type Coq__Init__Datatypes__13) A) -> + cupicef.Univ s) -> + case__Some : + (__ : cupicef.Term (cupicef.type Coq__Init__Datatypes__13) A -> + cupicef.Term s (P (Some A __))) -> + case__None : cupicef.Term s (P (None A)) -> + x : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type Coq__Init__Datatypes__13)) + (option (cupicef.type Coq__Init__Datatypes__13) A) -> + cupicef.Term s (P x). + +[ s, A, P, case__Some, case__None, __] + match____option s A P case__Some case__None (Some A __) + --> + case__Some __. + +[ s, A, P, case__Some, case__None] + match____option s A P case__Some case__None (None A) + --> + case__None. + +[ s, A, P, s'] + match____option s A + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____option s' A (x => P x). + +def option__rect : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (o : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (option (cupicef.type (cupicef.s cupicef.z)) A) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (Some A a))) -> + f0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (None A)) -> + o : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (option (cupicef.type (cupicef.s cupicef.z)) A) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P o) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (o : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (option (cupicef.type (cupicef.s cupicef.z)) A) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (Some A a))) => + f0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (None A)) => + o : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (option (cupicef.type (cupicef.s cupicef.z)) A) => + match____option (cupicef.type (cupicef.s cupicef.z)) A + (o0 : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (option (cupicef.type (cupicef.s cupicef.z)) A) => + P o0) + f f0 o. + +def option__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (o : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (option (cupicef.type (cupicef.s cupicef.z)) A) -> + cupicef.Univ cupicef.prop) -> + f : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (P (Some A a))) -> + f0 : cupicef.Term cupicef.prop (P (None A)) -> + o : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (option (cupicef.type (cupicef.s cupicef.z)) A) -> + cupicef.Term cupicef.prop (P o) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (o : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (option (cupicef.type (cupicef.s cupicef.z)) A) -> + cupicef.Univ cupicef.prop) => + f : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (P (Some A a))) => + f0 : cupicef.Term cupicef.prop (P (None A)) => + o : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (option (cupicef.type (cupicef.s cupicef.z)) A) => + match____option cupicef.prop A + (o0 : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (option (cupicef.type (cupicef.s cupicef.z)) A) => + P o0) + f f0 o. + +def option__rec : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (o : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (option (cupicef.type (cupicef.s cupicef.z)) A) -> + cupicef.Univ cupicef.set) -> + f : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set (P (Some A a))) -> + f0 : cupicef.Term cupicef.set (P (None A)) -> + o : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (option (cupicef.type (cupicef.s cupicef.z)) A) -> + cupicef.Term cupicef.set (P o) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (o : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (option (cupicef.type (cupicef.s cupicef.z)) A) -> + cupicef.Univ cupicef.set) => + option__rect A + (cupicef.cast + (cupicef.rule + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (option (cupicef.type (cupicef.s cupicef.z)) A) + (o : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (option (cupicef.type (cupicef.s cupicef.z)) A) => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (option (cupicef.type (cupicef.s cupicef.z)) A) + (o : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (option (cupicef.type (cupicef.s cupicef.z)) A) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P). + +def option__map : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + o : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (option (cupicef.type (cupicef.s cupicef.z)) A) -> + cupicef.Term (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (option (cupicef.type (cupicef.s cupicef.z)) B) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + o : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (option (cupicef.type (cupicef.s cupicef.z)) A) => + match____option + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) A + (o0 : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (option (cupicef.type (cupicef.s cupicef.z)) A) => + option (cupicef.type (cupicef.s cupicef.z)) B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Some B (f a)) + (None B) o. + +def sum : + Coq__Init__Datatypes__19 : cupicef.Sort -> + Coq__Init__Datatypes__20 : cupicef.Sort -> + A : cupicef.Univ Coq__Init__Datatypes__19 -> + B : cupicef.Univ Coq__Init__Datatypes__20 -> + cupicef.Univ + (cupicef.sup cupicef.set + (cupicef.sup Coq__Init__Datatypes__19 Coq__Init__Datatypes__20)). + +sum' : + A : cupicef.Univ cupicef.sinf -> + B : cupicef.Univ cupicef.sinf -> + cupicef.Univ cupicef.sinf. + +[ Coq__Init__Datatypes__19, Coq__Init__Datatypes__20, A, B] + cupicef.code (cupicef.cU _) + (sum Coq__Init__Datatypes__19 Coq__Init__Datatypes__20 A B) + --> + sum' (cupicef.code (cupicef.cU Coq__Init__Datatypes__19) A) + (cupicef.code (cupicef.cU Coq__Init__Datatypes__20) B). + +def Coq__Init__Datatypes__19 : + cupicef.Nat + +:= cupicef.s cupicef.z. + +def Coq__Init__Datatypes__20 : + cupicef.Nat + +:= cupicef.s cupicef.z. + +def inl : + A : cupicef.Univ (cupicef.type Coq__Init__Datatypes__19) -> + B : cupicef.Univ (cupicef.type Coq__Init__Datatypes__20) -> + __ : cupicef.Term (cupicef.type Coq__Init__Datatypes__19) A -> + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type Coq__Init__Datatypes__19) + (cupicef.type Coq__Init__Datatypes__20))) + (sum (cupicef.type Coq__Init__Datatypes__19) + (cupicef.type Coq__Init__Datatypes__20) A B). + +def inr : + A : cupicef.Univ (cupicef.type Coq__Init__Datatypes__19) -> + B : cupicef.Univ (cupicef.type Coq__Init__Datatypes__20) -> + __ : cupicef.Term (cupicef.type Coq__Init__Datatypes__20) B -> + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type Coq__Init__Datatypes__19) + (cupicef.type Coq__Init__Datatypes__20))) + (sum (cupicef.type Coq__Init__Datatypes__19) + (cupicef.type Coq__Init__Datatypes__20) A B). + +[ ] cupicef.code_guarded? _ (inl _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (inr _ _ _) --> cupicef.guarded. + +def match____sum : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type Coq__Init__Datatypes__19) -> + B : cupicef.Univ (cupicef.type Coq__Init__Datatypes__20) -> + P : + (cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type Coq__Init__Datatypes__19) + (cupicef.type Coq__Init__Datatypes__20))) + (sum (cupicef.type Coq__Init__Datatypes__19) + (cupicef.type Coq__Init__Datatypes__20) A B) -> + cupicef.Univ s) -> + case__inl : + (__ : cupicef.Term (cupicef.type Coq__Init__Datatypes__19) A -> + cupicef.Term s (P (inl A B __))) -> + case__inr : + (__ : cupicef.Term (cupicef.type Coq__Init__Datatypes__20) B -> + cupicef.Term s (P (inr A B __))) -> + x : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type Coq__Init__Datatypes__19) + (cupicef.type Coq__Init__Datatypes__20))) + (sum (cupicef.type Coq__Init__Datatypes__19) + (cupicef.type Coq__Init__Datatypes__20) A B) -> + cupicef.Term s (P x). + +[ s, A, B, P, case__inl, case__inr, __] + match____sum s A B P case__inl case__inr (inl A B __) + --> + case__inl __. + +[ s, A, B, P, case__inl, case__inr, __] + match____sum s A B P case__inl case__inr (inr A B __) + --> + case__inr __. + +[ s, A, B, P, s'] + match____sum s A B + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____sum s' A B (x => P x). + +def sum__rect : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (s : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (inl A B a))) -> + f0 : + (b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (inr A B b))) -> + s : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P s) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (s : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (inl A B a))) => + f0 : + (b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (inr A B b))) => + s : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + match____sum (cupicef.type (cupicef.s cupicef.z)) A B + (s0 : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + P s0) + f f0 s. + +def sum__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (s : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ cupicef.prop) -> + f : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (P (inl A B a))) -> + f0 : + (b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term cupicef.prop (P (inr A B b))) -> + s : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Term cupicef.prop (P s) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (s : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ cupicef.prop) => + f : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (P (inl A B a))) => + f0 : + (b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term cupicef.prop (P (inr A B b))) => + s : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + match____sum cupicef.prop A B + (s0 : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + P s0) + f f0 s. + +def sum__rec : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (s : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ cupicef.set) -> + f : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set (P (inl A B a))) -> + f0 : + (b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term cupicef.set (P (inr A B b))) -> + s : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Term cupicef.set (P s) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (s : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ cupicef.set) => + sum__rect A B + (cupicef.cast + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) + (s : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) + (s : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P). + +def prod : + Coq__Init__Datatypes__23 : cupicef.Sort -> + Coq__Init__Datatypes__24 : cupicef.Sort -> + A : cupicef.Univ Coq__Init__Datatypes__23 -> + B : cupicef.Univ Coq__Init__Datatypes__24 -> + cupicef.Univ + (cupicef.sup Coq__Init__Datatypes__23 Coq__Init__Datatypes__24). + +prod' : + A : cupicef.Univ cupicef.sinf -> + B : cupicef.Univ cupicef.sinf -> + cupicef.Univ cupicef.sinf. + +[ Coq__Init__Datatypes__23, Coq__Init__Datatypes__24, A, B] + cupicef.code (cupicef.cU _) + (prod Coq__Init__Datatypes__23 Coq__Init__Datatypes__24 A B) + --> + prod' (cupicef.code (cupicef.cU Coq__Init__Datatypes__23) A) + (cupicef.code (cupicef.cU Coq__Init__Datatypes__24) B). + +def Coq__Init__Datatypes__23 : + cupicef.Nat + +:= cupicef.s cupicef.z. + +def Coq__Init__Datatypes__24 : + cupicef.Nat + +:= cupicef.s cupicef.z. + +def pair : + A : cupicef.Univ (cupicef.type Coq__Init__Datatypes__23) -> + B : cupicef.Univ (cupicef.type Coq__Init__Datatypes__24) -> + __ : cupicef.Term (cupicef.type Coq__Init__Datatypes__23) A -> + __0 : cupicef.Term (cupicef.type Coq__Init__Datatypes__24) B -> + cupicef.Term + (cupicef.sup (cupicef.type Coq__Init__Datatypes__23) + (cupicef.type Coq__Init__Datatypes__24)) + (prod (cupicef.type Coq__Init__Datatypes__23) + (cupicef.type Coq__Init__Datatypes__24) A B). + +[ ] cupicef.code_guarded? _ (pair _ _ _ _) --> cupicef.guarded. + +def match____prod : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type Coq__Init__Datatypes__23) -> + B : cupicef.Univ (cupicef.type Coq__Init__Datatypes__24) -> + P : + (cupicef.Term + (cupicef.sup (cupicef.type Coq__Init__Datatypes__23) + (cupicef.type Coq__Init__Datatypes__24)) + (prod (cupicef.type Coq__Init__Datatypes__23) + (cupicef.type Coq__Init__Datatypes__24) A B) -> + cupicef.Univ s) -> + case__pair : + (__ : cupicef.Term (cupicef.type Coq__Init__Datatypes__23) A -> + __0 : cupicef.Term (cupicef.type Coq__Init__Datatypes__24) B -> + cupicef.Term s (P (pair A B __ __0))) -> + x : + cupicef.Term + (cupicef.sup (cupicef.type Coq__Init__Datatypes__23) + (cupicef.type Coq__Init__Datatypes__24)) + (prod (cupicef.type Coq__Init__Datatypes__23) + (cupicef.type Coq__Init__Datatypes__24) A B) -> + cupicef.Term s (P x). + +[ s, A, B, P, case__pair, __, __0] + match____prod s A B P case__pair (pair A B __ __0) + --> + case__pair __ __0. + +[ s, A, B, P, s'] + match____prod s A B + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____prod s' A B (x => P x). + +def prod__rect : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (pair A B a b))) -> + p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (pair A B a b))) => + p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + match____prod (cupicef.type (cupicef.s cupicef.z)) A B + (p0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + P p0) + f p. + +def prod__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ cupicef.prop) -> + f : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term cupicef.prop (P (pair A B a b))) -> + p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Term cupicef.prop (P p) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ cupicef.prop) => + f : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term cupicef.prop (P (pair A B a b))) => + p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + match____prod cupicef.prop A B + (p0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + P p0) + f p. + +def prod__rec : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ cupicef.set) -> + f : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term cupicef.set (P (pair A B a b))) -> + p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Term cupicef.set (P p) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ cupicef.set) => + prod__rect A B + (cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) + (p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) + (p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P). + +def fst : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + match____prod (cupicef.type (cupicef.s cupicef.z)) A B + (p0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + A) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + x) + p. + +def snd : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + match____prod (cupicef.type (cupicef.s cupicef.z)) A B + (p0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + B) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + y) + p. + +thm surjective__pairing : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B)) + p (pair A B (fst A B p) (snd A B p))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + match____prod cupicef.prop A B + (p0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B)) + p0 (pair A B (fst A B p0) (snd A B p0))) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B)) + (pair A B (fst A B (pair A B a b)) (snd A B (pair A B a b)))) + p. + +thm injective__projections : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + p1 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + p2 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (fst A B p1) (fst A B p2)) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (snd A B p1) (snd A B p2)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B)) + p1 p2) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + p1 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + match____prod cupicef.prop A B + (p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) + (p2 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (fst A B p) (fst A B p2)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (fst A B p) (fst A B p2)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (snd A B p) (snd A B p2)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (snd A B p) (snd A B p2)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B)) + p p2)))) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + p2 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + match____prod cupicef.prop A B + (p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (fst A B (pair A B a b)) (fst A B p)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (fst A B (pair A B a b)) (fst A B p)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (snd A B (pair A B a b)) (snd A B p)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (snd A B (pair A B a b)) (snd A B p)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B)) + (pair A B a b) p))) + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + b0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + cupicef.cast + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a a0) + (Hfst : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a a0) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + b b0) + (Hsnd : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + b b0) => + (a1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B)) + (pair A B a1 b) (pair A B a0 b0)) + a))) + (cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (fst A B (pair A B a b)) (fst A B (pair A B a0 b0))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (fst A B (pair A B a b)) (fst A B (pair A B a0 b0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (snd A B (pair A B a b)) (snd A B (pair A B a0 b0))) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (snd A B (pair A B a b)) (snd A B (pair A B a0 b0))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B)) + (pair A B a b) (pair A B a0 b0)))) + cupicef.I + (Hfst : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a a0) => + Hsnd : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + b b0) => + Coq__Init__Logic.eq__ind__r A a0 + (a1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B)) + (pair A B a1 b) (pair A B a0 b0)) + (Coq__Init__Logic.eq__ind__r B b0 + (b1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B)) + (pair A B a0 b1) (pair A B a0 b0)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B)) + (pair A B a0 b0)) + b Hsnd) + a Hfst)) + p2) + p1. + +def prod__uncurry : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + C : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + C : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + f (pair A B x y). + +def prod__curry : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + C : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C) -> + p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + C : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C) => + p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + match____prod (cupicef.type (cupicef.s cupicef.z)) A B + (p0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + C) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + f x y) + p. + +def list : + Coq__Init__Datatypes__44 : cupicef.Sort -> + A : cupicef.Univ Coq__Init__Datatypes__44 -> + cupicef.Univ (cupicef.sup cupicef.set Coq__Init__Datatypes__44). + +list' : + A : cupicef.Univ cupicef.sinf -> cupicef.Univ cupicef.sinf. + +[ Coq__Init__Datatypes__44, A] + cupicef.code (cupicef.cU _) (list Coq__Init__Datatypes__44 A) + --> + list' (cupicef.code (cupicef.cU Coq__Init__Datatypes__44) A). + +def Coq__Init__Datatypes__44 : + cupicef.Nat + +:= cupicef.s cupicef.z. + +def nil : + A : cupicef.Univ (cupicef.type Coq__Init__Datatypes__44) -> + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type Coq__Init__Datatypes__44)) + (list (cupicef.type Coq__Init__Datatypes__44) A). + +def cons : + A : cupicef.Univ (cupicef.type Coq__Init__Datatypes__44) -> + __ : cupicef.Term (cupicef.type Coq__Init__Datatypes__44) A -> + __0 : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type Coq__Init__Datatypes__44)) + (list (cupicef.type Coq__Init__Datatypes__44) A) -> + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type Coq__Init__Datatypes__44)) + (list (cupicef.type Coq__Init__Datatypes__44) A). + +[ ] cupicef.code_guarded? _ (nil _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (cons _ _ _) --> cupicef.guarded. + +def match____list : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type Coq__Init__Datatypes__44) -> + P : + (cupicef.Term + (cupicef.sup cupicef.set (cupicef.type Coq__Init__Datatypes__44)) + (list (cupicef.type Coq__Init__Datatypes__44) A) -> + cupicef.Univ s) -> + case__nil : cupicef.Term s (P (nil A)) -> + case__cons : + (__ : cupicef.Term (cupicef.type Coq__Init__Datatypes__44) A -> + __0 : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type Coq__Init__Datatypes__44)) + (list (cupicef.type Coq__Init__Datatypes__44) A) -> + cupicef.Term s (P (cons A __ __0))) -> + x : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type Coq__Init__Datatypes__44)) + (list (cupicef.type Coq__Init__Datatypes__44) A) -> + cupicef.Term s (P x). + +[ s, A, P, case__nil, case__cons] + match____list s A P case__nil case__cons (nil A) + --> + case__nil. + +[ s, A, P, case__nil, case__cons, __, __0] + match____list s A P case__nil case__cons (cons A __ __0) + --> + case__cons __ __0. + +[ s, A, P, s'] + match____list s A + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____list s' A (x => P x). + +def list__rect : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (l : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (nil A)) -> + f0 : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + l : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P l) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (cons A a l))) -> + l : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P l) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (l : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (nil A)) => + f0 : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + l : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P l) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (cons A a l))) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 + (cupicef.sup cupicef.set + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (list (cupicef.type (cupicef.s cupicef.z)) A) + (l : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) => + P l)))) + (c => + c + (F => + l : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) => + match____list (cupicef.type (cupicef.s cupicef.z)) A + (l0 : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) => + P l0) + f + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + l0 : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) => + f0 y l0 (F l0)) + l)) + cupicef.0. + +def list__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (l : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) -> + cupicef.Univ cupicef.prop) -> + f : cupicef.Term cupicef.prop (P (nil A)) -> + f0 : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + l : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) -> + __ : cupicef.Term cupicef.prop (P l) -> + cupicef.Term cupicef.prop (P (cons A a l))) -> + l : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) -> + cupicef.Term cupicef.prop (P l) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (l : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) -> + cupicef.Univ cupicef.prop) => + f : cupicef.Term cupicef.prop (P (nil A)) => + f0 : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + l : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) -> + __ : cupicef.Term cupicef.prop (P l) -> + cupicef.Term cupicef.prop (P (cons A a l))) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.prop + (cupicef.prod + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop) + cupicef.I (list (cupicef.type (cupicef.s cupicef.z)) A) + (l : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) => + P l)))) + (c => + c + (F => + l : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) => + match____list cupicef.prop A + (l0 : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) => + P l0) + f + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + l0 : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) => + f0 y l0 (F l0)) + l)) + cupicef.0. + +def list__rec : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (l : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) -> + cupicef.Univ cupicef.set) -> + f : cupicef.Term cupicef.set (P (nil A)) -> + f0 : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + l : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) -> + __ : cupicef.Term cupicef.set (P l) -> + cupicef.Term cupicef.set (P (cons A a l))) -> + l : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) -> + cupicef.Term cupicef.set (P l) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (l : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) -> + cupicef.Univ cupicef.set) => + list__rect A + (cupicef.cast + (cupicef.rule + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (list (cupicef.type (cupicef.s cupicef.z)) A) + (l : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (list (cupicef.type (cupicef.s cupicef.z)) A) + (l : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P). + +def length : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + __ : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) -> + cupicef.Term cupicef.set nat + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.prod + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + cupicef.set + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.type (cupicef.s cupicef.z))) + cupicef.set) + cupicef.I (list (cupicef.type (cupicef.s cupicef.z)) A) + (l : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) => + nat)))) + (c => + c + (length0 => + l : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) => + match____list cupicef.set A + (l0 : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) => + nat) + O + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + l_27_ : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) => + S (length0 l_27_)) + l)) + cupicef.0. + +def app : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + __ : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) -> + __0 : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) -> + cupicef.Term (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.prod + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup cupicef.set + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup cupicef.set + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I (list (cupicef.type (cupicef.s cupicef.z)) A) + (l : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) => + cupicef.prod + (cupicef.sup cupicef.set + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup cupicef.set + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup cupicef.set + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.sup cupicef.set + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (list (cupicef.type (cupicef.s cupicef.z)) A) + (m : + cupicef.Term + (cupicef.sup cupicef.set + (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) => + list (cupicef.type (cupicef.s cupicef.z)) A))))) + (c => + c + (app0 => + l : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) => + m : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) => + match____list + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) A + (l0 : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) => + list (cupicef.type (cupicef.s cupicef.z)) A) + m + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + l1 : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (list (cupicef.type (cupicef.s cupicef.z)) A) => + cons A a (app0 l1 m)) + l)) + cupicef.0. + +def comparison : + cupicef.Univ cupicef.set. + +def Eq : + cupicef.Term cupicef.set comparison. + +def Lt : + cupicef.Term cupicef.set comparison. + +def Gt : + cupicef.Term cupicef.set comparison. + +[ ] cupicef.code_guarded? _ Eq --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ Lt --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ Gt --> cupicef.guarded. + +def match____comparison : + s : cupicef.Sort -> + P : (cupicef.Term cupicef.set comparison -> cupicef.Univ s) -> + case__Eq : cupicef.Term s (P Eq) -> + case__Lt : cupicef.Term s (P Lt) -> + case__Gt : cupicef.Term s (P Gt) -> + x : cupicef.Term cupicef.set comparison -> + cupicef.Term s (P x). + +[ s, P, case__Eq, case__Lt, case__Gt] + match____comparison s P case__Eq case__Lt case__Gt Eq + --> + case__Eq. + +[ s, P, case__Eq, case__Lt, case__Gt] + match____comparison s P case__Eq case__Lt case__Gt Lt + --> + case__Lt. + +[ s, P, case__Eq, case__Lt, case__Gt] + match____comparison s P case__Eq case__Lt case__Gt Gt + --> + case__Gt. + +[ s, P, s'] + match____comparison s + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____comparison s' (x => P x). + +def comparison__rect : + P : + (c : cupicef.Term cupicef.set comparison -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P Eq) -> + f0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P Lt) -> + f1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P Gt) -> + c : cupicef.Term cupicef.set comparison -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P c) + +:= P : + (c : cupicef.Term cupicef.set comparison -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P Eq) => + f0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P Lt) => + f1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P Gt) => + c : cupicef.Term cupicef.set comparison => + match____comparison (cupicef.type (cupicef.s cupicef.z)) + (c0 : cupicef.Term cupicef.set comparison => P c0) f f0 f1 c. + +def comparison__ind : + P : (c : cupicef.Term cupicef.set comparison -> cupicef.Univ cupicef.prop) -> + f : cupicef.Term cupicef.prop (P Eq) -> + f0 : cupicef.Term cupicef.prop (P Lt) -> + f1 : cupicef.Term cupicef.prop (P Gt) -> + c : cupicef.Term cupicef.set comparison -> + cupicef.Term cupicef.prop (P c) + +:= P : (c : cupicef.Term cupicef.set comparison -> cupicef.Univ cupicef.prop) => + f : cupicef.Term cupicef.prop (P Eq) => + f0 : cupicef.Term cupicef.prop (P Lt) => + f1 : cupicef.Term cupicef.prop (P Gt) => + c : cupicef.Term cupicef.set comparison => + match____comparison cupicef.prop + (c0 : cupicef.Term cupicef.set comparison => P c0) f f0 f1 c. + +def comparison__rec : + P : (c : cupicef.Term cupicef.set comparison -> cupicef.Univ cupicef.set) -> + f : cupicef.Term cupicef.set (P Eq) -> + f0 : cupicef.Term cupicef.set (P Lt) -> + f1 : cupicef.Term cupicef.set (P Gt) -> + c : cupicef.Term cupicef.set comparison -> + cupicef.Term cupicef.set (P c) + +:= P : (c : cupicef.Term cupicef.set comparison -> cupicef.Univ cupicef.set) => + comparison__rect + (cupicef.cast + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.set (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + cupicef.I comparison + (c : cupicef.Term cupicef.set comparison => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I comparison + (c : cupicef.Term cupicef.set comparison => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P). + +def let__H0 : + c : cupicef.Term cupicef.set comparison -> + c_27_ : cupicef.Term cupicef.set comparison -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Lt))) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Lt) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= c : cupicef.Term cupicef.set comparison => + c_27_ : cupicef.Term cupicef.set comparison => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Lt))) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Lt) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq + (e : cupicef.Term cupicef.set comparison => + match____comparison (cupicef.type (cupicef.s cupicef.z)) + (c0 : cupicef.Term cupicef.set comparison => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True Coq__Init__Logic.False Coq__Init__Logic.False e) + Coq__Init__Logic.I Lt H0. + +def let__n : + c : cupicef.Term cupicef.set comparison -> + c_27_ : cupicef.Term cupicef.set comparison -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Lt))) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= c : cupicef.Term cupicef.set comparison => + c_27_ : cupicef.Term cupicef.set comparison => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Lt))) => + H + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Lt) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Lt) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Lt)) + cupicef.I + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Lt) => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H0 c c_27_ H H0))). + +def let__H1 : + c : cupicef.Term cupicef.set comparison -> + c_27_ : cupicef.Term cupicef.set comparison -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Gt))) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Gt) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= c : cupicef.Term cupicef.set comparison => + c_27_ : cupicef.Term cupicef.set comparison => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Gt))) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Gt) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq + (e : cupicef.Term cupicef.set comparison => + match____comparison (cupicef.type (cupicef.s cupicef.z)) + (c0 : cupicef.Term cupicef.set comparison => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True Coq__Init__Logic.False Coq__Init__Logic.False e) + Coq__Init__Logic.I Gt H0. + +def let__n0 : + c : cupicef.Term cupicef.set comparison -> + c_27_ : cupicef.Term cupicef.set comparison -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Gt))) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= c : cupicef.Term cupicef.set comparison => + c_27_ : cupicef.Term cupicef.set comparison => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Gt))) => + H + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Gt) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Gt) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Gt)) + cupicef.I + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Gt) => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H1 c c_27_ H H0))). + +def let__H2 : + c : cupicef.Term cupicef.set comparison -> + c_27_ : cupicef.Term cupicef.set comparison -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Eq))) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Eq) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= c : cupicef.Term cupicef.set comparison => + c_27_ : cupicef.Term cupicef.set comparison => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Eq))) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Eq) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt + (e : cupicef.Term cupicef.set comparison => + match____comparison (cupicef.type (cupicef.s cupicef.z)) + (c0 : cupicef.Term cupicef.set comparison => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False Coq__Init__Logic.True Coq__Init__Logic.False e) + Coq__Init__Logic.I Eq H0. + +def let__n1 : + c : cupicef.Term cupicef.set comparison -> + c_27_ : cupicef.Term cupicef.set comparison -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Eq))) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= c : cupicef.Term cupicef.set comparison => + c_27_ : cupicef.Term cupicef.set comparison => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Eq))) => + H + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Eq) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Eq) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Eq)) + cupicef.I + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Eq) => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H2 c c_27_ H H0))). + +def let__H3 : + c : cupicef.Term cupicef.set comparison -> + c_27_ : cupicef.Term cupicef.set comparison -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Gt))) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Gt) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= c : cupicef.Term cupicef.set comparison => + c_27_ : cupicef.Term cupicef.set comparison => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Gt))) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Gt) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt + (e : cupicef.Term cupicef.set comparison => + match____comparison (cupicef.type (cupicef.s cupicef.z)) + (c0 : cupicef.Term cupicef.set comparison => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False Coq__Init__Logic.True Coq__Init__Logic.False e) + Coq__Init__Logic.I Gt H0. + +def let__n2 : + c : cupicef.Term cupicef.set comparison -> + c_27_ : cupicef.Term cupicef.set comparison -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Gt))) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= c : cupicef.Term cupicef.set comparison => + c_27_ : cupicef.Term cupicef.set comparison => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Gt))) => + H + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Gt) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Gt) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Gt)) + cupicef.I + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Gt) => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H3 c c_27_ H H0))). + +def let__H4 : + c : cupicef.Term cupicef.set comparison -> + c_27_ : cupicef.Term cupicef.set comparison -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Eq))) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Eq) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= c : cupicef.Term cupicef.set comparison => + c_27_ : cupicef.Term cupicef.set comparison => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Eq))) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Eq) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt + (e : cupicef.Term cupicef.set comparison => + match____comparison (cupicef.type (cupicef.s cupicef.z)) + (c0 : cupicef.Term cupicef.set comparison => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False Coq__Init__Logic.False Coq__Init__Logic.True e) + Coq__Init__Logic.I Eq H0. + +def let__n3 : + c : cupicef.Term cupicef.set comparison -> + c_27_ : cupicef.Term cupicef.set comparison -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Eq))) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= c : cupicef.Term cupicef.set comparison => + c_27_ : cupicef.Term cupicef.set comparison => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Eq))) => + H + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Eq) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Eq) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Eq)) + cupicef.I + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Eq) => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H4 c c_27_ H H0))). + +def let__H5 : + c : cupicef.Term cupicef.set comparison -> + c_27_ : cupicef.Term cupicef.set comparison -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Lt))) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Lt) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= c : cupicef.Term cupicef.set comparison => + c_27_ : cupicef.Term cupicef.set comparison => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Lt))) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Lt) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt + (e : cupicef.Term cupicef.set comparison => + match____comparison (cupicef.type (cupicef.s cupicef.z)) + (c0 : cupicef.Term cupicef.set comparison => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False Coq__Init__Logic.False Coq__Init__Logic.True e) + Coq__Init__Logic.I Lt H0. + +def let__n4 : + c : cupicef.Term cupicef.set comparison -> + c_27_ : cupicef.Term cupicef.set comparison -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Lt))) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= c : cupicef.Term cupicef.set comparison => + c_27_ : cupicef.Term cupicef.set comparison => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Lt))) => + H + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Lt) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Lt) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Lt)) + cupicef.I + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Lt) => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H5 c c_27_ H H0))). + +thm comparison__eq__stable : + c : cupicef.Term cupicef.set comparison -> + c_27_ : cupicef.Term cupicef.set comparison -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + c c_27_))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + c c_27_) + +:= c : cupicef.Term cupicef.set comparison => + match____comparison cupicef.prop + (c0 : cupicef.Term cupicef.set comparison => + cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I comparison + (c_27_ : cupicef.Term cupicef.set comparison => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + c0 c_27_))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + c0 c_27_))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + c0 c_27_))) + (c_27_ : cupicef.Term cupicef.set comparison => + match____comparison cupicef.prop + (c0 : cupicef.Term cupicef.set comparison => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq c0))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq c0))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq c0)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Eq))) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Lt))) => + Coq__Init__Logic.match____False cupicef.prop + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Lt) + (let__n c c_27_ H)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Gt))) => + Coq__Init__Logic.match____False cupicef.prop + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Gt) + (let__n0 c c_27_ H)) + c_27_) + (c_27_ : cupicef.Term cupicef.set comparison => + match____comparison cupicef.prop + (c0 : cupicef.Term cupicef.set comparison => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt c0))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt c0))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt c0)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Eq))) => + Coq__Init__Logic.match____False cupicef.prop + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Eq) + (let__n1 c c_27_ H)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Lt))) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Gt))) => + Coq__Init__Logic.match____False cupicef.prop + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Gt) + (let__n2 c c_27_ H)) + c_27_) + (c_27_ : cupicef.Term cupicef.set comparison => + match____comparison cupicef.prop + (c0 : cupicef.Term cupicef.set comparison => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt c0))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt c0))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt c0)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Eq))) => + Coq__Init__Logic.match____False cupicef.prop + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Eq) + (let__n3 c c_27_ H)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Lt))) => + Coq__Init__Logic.match____False cupicef.prop + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Lt) + (let__n4 c c_27_ H)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Gt))) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt) + c_27_) + c. + +def CompOpp : + r : cupicef.Term cupicef.set comparison -> + cupicef.Term cupicef.set comparison + +:= r : cupicef.Term cupicef.set comparison => + match____comparison cupicef.set + (r0 : cupicef.Term cupicef.set comparison => comparison) Eq Gt Lt r. + +thm CompOpp__involutive : + c : cupicef.Term cupicef.set comparison -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp (CompOpp c)) c) + +:= c : cupicef.Term cupicef.set comparison => + match____comparison cupicef.prop + (c0 : cupicef.Term cupicef.set comparison => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp (CompOpp c0)) c0) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt) + c. + +def let__H6 : + c : cupicef.Term cupicef.set comparison -> + c_27_ : cupicef.Term cupicef.set comparison -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Eq) (CompOpp Lt)) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= c : cupicef.Term cupicef.set comparison => + c_27_ : cupicef.Term cupicef.set comparison => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Eq) (CompOpp Lt)) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Eq) + (e : cupicef.Term cupicef.set comparison => + match____comparison (cupicef.type (cupicef.s cupicef.z)) + (c0 : cupicef.Term cupicef.set comparison => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True Coq__Init__Logic.False Coq__Init__Logic.False e) + Coq__Init__Logic.I (CompOpp Lt) H. + +def let__H7 : + c : cupicef.Term cupicef.set comparison -> + c_27_ : cupicef.Term cupicef.set comparison -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Eq) (CompOpp Gt)) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= c : cupicef.Term cupicef.set comparison => + c_27_ : cupicef.Term cupicef.set comparison => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Eq) (CompOpp Gt)) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Eq) + (e : cupicef.Term cupicef.set comparison => + match____comparison (cupicef.type (cupicef.s cupicef.z)) + (c0 : cupicef.Term cupicef.set comparison => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True Coq__Init__Logic.False Coq__Init__Logic.False e) + Coq__Init__Logic.I (CompOpp Gt) H. + +def let__H8 : + c : cupicef.Term cupicef.set comparison -> + c_27_ : cupicef.Term cupicef.set comparison -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Lt) (CompOpp Eq)) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= c : cupicef.Term cupicef.set comparison => + c_27_ : cupicef.Term cupicef.set comparison => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Lt) (CompOpp Eq)) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Lt) + (e : cupicef.Term cupicef.set comparison => + match____comparison (cupicef.type (cupicef.s cupicef.z)) + (c0 : cupicef.Term cupicef.set comparison => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False Coq__Init__Logic.False Coq__Init__Logic.True e) + Coq__Init__Logic.I (CompOpp Eq) H. + +def let__H9 : + c : cupicef.Term cupicef.set comparison -> + c_27_ : cupicef.Term cupicef.set comparison -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Gt) (CompOpp Eq)) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= c : cupicef.Term cupicef.set comparison => + c_27_ : cupicef.Term cupicef.set comparison => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Gt) (CompOpp Eq)) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Gt) + (e : cupicef.Term cupicef.set comparison => + match____comparison (cupicef.type (cupicef.s cupicef.z)) + (c0 : cupicef.Term cupicef.set comparison => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False Coq__Init__Logic.True Coq__Init__Logic.False e) + Coq__Init__Logic.I (CompOpp Eq) H. + +thm CompOpp__inj : + c : cupicef.Term cupicef.set comparison -> + c_27_ : cupicef.Term cupicef.set comparison -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp c) (CompOpp c_27_)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + c c_27_) + +:= c : cupicef.Term cupicef.set comparison => + match____comparison cupicef.prop + (c0 : cupicef.Term cupicef.set comparison => + cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I comparison + (c_27_ : cupicef.Term cupicef.set comparison => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp c0) (CompOpp c_27_)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp c0) (CompOpp c_27_)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + c0 c_27_))) + (c_27_ : cupicef.Term cupicef.set comparison => + match____comparison cupicef.prop + (c0 : cupicef.Term cupicef.set comparison => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Eq) (CompOpp c0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Eq) (CompOpp c0)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq c0)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Eq) (CompOpp Eq)) => + H) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Eq) (CompOpp Lt)) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Lt) + (let__H6 c c_27_ H)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Eq) (CompOpp Gt)) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Gt) + (let__H7 c c_27_ H)) + c_27_) + (c_27_ : cupicef.Term cupicef.set comparison => + match____comparison cupicef.prop + (c0 : cupicef.Term cupicef.set comparison => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Lt) (CompOpp c0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Lt) (CompOpp c0)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt c0)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Lt) (CompOpp Eq)) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Eq) + (let__H8 c c_27_ H)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Lt) (CompOpp Lt)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Lt) (CompOpp Gt)) => + Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Lt H) + c_27_) + (c_27_ : cupicef.Term cupicef.set comparison => + match____comparison cupicef.prop + (c0 : cupicef.Term cupicef.set comparison => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Gt) (CompOpp c0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Gt) (CompOpp c0)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt c0)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Gt) (CompOpp Eq)) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Eq) + (let__H9 c c_27_ H)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Gt) (CompOpp Lt)) => + Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Gt H) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp Gt) (CompOpp Gt)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt) + c_27_) + c. + +thm CompOpp__iff : + c : cupicef.Term cupicef.set comparison -> + c_27_ : cupicef.Term cupicef.set comparison -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp c) c_27_) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + c (CompOpp c_27_))) + +:= c : cupicef.Term cupicef.set comparison => + c_27_ : cupicef.Term cupicef.set comparison => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp c) c_27_) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp c) c_27_) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + c (CompOpp c_27_))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + c (CompOpp c_27_)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + c (CompOpp c_27_)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp c) c_27_)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp c) c_27_) => + CompOpp__inj c (CompOpp c_27_) + (Coq__Init__Logic.eq__ind__r + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I comparison) + c_27_ + (c0 : cupicef.Term cupicef.set comparison => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + (CompOpp c) c0) + H (CompOpp (CompOpp c_27_)) (CompOpp__involutive c_27_))) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + c (CompOpp c_27_)) => + CompOpp__inj (CompOpp c) c_27_ + (Coq__Init__Logic.eq__ind__r + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I comparison) + c + (c0 : cupicef.Term cupicef.set comparison => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + c0 (CompOpp c_27_)) + H (CompOpp (CompOpp c)) (CompOpp__involutive c))). + +def CompareSpec : + Peq : cupicef.Univ cupicef.prop -> + Plt : cupicef.Univ cupicef.prop -> + Pgt : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.set comparison -> + cupicef.Univ cupicef.prop. + +def CompEq : + Peq : cupicef.Univ cupicef.prop -> + Plt : cupicef.Univ cupicef.prop -> + Pgt : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop Peq -> + cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt Eq). + +def CompLt : + Peq : cupicef.Univ cupicef.prop -> + Plt : cupicef.Univ cupicef.prop -> + Pgt : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop Plt -> + cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt Lt). + +def CompGt : + Peq : cupicef.Univ cupicef.prop -> + Plt : cupicef.Univ cupicef.prop -> + Pgt : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop Pgt -> + cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt Gt). + +[ ] cupicef.code_guarded? _ (CompEq _ _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (CompLt _ _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (CompGt _ _ _ _) --> cupicef.guarded. + +def match____CompareSpec : + s : cupicef.Sort -> + Peq : cupicef.Univ cupicef.prop -> + Plt : cupicef.Univ cupicef.prop -> + Pgt : cupicef.Univ cupicef.prop -> + P : + (__ : cupicef.Term cupicef.set comparison -> + cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt __) -> + cupicef.Univ s) -> + case__CompEq : + (__ : cupicef.Term cupicef.prop Peq -> + cupicef.Term s (P Eq (CompEq Peq Plt Pgt __))) -> + case__CompLt : + (__ : cupicef.Term cupicef.prop Plt -> + cupicef.Term s (P Lt (CompLt Peq Plt Pgt __))) -> + case__CompGt : + (__ : cupicef.Term cupicef.prop Pgt -> + cupicef.Term s (P Gt (CompGt Peq Plt Pgt __))) -> + __ : cupicef.Term cupicef.set comparison -> + x : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt __) -> + cupicef.Term s (P __ x). + +[ s, Peq, Plt, Pgt, P, case__CompEq, case__CompLt, case__CompGt, __] + match____CompareSpec s Peq Plt Pgt P case__CompEq case__CompLt case__CompGt + {Eq} (CompEq Peq Plt Pgt __) + --> + case__CompEq __. + +[ s, Peq, Plt, Pgt, P, case__CompEq, case__CompLt, case__CompGt, __] + match____CompareSpec s Peq Plt Pgt P case__CompEq case__CompLt case__CompGt + {Lt} (CompLt Peq Plt Pgt __) + --> + case__CompLt __. + +[ s, Peq, Plt, Pgt, P, case__CompEq, case__CompLt, case__CompGt, __] + match____CompareSpec s Peq Plt Pgt P case__CompEq case__CompLt case__CompGt + {Gt} (CompGt Peq Plt Pgt __) + --> + case__CompGt __. + +[ s, Peq, Plt, Pgt, P, s'] + match____CompareSpec s Peq Plt Pgt + (__ => x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P __ x))) + --> + match____CompareSpec s' Peq Plt Pgt (__ => x => P __ x). + +def CompareSpec__ind : + Peq : cupicef.Univ cupicef.prop -> + Plt : cupicef.Univ cupicef.prop -> + Pgt : cupicef.Univ cupicef.prop -> + P : (__ : cupicef.Term cupicef.set comparison -> cupicef.Univ cupicef.prop) -> + f : + (__ : cupicef.Term cupicef.prop Peq -> cupicef.Term cupicef.prop (P Eq)) -> + f0 : + (__ : cupicef.Term cupicef.prop Plt -> cupicef.Term cupicef.prop (P Lt)) -> + f1 : + (__ : cupicef.Term cupicef.prop Pgt -> cupicef.Term cupicef.prop (P Gt)) -> + c : cupicef.Term cupicef.set comparison -> + c0 : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt c) -> + cupicef.Term cupicef.prop (P c) + +:= Peq : cupicef.Univ cupicef.prop => + Plt : cupicef.Univ cupicef.prop => + Pgt : cupicef.Univ cupicef.prop => + P : + (__ : cupicef.Term cupicef.set comparison -> cupicef.Univ cupicef.prop) => + f : + (__ : cupicef.Term cupicef.prop Peq -> cupicef.Term cupicef.prop (P Eq)) => + f0 : + (__ : cupicef.Term cupicef.prop Plt -> cupicef.Term cupicef.prop (P Lt)) => + f1 : + (__ : cupicef.Term cupicef.prop Pgt -> cupicef.Term cupicef.prop (P Gt)) => + c : cupicef.Term cupicef.set comparison => + c0 : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt c) => + match____CompareSpec cupicef.prop Peq Plt Pgt + (c1 : cupicef.Term cupicef.set comparison => + __ : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt c1) => + P c1) + f f0 f1 c c0. + +def CompareSpecT : + Peq : cupicef.Univ cupicef.prop -> + Plt : cupicef.Univ cupicef.prop -> + Pgt : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.set comparison -> + cupicef.Univ cupicef.set. + +def CompEqT : + Peq : cupicef.Univ cupicef.prop -> + Plt : cupicef.Univ cupicef.prop -> + Pgt : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop Peq -> + cupicef.Term cupicef.set (CompareSpecT Peq Plt Pgt Eq). + +def CompLtT : + Peq : cupicef.Univ cupicef.prop -> + Plt : cupicef.Univ cupicef.prop -> + Pgt : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop Plt -> + cupicef.Term cupicef.set (CompareSpecT Peq Plt Pgt Lt). + +def CompGtT : + Peq : cupicef.Univ cupicef.prop -> + Plt : cupicef.Univ cupicef.prop -> + Pgt : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop Pgt -> + cupicef.Term cupicef.set (CompareSpecT Peq Plt Pgt Gt). + +[ ] cupicef.code_guarded? _ (CompEqT _ _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (CompLtT _ _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (CompGtT _ _ _ _) --> cupicef.guarded. + +def match____CompareSpecT : + s : cupicef.Sort -> + Peq : cupicef.Univ cupicef.prop -> + Plt : cupicef.Univ cupicef.prop -> + Pgt : cupicef.Univ cupicef.prop -> + P : + (__ : cupicef.Term cupicef.set comparison -> + cupicef.Term cupicef.set (CompareSpecT Peq Plt Pgt __) -> + cupicef.Univ s) -> + case__CompEqT : + (__ : cupicef.Term cupicef.prop Peq -> + cupicef.Term s (P Eq (CompEqT Peq Plt Pgt __))) -> + case__CompLtT : + (__ : cupicef.Term cupicef.prop Plt -> + cupicef.Term s (P Lt (CompLtT Peq Plt Pgt __))) -> + case__CompGtT : + (__ : cupicef.Term cupicef.prop Pgt -> + cupicef.Term s (P Gt (CompGtT Peq Plt Pgt __))) -> + __ : cupicef.Term cupicef.set comparison -> + x : cupicef.Term cupicef.set (CompareSpecT Peq Plt Pgt __) -> + cupicef.Term s (P __ x). + +[ s, Peq, Plt, Pgt, P, case__CompEqT, case__CompLtT, case__CompGtT, __] + match____CompareSpecT s Peq Plt Pgt P case__CompEqT case__CompLtT + case__CompGtT {Eq} (CompEqT Peq Plt Pgt __) + --> + case__CompEqT __. + +[ s, Peq, Plt, Pgt, P, case__CompEqT, case__CompLtT, case__CompGtT, __] + match____CompareSpecT s Peq Plt Pgt P case__CompEqT case__CompLtT + case__CompGtT {Lt} (CompLtT Peq Plt Pgt __) + --> + case__CompLtT __. + +[ s, Peq, Plt, Pgt, P, case__CompEqT, case__CompLtT, case__CompGtT, __] + match____CompareSpecT s Peq Plt Pgt P case__CompEqT case__CompLtT + case__CompGtT {Gt} (CompGtT Peq Plt Pgt __) + --> + case__CompGtT __. + +[ s, Peq, Plt, Pgt, P, s'] + match____CompareSpecT s Peq Plt Pgt + (__ => x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P __ x))) + --> + match____CompareSpecT s' Peq Plt Pgt (__ => x => P __ x). + +def CompareSpecT__rect : + Peq : cupicef.Univ cupicef.prop -> + Plt : cupicef.Univ cupicef.prop -> + Pgt : cupicef.Univ cupicef.prop -> + P : + (c : cupicef.Term cupicef.set comparison -> + c0 : cupicef.Term cupicef.set (CompareSpecT Peq Plt Pgt c) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : + (p : cupicef.Term cupicef.prop Peq -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P Eq (CompEqT Peq Plt Pgt p))) -> + f0 : + (p : cupicef.Term cupicef.prop Plt -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P Lt (CompLtT Peq Plt Pgt p))) -> + f1 : + (p : cupicef.Term cupicef.prop Pgt -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P Gt (CompGtT Peq Plt Pgt p))) -> + c : cupicef.Term cupicef.set comparison -> + c0 : cupicef.Term cupicef.set (CompareSpecT Peq Plt Pgt c) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P c c0) + +:= Peq : cupicef.Univ cupicef.prop => + Plt : cupicef.Univ cupicef.prop => + Pgt : cupicef.Univ cupicef.prop => + P : + (c : cupicef.Term cupicef.set comparison -> + c0 : cupicef.Term cupicef.set (CompareSpecT Peq Plt Pgt c) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : + (p : cupicef.Term cupicef.prop Peq -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P Eq (CompEqT Peq Plt Pgt p))) => + f0 : + (p : cupicef.Term cupicef.prop Plt -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P Lt (CompLtT Peq Plt Pgt p))) => + f1 : + (p : cupicef.Term cupicef.prop Pgt -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P Gt (CompGtT Peq Plt Pgt p))) => + c : cupicef.Term cupicef.set comparison => + c0 : cupicef.Term cupicef.set (CompareSpecT Peq Plt Pgt c) => + match____CompareSpecT (cupicef.type (cupicef.s cupicef.z)) Peq Plt Pgt + (c1 : cupicef.Term cupicef.set comparison => + c2 : cupicef.Term cupicef.set (CompareSpecT Peq Plt Pgt c1) => + P c1 c2) + f f0 f1 c c0. + +def CompareSpecT__ind : + Peq : cupicef.Univ cupicef.prop -> + Plt : cupicef.Univ cupicef.prop -> + Pgt : cupicef.Univ cupicef.prop -> + P : + (c : cupicef.Term cupicef.set comparison -> + c0 : cupicef.Term cupicef.set (CompareSpecT Peq Plt Pgt c) -> + cupicef.Univ cupicef.prop) -> + f : + (p : cupicef.Term cupicef.prop Peq -> + cupicef.Term cupicef.prop (P Eq (CompEqT Peq Plt Pgt p))) -> + f0 : + (p : cupicef.Term cupicef.prop Plt -> + cupicef.Term cupicef.prop (P Lt (CompLtT Peq Plt Pgt p))) -> + f1 : + (p : cupicef.Term cupicef.prop Pgt -> + cupicef.Term cupicef.prop (P Gt (CompGtT Peq Plt Pgt p))) -> + c : cupicef.Term cupicef.set comparison -> + c0 : cupicef.Term cupicef.set (CompareSpecT Peq Plt Pgt c) -> + cupicef.Term cupicef.prop (P c c0) + +:= Peq : cupicef.Univ cupicef.prop => + Plt : cupicef.Univ cupicef.prop => + Pgt : cupicef.Univ cupicef.prop => + P : + (c : cupicef.Term cupicef.set comparison -> + c0 : cupicef.Term cupicef.set (CompareSpecT Peq Plt Pgt c) -> + cupicef.Univ cupicef.prop) => + f : + (p : cupicef.Term cupicef.prop Peq -> + cupicef.Term cupicef.prop (P Eq (CompEqT Peq Plt Pgt p))) => + f0 : + (p : cupicef.Term cupicef.prop Plt -> + cupicef.Term cupicef.prop (P Lt (CompLtT Peq Plt Pgt p))) => + f1 : + (p : cupicef.Term cupicef.prop Pgt -> + cupicef.Term cupicef.prop (P Gt (CompGtT Peq Plt Pgt p))) => + c : cupicef.Term cupicef.set comparison => + c0 : cupicef.Term cupicef.set (CompareSpecT Peq Plt Pgt c) => + match____CompareSpecT cupicef.prop Peq Plt Pgt + (c1 : cupicef.Term cupicef.set comparison => + c2 : cupicef.Term cupicef.set (CompareSpecT Peq Plt Pgt c1) => + P c1 c2) + f f0 f1 c c0. + +def CompareSpecT__rec : + Peq : cupicef.Univ cupicef.prop -> + Plt : cupicef.Univ cupicef.prop -> + Pgt : cupicef.Univ cupicef.prop -> + P : + (c : cupicef.Term cupicef.set comparison -> + c0 : cupicef.Term cupicef.set (CompareSpecT Peq Plt Pgt c) -> + cupicef.Univ cupicef.set) -> + f : + (p : cupicef.Term cupicef.prop Peq -> + cupicef.Term cupicef.set (P Eq (CompEqT Peq Plt Pgt p))) -> + f0 : + (p : cupicef.Term cupicef.prop Plt -> + cupicef.Term cupicef.set (P Lt (CompLtT Peq Plt Pgt p))) -> + f1 : + (p : cupicef.Term cupicef.prop Pgt -> + cupicef.Term cupicef.set (P Gt (CompGtT Peq Plt Pgt p))) -> + c : cupicef.Term cupicef.set comparison -> + c0 : cupicef.Term cupicef.set (CompareSpecT Peq Plt Pgt c) -> + cupicef.Term cupicef.set (P c c0) + +:= Peq : cupicef.Univ cupicef.prop => + Plt : cupicef.Univ cupicef.prop => + Pgt : cupicef.Univ cupicef.prop => + P : + (c : cupicef.Term cupicef.set comparison -> + c0 : cupicef.Term cupicef.set (CompareSpecT Peq Plt Pgt c) -> + cupicef.Univ cupicef.set) => + CompareSpecT__rect Peq Plt Pgt + (cupicef.cast + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I comparison + (c : cupicef.Term cupicef.set comparison => + cupicef.prod cupicef.set (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (CompareSpecT Peq Plt Pgt c) + (c0 : cupicef.Term cupicef.set (CompareSpecT Peq Plt Pgt c) => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I comparison + (c : cupicef.Term cupicef.set comparison => + cupicef.prod cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (CompareSpecT Peq Plt Pgt c) + (c0 : cupicef.Term cupicef.set (CompareSpecT Peq Plt Pgt c) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I))) + cupicef.I P). + +def let__H11 : + Peq : cupicef.Univ cupicef.prop -> + Plt : cupicef.Univ cupicef.prop -> + Pgt : cupicef.Univ cupicef.prop -> + c : cupicef.Term cupicef.set comparison -> + H : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt Eq) -> + H0 : cupicef.Term cupicef.prop Plt -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Eq) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Eq) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= Peq : cupicef.Univ cupicef.prop => + Plt : cupicef.Univ cupicef.prop => + Pgt : cupicef.Univ cupicef.prop => + c : cupicef.Term cupicef.set comparison => + H : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt Eq) => + H0 : cupicef.Term cupicef.prop Plt => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Eq) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Eq) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt + (e : cupicef.Term cupicef.set comparison => + match____comparison (cupicef.type (cupicef.s cupicef.z)) + (c0 : cupicef.Term cupicef.set comparison => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False Coq__Init__Logic.True Coq__Init__Logic.False e) + Coq__Init__Logic.I Eq H2. + +def let__H12 : + Peq : cupicef.Univ cupicef.prop -> + Plt : cupicef.Univ cupicef.prop -> + Pgt : cupicef.Univ cupicef.prop -> + c : cupicef.Term cupicef.set comparison -> + H : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt Eq) -> + H0 : cupicef.Term cupicef.prop Pgt -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Eq) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Eq) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= Peq : cupicef.Univ cupicef.prop => + Plt : cupicef.Univ cupicef.prop => + Pgt : cupicef.Univ cupicef.prop => + c : cupicef.Term cupicef.set comparison => + H : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt Eq) => + H0 : cupicef.Term cupicef.prop Pgt => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Eq) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Eq) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt + (e : cupicef.Term cupicef.set comparison => + match____comparison (cupicef.type (cupicef.s cupicef.z)) + (c0 : cupicef.Term cupicef.set comparison => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False Coq__Init__Logic.False Coq__Init__Logic.True e) + Coq__Init__Logic.I Eq H2. + +def let__H10 : + Peq : cupicef.Univ cupicef.prop -> + Plt : cupicef.Univ cupicef.prop -> + Pgt : cupicef.Univ cupicef.prop -> + c : cupicef.Term cupicef.set comparison -> + H : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt Eq) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Eq) -> + cupicef.Term cupicef.prop Peq + +:= Peq : cupicef.Univ cupicef.prop => + Plt : cupicef.Univ cupicef.prop => + Pgt : cupicef.Univ cupicef.prop => + c : cupicef.Term cupicef.set comparison => + H : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt Eq) => + match____CompareSpec cupicef.prop Peq Plt Pgt + (c0 : cupicef.Term cupicef.set comparison => + __ : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt c0) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + c0 Eq) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + c0 Eq) => + Peq)) + (H0 : cupicef.Term cupicef.prop Peq => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Eq) => + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Eq) => + H3 : cupicef.Term cupicef.prop Peq => + H3) H1 H0) + (H0 : cupicef.Term cupicef.prop Plt => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Eq) => + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Eq) => + Coq__Init__Logic.False__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I Plt + (H3 : cupicef.Term cupicef.prop Plt => Peq)) + (let__H11 Peq Plt Pgt c H H0 H1 H2)) + H1 H0) + (H0 : cupicef.Term cupicef.prop Pgt => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Eq) => + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Eq) => + Coq__Init__Logic.False__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I Pgt + (H3 : cupicef.Term cupicef.prop Pgt => Peq)) + (let__H12 Peq Plt Pgt c H H0 H1 H2)) + H1 H0) + Eq H. + +def let__H14 : + Peq : cupicef.Univ cupicef.prop -> + Plt : cupicef.Univ cupicef.prop -> + Pgt : cupicef.Univ cupicef.prop -> + c : cupicef.Term cupicef.set comparison -> + H : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt Lt) -> + H0 : cupicef.Term cupicef.prop Peq -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Lt) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Lt) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= Peq : cupicef.Univ cupicef.prop => + Plt : cupicef.Univ cupicef.prop => + Pgt : cupicef.Univ cupicef.prop => + c : cupicef.Term cupicef.set comparison => + H : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt Lt) => + H0 : cupicef.Term cupicef.prop Peq => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Lt) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Lt) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq + (e : cupicef.Term cupicef.set comparison => + match____comparison (cupicef.type (cupicef.s cupicef.z)) + (c0 : cupicef.Term cupicef.set comparison => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True Coq__Init__Logic.False Coq__Init__Logic.False e) + Coq__Init__Logic.I Lt H2. + +def let__H15 : + Peq : cupicef.Univ cupicef.prop -> + Plt : cupicef.Univ cupicef.prop -> + Pgt : cupicef.Univ cupicef.prop -> + c : cupicef.Term cupicef.set comparison -> + H : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt Lt) -> + H0 : cupicef.Term cupicef.prop Pgt -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Lt) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Lt) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= Peq : cupicef.Univ cupicef.prop => + Plt : cupicef.Univ cupicef.prop => + Pgt : cupicef.Univ cupicef.prop => + c : cupicef.Term cupicef.set comparison => + H : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt Lt) => + H0 : cupicef.Term cupicef.prop Pgt => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Lt) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Lt) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt + (e : cupicef.Term cupicef.set comparison => + match____comparison (cupicef.type (cupicef.s cupicef.z)) + (c0 : cupicef.Term cupicef.set comparison => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False Coq__Init__Logic.False Coq__Init__Logic.True e) + Coq__Init__Logic.I Lt H2. + +def let__H13 : + Peq : cupicef.Univ cupicef.prop -> + Plt : cupicef.Univ cupicef.prop -> + Pgt : cupicef.Univ cupicef.prop -> + c : cupicef.Term cupicef.set comparison -> + H : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt Lt) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Lt) -> + cupicef.Term cupicef.prop Plt + +:= Peq : cupicef.Univ cupicef.prop => + Plt : cupicef.Univ cupicef.prop => + Pgt : cupicef.Univ cupicef.prop => + c : cupicef.Term cupicef.set comparison => + H : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt Lt) => + match____CompareSpec cupicef.prop Peq Plt Pgt + (c0 : cupicef.Term cupicef.set comparison => + __ : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt c0) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + c0 Lt) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + c0 Lt) => + Plt)) + (H0 : cupicef.Term cupicef.prop Peq => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Lt) => + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Lt) => + Coq__Init__Logic.False__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I Peq + (H3 : cupicef.Term cupicef.prop Peq => Plt)) + (let__H14 Peq Plt Pgt c H H0 H1 H2)) + H1 H0) + (H0 : cupicef.Term cupicef.prop Plt => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Lt) => + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Lt) => + H3 : cupicef.Term cupicef.prop Plt => + H3) H1 H0) + (H0 : cupicef.Term cupicef.prop Pgt => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Lt) => + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Lt) => + Coq__Init__Logic.False__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I Pgt + (H3 : cupicef.Term cupicef.prop Pgt => Plt)) + (let__H15 Peq Plt Pgt c H H0 H1 H2)) + H1 H0) + Lt H. + +def let__H17 : + Peq : cupicef.Univ cupicef.prop -> + Plt : cupicef.Univ cupicef.prop -> + Pgt : cupicef.Univ cupicef.prop -> + c : cupicef.Term cupicef.set comparison -> + H : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt Gt) -> + H0 : cupicef.Term cupicef.prop Peq -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Gt) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Gt) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= Peq : cupicef.Univ cupicef.prop => + Plt : cupicef.Univ cupicef.prop => + Pgt : cupicef.Univ cupicef.prop => + c : cupicef.Term cupicef.set comparison => + H : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt Gt) => + H0 : cupicef.Term cupicef.prop Peq => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Gt) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Gt) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq + (e : cupicef.Term cupicef.set comparison => + match____comparison (cupicef.type (cupicef.s cupicef.z)) + (c0 : cupicef.Term cupicef.set comparison => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True Coq__Init__Logic.False Coq__Init__Logic.False e) + Coq__Init__Logic.I Gt H2. + +def let__H18 : + Peq : cupicef.Univ cupicef.prop -> + Plt : cupicef.Univ cupicef.prop -> + Pgt : cupicef.Univ cupicef.prop -> + c : cupicef.Term cupicef.set comparison -> + H : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt Gt) -> + H0 : cupicef.Term cupicef.prop Plt -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Gt) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Gt) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= Peq : cupicef.Univ cupicef.prop => + Plt : cupicef.Univ cupicef.prop => + Pgt : cupicef.Univ cupicef.prop => + c : cupicef.Term cupicef.set comparison => + H : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt Gt) => + H0 : cupicef.Term cupicef.prop Plt => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Gt) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Gt) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt + (e : cupicef.Term cupicef.set comparison => + match____comparison (cupicef.type (cupicef.s cupicef.z)) + (c0 : cupicef.Term cupicef.set comparison => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False Coq__Init__Logic.True Coq__Init__Logic.False e) + Coq__Init__Logic.I Gt H2. + +def let__H16 : + Peq : cupicef.Univ cupicef.prop -> + Plt : cupicef.Univ cupicef.prop -> + Pgt : cupicef.Univ cupicef.prop -> + c : cupicef.Term cupicef.set comparison -> + H : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt Gt) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Gt) -> + cupicef.Term cupicef.prop Pgt + +:= Peq : cupicef.Univ cupicef.prop => + Plt : cupicef.Univ cupicef.prop => + Pgt : cupicef.Univ cupicef.prop => + c : cupicef.Term cupicef.set comparison => + H : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt Gt) => + match____CompareSpec cupicef.prop Peq Plt Pgt + (c0 : cupicef.Term cupicef.set comparison => + __ : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt c0) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + c0 Gt) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + c0 Gt) => + Pgt)) + (H0 : cupicef.Term cupicef.prop Peq => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Gt) => + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq Gt) => + Coq__Init__Logic.False__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I Peq + (H3 : cupicef.Term cupicef.prop Peq => Pgt)) + (let__H17 Peq Plt Pgt c H H0 H1 H2)) + H1 H0) + (H0 : cupicef.Term cupicef.prop Plt => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Gt) => + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt Gt) => + Coq__Init__Logic.False__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I Plt + (H3 : cupicef.Term cupicef.prop Plt => Pgt)) + (let__H18 Peq Plt Pgt c H H0 H1 H2)) + H1 H0) + (H0 : cupicef.Term cupicef.prop Pgt => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Gt) => + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt Gt) => + H3 : cupicef.Term cupicef.prop Pgt => + H3) H1 H0) + Gt H. + +def CompareSpec2Type : + Peq : cupicef.Univ cupicef.prop -> + Plt : cupicef.Univ cupicef.prop -> + Pgt : cupicef.Univ cupicef.prop -> + c : cupicef.Term cupicef.set comparison -> + __ : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt c) -> + cupicef.Term cupicef.set (CompareSpecT Peq Plt Pgt c) + +:= Peq : cupicef.Univ cupicef.prop => + Plt : cupicef.Univ cupicef.prop => + Pgt : cupicef.Univ cupicef.prop => + c : cupicef.Term cupicef.set comparison => + match____comparison cupicef.set + (c0 : cupicef.Term cupicef.set comparison => + cupicef.prod cupicef.prop cupicef.set + (cupicef.rule cupicef.prop cupicef.set) cupicef.I + (CompareSpec Peq Plt Pgt c0) + (__ : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt c0) => + CompareSpecT Peq Plt Pgt c0)) + (H : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt Eq) => + CompEqT Peq Plt Pgt + (let__H10 Peq Plt Pgt c H + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Eq))) + (H : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt Lt) => + CompLtT Peq Plt Pgt + (let__H13 Peq Plt Pgt c H + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Lt))) + (H : cupicef.Term cupicef.prop (CompareSpec Peq Plt Pgt Gt) => + CompGtT Peq Plt Pgt + (let__H16 Peq Plt Pgt c H + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I comparison) + Gt))) + c. + +def CompSpec : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eq : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + lt : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.set comparison -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eq : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + lt : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + CompareSpec (eq x y) (lt x y) (lt y x). + +def CompSpecT : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eq : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + lt : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.set comparison -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) + +:= cupicef.cast + (cupicef.rule (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set + (cupicef.type (cupicef.s cupicef.z)))))))) + (cupicef.rule (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))))))))) + (cupicef.prod (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set + (cupicef.type (cupicef.s cupicef.z))))))) + (cupicef.rule (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set + (cupicef.type (cupicef.s cupicef.z)))))))) + cupicef.I + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set + (cupicef.type (cupicef.s cupicef.z)))))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set + (cupicef.type (cupicef.s cupicef.z))))))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (eq : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set + (cupicef.type (cupicef.s cupicef.z)))))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (lt : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I comparison + (__ : cupicef.Term cupicef.set comparison => + cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))))))) + (cupicef.prod (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))))))) + (cupicef.rule (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))))))))) + cupicef.I + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))))))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))))))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (eq : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))))))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (lt : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I comparison + (__ : cupicef.Term cupicef.set comparison => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I))))))) + cupicef.I + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eq : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + lt : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + CompareSpecT (eq x y) (lt x y) (lt y x)). + +def CompSpec2Type : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eq : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + lt : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + c : cupicef.Term cupicef.set comparison -> + __ : cupicef.Term cupicef.prop (CompSpec A eq lt x y c) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (CompSpecT A eq lt x y c) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eq : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + lt : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + c : cupicef.Term cupicef.set comparison => + H : cupicef.Term cupicef.prop (CompSpec A eq lt x y c) => + CompareSpec2Type (eq x y) (lt x y) (lt y x) c H. + +def identity : + Coq__Init__Datatypes__80 : cupicef.Sort -> + A : cupicef.Univ Coq__Init__Datatypes__80 -> + a : cupicef.Term Coq__Init__Datatypes__80 A -> + __ : cupicef.Term Coq__Init__Datatypes__80 A -> + cupicef.Univ cupicef.prop. + +identity' : + A : cupicef.Univ cupicef.sinf -> + a : cupicef.Term cupicef.sinf A -> + __ : cupicef.Term cupicef.sinf A -> + cupicef.Univ cupicef.sinf. + +[ Coq__Init__Datatypes__80, A, a, __] + cupicef.code (cupicef.cU _) (identity Coq__Init__Datatypes__80 A a __) + --> + identity' (cupicef.code (cupicef.cU Coq__Init__Datatypes__80) A) a __. + +def Coq__Init__Datatypes__80 : + cupicef.Nat + +:= cupicef.s cupicef.z. + +def identity__refl : + A : cupicef.Univ (cupicef.type Coq__Init__Datatypes__80) -> + a : cupicef.Term (cupicef.type Coq__Init__Datatypes__80) A -> + cupicef.Term cupicef.prop + (identity (cupicef.type Coq__Init__Datatypes__80) A a a). + +[ ] cupicef.code_guarded? _ (identity__refl _ _) --> cupicef.guarded. + +def match____identity : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type Coq__Init__Datatypes__80) -> + a : cupicef.Term (cupicef.type Coq__Init__Datatypes__80) A -> + P : + (__ : cupicef.Term (cupicef.type Coq__Init__Datatypes__80) A -> + cupicef.Term cupicef.prop + (identity (cupicef.type Coq__Init__Datatypes__80) A a __) -> + cupicef.Univ s) -> + case__identity__refl : cupicef.Term s (P a (identity__refl A a)) -> + __ : cupicef.Term (cupicef.type Coq__Init__Datatypes__80) A -> + x : + cupicef.Term cupicef.prop + (identity (cupicef.type Coq__Init__Datatypes__80) A a __) -> + cupicef.Term s (P __ x). + +[ s, A, a, P, case__identity__refl] + match____identity s A a P case__identity__refl {a} (identity__refl A a) + --> + case__identity__refl. + +[ s, A, a, P, s'] + match____identity s A a + (__ => x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P __ x))) + --> + match____identity s' A a (__ => x => P __ x). + +def identity__rect : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + i : + cupicef.Term cupicef.prop + (identity (cupicef.type (cupicef.s cupicef.z)) A a a0) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P a (identity__refl A a)) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + i : + cupicef.Term cupicef.prop + (identity (cupicef.type (cupicef.s cupicef.z)) A a y) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y i) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P : + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + i : + cupicef.Term cupicef.prop + (identity (cupicef.type (cupicef.s cupicef.z)) A a a0) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P a (identity__refl A a)) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + i : + cupicef.Term cupicef.prop + (identity (cupicef.type (cupicef.s cupicef.z)) A a y) => + match____identity (cupicef.type (cupicef.s cupicef.z)) A a + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + i0 : + cupicef.Term cupicef.prop + (identity (cupicef.type (cupicef.s cupicef.z)) A a y0) => + P y0 i0) + f y i. + +def identity__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + i : + cupicef.Term cupicef.prop + (identity (cupicef.type (cupicef.s cupicef.z)) A a a0) -> + cupicef.Univ cupicef.prop) -> + f : cupicef.Term cupicef.prop (P a (identity__refl A a)) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + i : + cupicef.Term cupicef.prop + (identity (cupicef.type (cupicef.s cupicef.z)) A a y) -> + cupicef.Term cupicef.prop (P y i) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P : + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + i : + cupicef.Term cupicef.prop + (identity (cupicef.type (cupicef.s cupicef.z)) A a a0) -> + cupicef.Univ cupicef.prop) => + f : cupicef.Term cupicef.prop (P a (identity__refl A a)) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + i : + cupicef.Term cupicef.prop + (identity (cupicef.type (cupicef.s cupicef.z)) A a y) => + match____identity cupicef.prop A a + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + i0 : + cupicef.Term cupicef.prop + (identity (cupicef.type (cupicef.s cupicef.z)) A a y0) => + P y0 i0) + f y i. + +def identity__rec : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + i : + cupicef.Term cupicef.prop + (identity (cupicef.type (cupicef.s cupicef.z)) A a a0) -> + cupicef.Univ cupicef.set) -> + f : cupicef.Term cupicef.set (P a (identity__refl A a)) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + i : + cupicef.Term cupicef.prop + (identity (cupicef.type (cupicef.s cupicef.z)) A a y) -> + cupicef.Term cupicef.set (P y i) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P : + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + i : + cupicef.Term cupicef.prop + (identity (cupicef.type (cupicef.s cupicef.z)) A a a0) -> + cupicef.Univ cupicef.set) => + identity__rect A a + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (identity (cupicef.type (cupicef.s cupicef.z)) A a a0) + (i : + cupicef.Term cupicef.prop + (identity (cupicef.type (cupicef.s cupicef.z)) A a a0) => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I A + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (identity (cupicef.type (cupicef.s cupicef.z)) A a a0) + (i : + cupicef.Term cupicef.prop + (identity (cupicef.type (cupicef.s cupicef.z)) A a a0) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I))) + cupicef.I P). + +def ID : + cupicef.Univ (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + +:= cupicef.prod (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => A)). + +def id : + cupicef.Term (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) ID + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x. + +def IDProp : + cupicef.Univ cupicef.prop + +:= cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => A)). + +def idProp : + cupicef.Term cupicef.prop IDProp + +:= A : cupicef.Univ cupicef.prop => x : cupicef.Term cupicef.prop A => x. + +(; End of translation. ;) diff --git a/out/Coq__Init__Decimal.dk b/out/Coq__Init__Decimal.dk new file mode 100644 index 000000000..15a4a5689 --- /dev/null +++ b/out/Coq__Init__Decimal.dk @@ -0,0 +1,816 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def uint : + cupicef.Univ cupicef.set. + +def Nil : + cupicef.Term cupicef.set uint. + +def D0 : + __ : cupicef.Term cupicef.set uint -> cupicef.Term cupicef.set uint. + +def D1 : + __ : cupicef.Term cupicef.set uint -> cupicef.Term cupicef.set uint. + +def D2 : + __ : cupicef.Term cupicef.set uint -> cupicef.Term cupicef.set uint. + +def D3 : + __ : cupicef.Term cupicef.set uint -> cupicef.Term cupicef.set uint. + +def D4 : + __ : cupicef.Term cupicef.set uint -> cupicef.Term cupicef.set uint. + +def D5 : + __ : cupicef.Term cupicef.set uint -> cupicef.Term cupicef.set uint. + +def D6 : + __ : cupicef.Term cupicef.set uint -> cupicef.Term cupicef.set uint. + +def D7 : + __ : cupicef.Term cupicef.set uint -> cupicef.Term cupicef.set uint. + +def D8 : + __ : cupicef.Term cupicef.set uint -> cupicef.Term cupicef.set uint. + +def D9 : + __ : cupicef.Term cupicef.set uint -> cupicef.Term cupicef.set uint. + +[ ] cupicef.code_guarded? _ Nil --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (D0 _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (D1 _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (D2 _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (D3 _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (D4 _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (D5 _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (D6 _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (D7 _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (D8 _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (D9 _) --> cupicef.guarded. + +def match____uint : + s : cupicef.Sort -> + P : (cupicef.Term cupicef.set uint -> cupicef.Univ s) -> + case__Nil : cupicef.Term s (P Nil) -> + case__D0 : + (__ : cupicef.Term cupicef.set uint -> cupicef.Term s (P (D0 __))) -> + case__D1 : + (__ : cupicef.Term cupicef.set uint -> cupicef.Term s (P (D1 __))) -> + case__D2 : + (__ : cupicef.Term cupicef.set uint -> cupicef.Term s (P (D2 __))) -> + case__D3 : + (__ : cupicef.Term cupicef.set uint -> cupicef.Term s (P (D3 __))) -> + case__D4 : + (__ : cupicef.Term cupicef.set uint -> cupicef.Term s (P (D4 __))) -> + case__D5 : + (__ : cupicef.Term cupicef.set uint -> cupicef.Term s (P (D5 __))) -> + case__D6 : + (__ : cupicef.Term cupicef.set uint -> cupicef.Term s (P (D6 __))) -> + case__D7 : + (__ : cupicef.Term cupicef.set uint -> cupicef.Term s (P (D7 __))) -> + case__D8 : + (__ : cupicef.Term cupicef.set uint -> cupicef.Term s (P (D8 __))) -> + case__D9 : + (__ : cupicef.Term cupicef.set uint -> cupicef.Term s (P (D9 __))) -> + x : cupicef.Term cupicef.set uint -> + cupicef.Term s (P x). + +[ s, P, case__Nil, case__D0, case__D1, case__D2, case__D3, case__D4, case__D5, case__D6, case__D7, case__D8, case__D9] + match____uint s P case__Nil case__D0 case__D1 case__D2 case__D3 case__D4 + case__D5 case__D6 case__D7 case__D8 case__D9 Nil + --> + case__Nil. + +[ s, P, case__Nil, case__D0, case__D1, case__D2, case__D3, case__D4, case__D5, case__D6, case__D7, case__D8, case__D9, __] + match____uint s P case__Nil case__D0 case__D1 case__D2 case__D3 case__D4 + case__D5 case__D6 case__D7 case__D8 case__D9 (D0 __) + --> + case__D0 __. + +[ s, P, case__Nil, case__D0, case__D1, case__D2, case__D3, case__D4, case__D5, case__D6, case__D7, case__D8, case__D9, __] + match____uint s P case__Nil case__D0 case__D1 case__D2 case__D3 case__D4 + case__D5 case__D6 case__D7 case__D8 case__D9 (D1 __) + --> + case__D1 __. + +[ s, P, case__Nil, case__D0, case__D1, case__D2, case__D3, case__D4, case__D5, case__D6, case__D7, case__D8, case__D9, __] + match____uint s P case__Nil case__D0 case__D1 case__D2 case__D3 case__D4 + case__D5 case__D6 case__D7 case__D8 case__D9 (D2 __) + --> + case__D2 __. + +[ s, P, case__Nil, case__D0, case__D1, case__D2, case__D3, case__D4, case__D5, case__D6, case__D7, case__D8, case__D9, __] + match____uint s P case__Nil case__D0 case__D1 case__D2 case__D3 case__D4 + case__D5 case__D6 case__D7 case__D8 case__D9 (D3 __) + --> + case__D3 __. + +[ s, P, case__Nil, case__D0, case__D1, case__D2, case__D3, case__D4, case__D5, case__D6, case__D7, case__D8, case__D9, __] + match____uint s P case__Nil case__D0 case__D1 case__D2 case__D3 case__D4 + case__D5 case__D6 case__D7 case__D8 case__D9 (D4 __) + --> + case__D4 __. + +[ s, P, case__Nil, case__D0, case__D1, case__D2, case__D3, case__D4, case__D5, case__D6, case__D7, case__D8, case__D9, __] + match____uint s P case__Nil case__D0 case__D1 case__D2 case__D3 case__D4 + case__D5 case__D6 case__D7 case__D8 case__D9 (D5 __) + --> + case__D5 __. + +[ s, P, case__Nil, case__D0, case__D1, case__D2, case__D3, case__D4, case__D5, case__D6, case__D7, case__D8, case__D9, __] + match____uint s P case__Nil case__D0 case__D1 case__D2 case__D3 case__D4 + case__D5 case__D6 case__D7 case__D8 case__D9 (D6 __) + --> + case__D6 __. + +[ s, P, case__Nil, case__D0, case__D1, case__D2, case__D3, case__D4, case__D5, case__D6, case__D7, case__D8, case__D9, __] + match____uint s P case__Nil case__D0 case__D1 case__D2 case__D3 case__D4 + case__D5 case__D6 case__D7 case__D8 case__D9 (D7 __) + --> + case__D7 __. + +[ s, P, case__Nil, case__D0, case__D1, case__D2, case__D3, case__D4, case__D5, case__D6, case__D7, case__D8, case__D9, __] + match____uint s P case__Nil case__D0 case__D1 case__D2 case__D3 case__D4 + case__D5 case__D6 case__D7 case__D8 case__D9 (D8 __) + --> + case__D8 __. + +[ s, P, case__Nil, case__D0, case__D1, case__D2, case__D3, case__D4, case__D5, case__D6, case__D7, case__D8, case__D9, __] + match____uint s P case__Nil case__D0 case__D1 case__D2 case__D3 case__D4 + case__D5 case__D6 case__D7 case__D8 case__D9 (D9 __) + --> + case__D9 __. + +[ s, P, s'] + match____uint s + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____uint s' (x => P x). + +def uint__rect : + P : + (u : cupicef.Term cupicef.set uint -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P Nil) -> + f0 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P u) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (D0 u))) -> + f1 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P u) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (D1 u))) -> + f2 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P u) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (D2 u))) -> + f3 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P u) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (D3 u))) -> + f4 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P u) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (D4 u))) -> + f5 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P u) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (D5 u))) -> + f6 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P u) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (D6 u))) -> + f7 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P u) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (D7 u))) -> + f8 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P u) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (D8 u))) -> + f9 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P u) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (D9 u))) -> + u : cupicef.Term cupicef.set uint -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P u) + +:= P : + (u : cupicef.Term cupicef.set uint -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P Nil) => + f0 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P u) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (D0 u))) => + f1 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P u) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (D1 u))) => + f2 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P u) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (D2 u))) => + f3 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P u) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (D3 u))) => + f4 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P u) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (D4 u))) => + f5 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P u) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (D5 u))) => + f6 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P u) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (D6 u))) => + f7 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P u) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (D7 u))) => + f8 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P u) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (D8 u))) => + f9 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P u) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (D9 u))) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.prod cupicef.set (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + cupicef.I uint (u : cupicef.Term cupicef.set uint => P u)))) + (c => + c + (F => + u : cupicef.Term cupicef.set uint => + match____uint (cupicef.type (cupicef.s cupicef.z)) + (u0 : cupicef.Term cupicef.set uint => P u0) f + (u0 : cupicef.Term cupicef.set uint => f0 u0 (F u0)) + (u0 : cupicef.Term cupicef.set uint => f1 u0 (F u0)) + (u0 : cupicef.Term cupicef.set uint => f2 u0 (F u0)) + (u0 : cupicef.Term cupicef.set uint => f3 u0 (F u0)) + (u0 : cupicef.Term cupicef.set uint => f4 u0 (F u0)) + (u0 : cupicef.Term cupicef.set uint => f5 u0 (F u0)) + (u0 : cupicef.Term cupicef.set uint => f6 u0 (F u0)) + (u0 : cupicef.Term cupicef.set uint => f7 u0 (F u0)) + (u0 : cupicef.Term cupicef.set uint => f8 u0 (F u0)) + (u0 : cupicef.Term cupicef.set uint => f9 u0 (F u0)) u)) + cupicef.0. + +def uint__ind : + P : (u : cupicef.Term cupicef.set uint -> cupicef.Univ cupicef.prop) -> + f : cupicef.Term cupicef.prop (P Nil) -> + f0 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.prop (P u) -> + cupicef.Term cupicef.prop (P (D0 u))) -> + f1 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.prop (P u) -> + cupicef.Term cupicef.prop (P (D1 u))) -> + f2 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.prop (P u) -> + cupicef.Term cupicef.prop (P (D2 u))) -> + f3 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.prop (P u) -> + cupicef.Term cupicef.prop (P (D3 u))) -> + f4 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.prop (P u) -> + cupicef.Term cupicef.prop (P (D4 u))) -> + f5 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.prop (P u) -> + cupicef.Term cupicef.prop (P (D5 u))) -> + f6 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.prop (P u) -> + cupicef.Term cupicef.prop (P (D6 u))) -> + f7 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.prop (P u) -> + cupicef.Term cupicef.prop (P (D7 u))) -> + f8 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.prop (P u) -> + cupicef.Term cupicef.prop (P (D8 u))) -> + f9 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.prop (P u) -> + cupicef.Term cupicef.prop (P (D9 u))) -> + u : cupicef.Term cupicef.set uint -> + cupicef.Term cupicef.prop (P u) + +:= P : (u : cupicef.Term cupicef.set uint -> cupicef.Univ cupicef.prop) => + f : cupicef.Term cupicef.prop (P Nil) => + f0 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.prop (P u) -> + cupicef.Term cupicef.prop (P (D0 u))) => + f1 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.prop (P u) -> + cupicef.Term cupicef.prop (P (D1 u))) => + f2 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.prop (P u) -> + cupicef.Term cupicef.prop (P (D2 u))) => + f3 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.prop (P u) -> + cupicef.Term cupicef.prop (P (D3 u))) => + f4 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.prop (P u) -> + cupicef.Term cupicef.prop (P (D4 u))) => + f5 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.prop (P u) -> + cupicef.Term cupicef.prop (P (D5 u))) => + f6 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.prop (P u) -> + cupicef.Term cupicef.prop (P (D6 u))) => + f7 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.prop (P u) -> + cupicef.Term cupicef.prop (P (D7 u))) => + f8 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.prop (P u) -> + cupicef.Term cupicef.prop (P (D8 u))) => + f9 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.prop (P u) -> + cupicef.Term cupicef.prop (P (D9 u))) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.prop + (cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I uint + (u : cupicef.Term cupicef.set uint => P u)))) + (c => + c + (F => + u : cupicef.Term cupicef.set uint => + match____uint cupicef.prop + (u0 : cupicef.Term cupicef.set uint => P u0) f + (u0 : cupicef.Term cupicef.set uint => f0 u0 (F u0)) + (u0 : cupicef.Term cupicef.set uint => f1 u0 (F u0)) + (u0 : cupicef.Term cupicef.set uint => f2 u0 (F u0)) + (u0 : cupicef.Term cupicef.set uint => f3 u0 (F u0)) + (u0 : cupicef.Term cupicef.set uint => f4 u0 (F u0)) + (u0 : cupicef.Term cupicef.set uint => f5 u0 (F u0)) + (u0 : cupicef.Term cupicef.set uint => f6 u0 (F u0)) + (u0 : cupicef.Term cupicef.set uint => f7 u0 (F u0)) + (u0 : cupicef.Term cupicef.set uint => f8 u0 (F u0)) + (u0 : cupicef.Term cupicef.set uint => f9 u0 (F u0)) u)) + cupicef.0. + +def uint__rec : + P : (u : cupicef.Term cupicef.set uint -> cupicef.Univ cupicef.set) -> + f : cupicef.Term cupicef.set (P Nil) -> + f0 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.set (P u) -> + cupicef.Term cupicef.set (P (D0 u))) -> + f1 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.set (P u) -> + cupicef.Term cupicef.set (P (D1 u))) -> + f2 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.set (P u) -> + cupicef.Term cupicef.set (P (D2 u))) -> + f3 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.set (P u) -> + cupicef.Term cupicef.set (P (D3 u))) -> + f4 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.set (P u) -> + cupicef.Term cupicef.set (P (D4 u))) -> + f5 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.set (P u) -> + cupicef.Term cupicef.set (P (D5 u))) -> + f6 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.set (P u) -> + cupicef.Term cupicef.set (P (D6 u))) -> + f7 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.set (P u) -> + cupicef.Term cupicef.set (P (D7 u))) -> + f8 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.set (P u) -> + cupicef.Term cupicef.set (P (D8 u))) -> + f9 : + (u : cupicef.Term cupicef.set uint -> + __ : cupicef.Term cupicef.set (P u) -> + cupicef.Term cupicef.set (P (D9 u))) -> + u : cupicef.Term cupicef.set uint -> + cupicef.Term cupicef.set (P u) + +:= P : (u : cupicef.Term cupicef.set uint -> cupicef.Univ cupicef.set) => + uint__rect + (cupicef.cast + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.set (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + cupicef.I uint + (u : cupicef.Term cupicef.set uint => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I uint + (u : cupicef.Term cupicef.set uint => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P). + +def int : + cupicef.Univ cupicef.set. + +def Pos : + d : cupicef.Term cupicef.set uint -> cupicef.Term cupicef.set int. + +def Neg : + d : cupicef.Term cupicef.set uint -> cupicef.Term cupicef.set int. + +[ ] cupicef.code_guarded? _ (Pos _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (Neg _) --> cupicef.guarded. + +def match____int : + s : cupicef.Sort -> + P : (cupicef.Term cupicef.set int -> cupicef.Univ s) -> + case__Pos : + (d : cupicef.Term cupicef.set uint -> cupicef.Term s (P (Pos d))) -> + case__Neg : + (d : cupicef.Term cupicef.set uint -> cupicef.Term s (P (Neg d))) -> + x : cupicef.Term cupicef.set int -> + cupicef.Term s (P x). + +[ s, P, case__Pos, case__Neg, d] + match____int s P case__Pos case__Neg (Pos d) + --> + case__Pos d. + +[ s, P, case__Pos, case__Neg, d] + match____int s P case__Pos case__Neg (Neg d) + --> + case__Neg d. + +[ s, P, s'] + match____int s (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____int s' (x => P x). + +def int__rect : + P : + (i : cupicef.Term cupicef.set int -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : + (d : cupicef.Term cupicef.set uint -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (Pos d))) -> + f0 : + (d : cupicef.Term cupicef.set uint -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (Neg d))) -> + i : cupicef.Term cupicef.set int -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P i) + +:= P : + (i : cupicef.Term cupicef.set int -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : + (d : cupicef.Term cupicef.set uint -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (Pos d))) => + f0 : + (d : cupicef.Term cupicef.set uint -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (Neg d))) => + i : cupicef.Term cupicef.set int => + match____int (cupicef.type (cupicef.s cupicef.z)) + (i0 : cupicef.Term cupicef.set int => P i0) f f0 i. + +def int__ind : + P : (i : cupicef.Term cupicef.set int -> cupicef.Univ cupicef.prop) -> + f : + (d : cupicef.Term cupicef.set uint -> + cupicef.Term cupicef.prop (P (Pos d))) -> + f0 : + (d : cupicef.Term cupicef.set uint -> + cupicef.Term cupicef.prop (P (Neg d))) -> + i : cupicef.Term cupicef.set int -> + cupicef.Term cupicef.prop (P i) + +:= P : (i : cupicef.Term cupicef.set int -> cupicef.Univ cupicef.prop) => + f : + (d : cupicef.Term cupicef.set uint -> + cupicef.Term cupicef.prop (P (Pos d))) => + f0 : + (d : cupicef.Term cupicef.set uint -> + cupicef.Term cupicef.prop (P (Neg d))) => + i : cupicef.Term cupicef.set int => + match____int cupicef.prop (i0 : cupicef.Term cupicef.set int => P i0) f f0 + i. + +def int__rec : + P : (i : cupicef.Term cupicef.set int -> cupicef.Univ cupicef.set) -> + f : + (d : cupicef.Term cupicef.set uint -> + cupicef.Term cupicef.set (P (Pos d))) -> + f0 : + (d : cupicef.Term cupicef.set uint -> + cupicef.Term cupicef.set (P (Neg d))) -> + i : cupicef.Term cupicef.set int -> + cupicef.Term cupicef.set (P i) + +:= P : (i : cupicef.Term cupicef.set int -> cupicef.Univ cupicef.set) => + int__rect + (cupicef.cast + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.set (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + cupicef.I int + (i : cupicef.Term cupicef.set int => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I int + (i : cupicef.Term cupicef.set int => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P). + +def nzhead : + d : cupicef.Term cupicef.set uint -> cupicef.Term cupicef.set uint + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I uint + (d : cupicef.Term cupicef.set uint => uint)))) + (c => + c + (nzhead0 => + d : cupicef.Term cupicef.set uint => + match____uint cupicef.set + (d0 : cupicef.Term cupicef.set uint => uint) d + (d0 : cupicef.Term cupicef.set uint => nzhead0 d0) + (u : cupicef.Term cupicef.set uint => d) + (u : cupicef.Term cupicef.set uint => d) + (u : cupicef.Term cupicef.set uint => d) + (u : cupicef.Term cupicef.set uint => d) + (u : cupicef.Term cupicef.set uint => d) + (u : cupicef.Term cupicef.set uint => d) + (u : cupicef.Term cupicef.set uint => d) + (u : cupicef.Term cupicef.set uint => d) + (u : cupicef.Term cupicef.set uint => d) d)) + cupicef.0. + +def unorm : + d : cupicef.Term cupicef.set uint -> cupicef.Term cupicef.set uint + +:= d : cupicef.Term cupicef.set uint => + match____uint cupicef.set (x : cupicef.Term cupicef.set uint => uint) + (D0 Nil) (u : cupicef.Term cupicef.set uint => D0 u) + (u : cupicef.Term cupicef.set uint => D1 u) + (u : cupicef.Term cupicef.set uint => D2 u) + (u : cupicef.Term cupicef.set uint => D3 u) + (u : cupicef.Term cupicef.set uint => D4 u) + (u : cupicef.Term cupicef.set uint => D5 u) + (u : cupicef.Term cupicef.set uint => D6 u) + (u : cupicef.Term cupicef.set uint => D7 u) + (u : cupicef.Term cupicef.set uint => D8 u) + (u : cupicef.Term cupicef.set uint => D9 u) (nzhead d). + +def norm : + d : cupicef.Term cupicef.set int -> cupicef.Term cupicef.set int + +:= d : cupicef.Term cupicef.set int => + match____int cupicef.set (d0 : cupicef.Term cupicef.set int => int) + (d0 : cupicef.Term cupicef.set uint => Pos (unorm d0)) + (d0 : cupicef.Term cupicef.set uint => + match____uint cupicef.set (x : cupicef.Term cupicef.set uint => int) + (Pos (D0 Nil)) (u : cupicef.Term cupicef.set uint => Neg (D0 u)) + (u : cupicef.Term cupicef.set uint => Neg (D1 u)) + (u : cupicef.Term cupicef.set uint => Neg (D2 u)) + (u : cupicef.Term cupicef.set uint => Neg (D3 u)) + (u : cupicef.Term cupicef.set uint => Neg (D4 u)) + (u : cupicef.Term cupicef.set uint => Neg (D5 u)) + (u : cupicef.Term cupicef.set uint => Neg (D6 u)) + (u : cupicef.Term cupicef.set uint => Neg (D7 u)) + (u : cupicef.Term cupicef.set uint => Neg (D8 u)) + (u : cupicef.Term cupicef.set uint => Neg (D9 u)) (nzhead d0)) + d. + +def opp : + d : cupicef.Term cupicef.set int -> cupicef.Term cupicef.set int + +:= d : cupicef.Term cupicef.set int => + match____int cupicef.set (d0 : cupicef.Term cupicef.set int => int) + (d0 : cupicef.Term cupicef.set uint => Neg d0) + (d0 : cupicef.Term cupicef.set uint => Pos d0) d. + +def revapp : + d : cupicef.Term cupicef.set uint -> + d_27_ : cupicef.Term cupicef.set uint -> + cupicef.Term cupicef.set uint + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + cupicef.I uint + (d : cupicef.Term cupicef.set uint => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I uint + (d_27_ : cupicef.Term cupicef.set uint => uint))))) + (c => + c + (revapp0 => + d : cupicef.Term cupicef.set uint => + d_27_ : cupicef.Term cupicef.set uint => + match____uint cupicef.set + (d0 : cupicef.Term cupicef.set uint => uint) d_27_ + (d0 : cupicef.Term cupicef.set uint => revapp0 d0 (D0 d_27_)) + (d0 : cupicef.Term cupicef.set uint => revapp0 d0 (D1 d_27_)) + (d0 : cupicef.Term cupicef.set uint => revapp0 d0 (D2 d_27_)) + (d0 : cupicef.Term cupicef.set uint => revapp0 d0 (D3 d_27_)) + (d0 : cupicef.Term cupicef.set uint => revapp0 d0 (D4 d_27_)) + (d0 : cupicef.Term cupicef.set uint => revapp0 d0 (D5 d_27_)) + (d0 : cupicef.Term cupicef.set uint => revapp0 d0 (D6 d_27_)) + (d0 : cupicef.Term cupicef.set uint => revapp0 d0 (D7 d_27_)) + (d0 : cupicef.Term cupicef.set uint => revapp0 d0 (D8 d_27_)) + (d0 : cupicef.Term cupicef.set uint => revapp0 d0 (D9 d_27_)) d)) + cupicef.0. + +def rev : + d : cupicef.Term cupicef.set uint -> cupicef.Term cupicef.set uint + +:= d : cupicef.Term cupicef.set uint => revapp d Nil. + +def Little__succ : + d : cupicef.Term cupicef.set uint -> cupicef.Term cupicef.set uint + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I uint + (d : cupicef.Term cupicef.set uint => uint)))) + (c => + c + (succ0 => + d : cupicef.Term cupicef.set uint => + match____uint cupicef.set + (d0 : cupicef.Term cupicef.set uint => uint) (D1 Nil) + (d0 : cupicef.Term cupicef.set uint => D1 d0) + (d0 : cupicef.Term cupicef.set uint => D2 d0) + (d0 : cupicef.Term cupicef.set uint => D3 d0) + (d0 : cupicef.Term cupicef.set uint => D4 d0) + (d0 : cupicef.Term cupicef.set uint => D5 d0) + (d0 : cupicef.Term cupicef.set uint => D6 d0) + (d0 : cupicef.Term cupicef.set uint => D7 d0) + (d0 : cupicef.Term cupicef.set uint => D8 d0) + (d0 : cupicef.Term cupicef.set uint => D9 d0) + (d0 : cupicef.Term cupicef.set uint => D0 (succ0 d0)) d)) + cupicef.0. + +def Little__double : + d : cupicef.Term cupicef.set uint -> cupicef.Term cupicef.set uint + +:= cupicef.fixproj cupicef.2 + (c => + c + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I uint + (d : cupicef.Term cupicef.set uint => uint))) + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I uint + (d : cupicef.Term cupicef.set uint => uint)))) + (c => + c + (double0 => + succ__double0 => + d : cupicef.Term cupicef.set uint => + match____uint cupicef.set + (d0 : cupicef.Term cupicef.set uint => uint) Nil + (d0 : cupicef.Term cupicef.set uint => D0 (double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D2 (double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D4 (double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D6 (double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D8 (double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D0 (succ__double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D2 (succ__double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D4 (succ__double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D6 (succ__double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D8 (succ__double0 d0)) d) + (double0 => + succ__double0 => + d : cupicef.Term cupicef.set uint => + match____uint cupicef.set + (d0 : cupicef.Term cupicef.set uint => uint) (D1 Nil) + (d0 : cupicef.Term cupicef.set uint => D1 (double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D3 (double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D5 (double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D7 (double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D9 (double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D1 (succ__double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D3 (succ__double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D5 (succ__double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D7 (succ__double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D9 (succ__double0 d0)) d)) + cupicef.0. + +def Little__succ__double : + d : cupicef.Term cupicef.set uint -> cupicef.Term cupicef.set uint + +:= cupicef.fixproj cupicef.2 + (c => + c + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I uint + (d : cupicef.Term cupicef.set uint => uint))) + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I uint + (d : cupicef.Term cupicef.set uint => uint)))) + (c => + c + (double0 => + succ__double0 => + d : cupicef.Term cupicef.set uint => + match____uint cupicef.set + (d0 : cupicef.Term cupicef.set uint => uint) Nil + (d0 : cupicef.Term cupicef.set uint => D0 (double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D2 (double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D4 (double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D6 (double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D8 (double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D0 (succ__double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D2 (succ__double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D4 (succ__double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D6 (succ__double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D8 (succ__double0 d0)) d) + (double0 => + succ__double0 => + d : cupicef.Term cupicef.set uint => + match____uint cupicef.set + (d0 : cupicef.Term cupicef.set uint => uint) (D1 Nil) + (d0 : cupicef.Term cupicef.set uint => D1 (double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D3 (double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D5 (double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D7 (double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D9 (double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D1 (succ__double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D3 (succ__double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D5 (succ__double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D7 (succ__double0 d0)) + (d0 : cupicef.Term cupicef.set uint => D9 (succ__double0 d0)) d)) + cupicef.1. + +(; End of translation. ;) diff --git a/out/Coq__Init__Logic.dk b/out/Coq__Init__Logic.dk new file mode 100644 index 000000000..35d8c288f --- /dev/null +++ b/out/Coq__Init__Logic.dk @@ -0,0 +1,20146 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +#REQUIRE cupicef. + +def True : + cupicef.Univ cupicef.prop. + +def I : + cupicef.Term cupicef.prop True. + +[ ] cupicef.code_guarded? _ I --> cupicef.guarded. + +def match____True : + s : cupicef.Sort -> + P : (cupicef.Term cupicef.prop True -> cupicef.Univ s) -> + case__I : cupicef.Term s (P I) -> + x : cupicef.Term cupicef.prop True -> + cupicef.Term s (P x). + +[ s, P, case__I] + match____True s P case__I I + --> + case__I. + +[ s, P, s'] + match____True s + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____True s' (x => P x). + +def True__rect : + P : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) P -> + t : cupicef.Term cupicef.prop True -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) P + +:= P : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) P => + t : cupicef.Term cupicef.prop True => + match____True (cupicef.type (cupicef.s cupicef.z)) + (__ : cupicef.Term cupicef.prop True => P) f t. + +def True__ind : + P : cupicef.Univ cupicef.prop -> + f : cupicef.Term cupicef.prop P -> + t : cupicef.Term cupicef.prop True -> + cupicef.Term cupicef.prop P + +:= P : cupicef.Univ cupicef.prop => + f : cupicef.Term cupicef.prop P => + t : cupicef.Term cupicef.prop True => + match____True cupicef.prop (__ : cupicef.Term cupicef.prop True => P) f t. + +def True__rec : + P : cupicef.Univ cupicef.set -> + f : cupicef.Term cupicef.set P -> + t : cupicef.Term cupicef.prop True -> + cupicef.Term cupicef.set P + +:= P : cupicef.Univ cupicef.set => + True__rect + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I P). + +def False : + cupicef.Univ cupicef.prop. + +def match____False : + s : cupicef.Sort -> + P : (cupicef.Term cupicef.prop False -> cupicef.Univ s) -> + x : cupicef.Term cupicef.prop False -> + cupicef.Term s (P x). + +[ s, P, s'] + match____False s + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____False s' (x => P x). + +def False__rect : + P : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : cupicef.Term cupicef.prop False -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) P + +:= P : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : cupicef.Term cupicef.prop False => + match____False (cupicef.type (cupicef.s cupicef.z)) + (__ : cupicef.Term cupicef.prop False => P) f. + +def False__ind : + P : cupicef.Univ cupicef.prop -> + f : cupicef.Term cupicef.prop False -> + cupicef.Term cupicef.prop P + +:= P : cupicef.Univ cupicef.prop => + f : cupicef.Term cupicef.prop False => + match____False cupicef.prop (__ : cupicef.Term cupicef.prop False => P) f. + +def False__rec : + P : cupicef.Univ cupicef.set -> + f : cupicef.Term cupicef.prop False -> + cupicef.Term cupicef.set P + +:= P : cupicef.Univ cupicef.set => + False__rect + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I P). + +def not : + A : cupicef.Univ cupicef.prop -> cupicef.Univ cupicef.prop + +:= A : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => False). + +def and : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + cupicef.Univ cupicef.prop. + +def conj : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop A -> + __0 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop (and A B). + +[ ] cupicef.code_guarded? _ (conj _ _ _ _) --> cupicef.guarded. + +def match____and : + s : cupicef.Sort -> + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + P : (cupicef.Term cupicef.prop (and A B) -> cupicef.Univ s) -> + case__conj : + (__ : cupicef.Term cupicef.prop A -> + __0 : cupicef.Term cupicef.prop B -> + cupicef.Term s (P (conj A B __ __0))) -> + x : cupicef.Term cupicef.prop (and A B) -> + cupicef.Term s (P x). + +[ s, A, B, P, case__conj, __, __0] + match____and s A B P case__conj (conj A B __ __0) + --> + case__conj __ __0. + +[ s, A, B, P, s'] + match____and s A B + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____and s' A B (x => P x). + +def and__rect : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + P : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term cupicef.prop A -> + __0 : cupicef.Term cupicef.prop B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) P) -> + a : cupicef.Term cupicef.prop (and A B) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) P + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + P : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term cupicef.prop A -> + __0 : cupicef.Term cupicef.prop B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) P) => + a : cupicef.Term cupicef.prop (and A B) => + match____and (cupicef.type (cupicef.s cupicef.z)) A B + (__ : cupicef.Term cupicef.prop (and A B) => P) f a. + +def and__ind : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + P : cupicef.Univ cupicef.prop -> + f : + (__ : cupicef.Term cupicef.prop A -> + __0 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop P) -> + a : cupicef.Term cupicef.prop (and A B) -> + cupicef.Term cupicef.prop P + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + P : cupicef.Univ cupicef.prop => + f : + (__ : cupicef.Term cupicef.prop A -> + __0 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop P) => + a : cupicef.Term cupicef.prop (and A B) => + match____and cupicef.prop A B + (__ : cupicef.Term cupicef.prop (and A B) => P) f a. + +def and__rec : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + P : cupicef.Univ cupicef.set -> + f : + (__ : cupicef.Term cupicef.prop A -> + __0 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.set P) -> + a : cupicef.Term cupicef.prop (and A B) -> + cupicef.Term cupicef.set P + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + P : cupicef.Univ cupicef.set => + and__rect A B + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I P). + +thm proj1 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (and A B) -> + cupicef.Term cupicef.prop A + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (and A B) => + match____and cupicef.prop A B + (__ : cupicef.Term cupicef.prop (and A B) => A) + (H0 : cupicef.Term cupicef.prop A => + H1 : cupicef.Term cupicef.prop B => + H0) + H. + +thm proj2 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (and A B) -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (and A B) => + match____and cupicef.prop A B + (__ : cupicef.Term cupicef.prop (and A B) => B) + (H0 : cupicef.Term cupicef.prop A => + H1 : cupicef.Term cupicef.prop B => + H1) + H. + +def or : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + cupicef.Univ cupicef.prop. + +def or__introl : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop (or A B). + +def or__intror : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop (or A B). + +[ ] cupicef.code_guarded? _ (or__introl _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (or__intror _ _ _) --> cupicef.guarded. + +def match____or : + s : cupicef.Sort -> + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + P : (cupicef.Term cupicef.prop (or A B) -> cupicef.Univ s) -> + case__or__introl : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term s (P (or__introl A B __))) -> + case__or__intror : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term s (P (or__intror A B __))) -> + x : cupicef.Term cupicef.prop (or A B) -> + cupicef.Term s (P x). + +[ s, A, B, P, case__or__introl, case__or__intror, __] + match____or s A B P case__or__introl case__or__intror (or__introl A B __) + --> + case__or__introl __. + +[ s, A, B, P, case__or__introl, case__or__intror, __] + match____or s A B P case__or__introl case__or__intror (or__intror A B __) + --> + case__or__intror __. + +[ s, A, B, P, s'] + match____or s A B + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____or s' A B (x => P x). + +def or__ind : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + P : cupicef.Univ cupicef.prop -> + f : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop P) -> + f0 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop P) -> + o : cupicef.Term cupicef.prop (or A B) -> + cupicef.Term cupicef.prop P + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + P : cupicef.Univ cupicef.prop => + f : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop P) => + f0 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop P) => + o : cupicef.Term cupicef.prop (or A B) => + match____or cupicef.prop A B + (__ : cupicef.Term cupicef.prop (or A B) => P) f f0 o. + +def iff : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)). + +thm iff__refl : + A : cupicef.Univ cupicef.prop -> cupicef.Term cupicef.prop (iff A A) + +:= A : cupicef.Univ cupicef.prop => + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => A)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => A)) + (H : cupicef.Term cupicef.prop A => H) + (H : cupicef.Term cupicef.prop A => H). + +thm iff__trans : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (iff A B) -> + __0 : cupicef.Term cupicef.prop (iff B C) -> + cupicef.Term cupicef.prop (iff A C) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (iff A B) => + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)) + (__ : + cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (iff B C) + (__0 : cupicef.Term cupicef.prop (iff B C) => iff A C)) + (H1 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H2 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + H0 : cupicef.Term cupicef.prop (iff B C) => + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__ : cupicef.Term cupicef.prop C => B)) + (__ : + cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__ : cupicef.Term cupicef.prop C => B))) => + iff A C) + (H3 : + (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop C) => + H4 : + (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop B) => + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__ : cupicef.Term cupicef.prop C => A)) + (H5 : cupicef.Term cupicef.prop A => H3 (H1 H5)) + (H5 : cupicef.Term cupicef.prop C => H2 (H1 (H2 (H4 H5))))) + H0) + H. + +thm iff__sym : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (iff A B) -> + cupicef.Term cupicef.prop (iff B A) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (iff A B) => + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)) + (__ : + cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A))) => + iff B A) + (H1 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H2 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + H2 H1) + H. + +thm and__iff__compat__l : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (iff B C) -> + cupicef.Term cupicef.prop (iff (and A B) (and A C)) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (iff B C) => + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__ : cupicef.Term cupicef.prop C => B)) + (__ : + cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__ : cupicef.Term cupicef.prop C => B))) => + iff (and A B) (and A C)) + (Hl : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop C) => + Hr : (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop B) => + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (and A B) + (__ : cupicef.Term cupicef.prop (and A B) => and A C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (and A C) + (__ : cupicef.Term cupicef.prop (and A C) => and A B)) + (H0 : cupicef.Term cupicef.prop (and A B) => + match____and cupicef.prop A B + (__ : cupicef.Term cupicef.prop (and A B) => and A C) + (H1 : cupicef.Term cupicef.prop A => + H2 : cupicef.Term cupicef.prop B => + conj A C H1 (Hl H2)) + H0) + (H0 : cupicef.Term cupicef.prop (and A C) => + match____and cupicef.prop A C + (__ : cupicef.Term cupicef.prop (and A C) => and A B) + (H1 : cupicef.Term cupicef.prop A => + H2 : cupicef.Term cupicef.prop C => + conj A B H1 (Hr H2)) + H0)) + H. + +thm and__iff__compat__r : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (iff B C) -> + cupicef.Term cupicef.prop (iff (and B A) (and C A)) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (iff B C) => + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__ : cupicef.Term cupicef.prop C => B)) + (__ : + cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__ : cupicef.Term cupicef.prop C => B))) => + iff (and B A) (and C A)) + (Hl : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop C) => + Hr : (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop B) => + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (and B A) + (__ : cupicef.Term cupicef.prop (and B A) => and C A)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (and C A) + (__ : cupicef.Term cupicef.prop (and C A) => and B A)) + (H0 : cupicef.Term cupicef.prop (and B A) => + match____and cupicef.prop B A + (__ : cupicef.Term cupicef.prop (and B A) => and C A) + (H1 : cupicef.Term cupicef.prop B => + H2 : cupicef.Term cupicef.prop A => + conj C A (Hl H1) H2) + H0) + (H0 : cupicef.Term cupicef.prop (and C A) => + match____and cupicef.prop C A + (__ : cupicef.Term cupicef.prop (and C A) => and B A) + (H1 : cupicef.Term cupicef.prop C => + H2 : cupicef.Term cupicef.prop A => + conj B A (Hr H1) H2) + H0)) + H. + +thm or__iff__compat__l : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (iff B C) -> + cupicef.Term cupicef.prop (iff (or A B) (or A C)) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (iff B C) => + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__ : cupicef.Term cupicef.prop C => B)) + (__ : + cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__ : cupicef.Term cupicef.prop C => B))) => + iff (or A B) (or A C)) + (Hl : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop C) => + Hr : (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop B) => + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (or A B) + (__ : cupicef.Term cupicef.prop (or A B) => or A C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (or A C) + (__ : cupicef.Term cupicef.prop (or A C) => or A B)) + (H0 : cupicef.Term cupicef.prop (or A B) => + match____or cupicef.prop A B + (__ : cupicef.Term cupicef.prop (or A B) => or A C) + (H1 : cupicef.Term cupicef.prop A => or__introl A C H1) + (H1 : cupicef.Term cupicef.prop B => or__intror A C (Hl H1)) H0) + (H0 : cupicef.Term cupicef.prop (or A C) => + match____or cupicef.prop A C + (__ : cupicef.Term cupicef.prop (or A C) => or A B) + (H1 : cupicef.Term cupicef.prop A => or__introl A B H1) + (H1 : cupicef.Term cupicef.prop C => or__intror A B (Hr H1)) H0)) + H. + +thm or__iff__compat__r : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (iff B C) -> + cupicef.Term cupicef.prop (iff (or B A) (or C A)) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (iff B C) => + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__ : cupicef.Term cupicef.prop C => B)) + (__ : + cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__ : cupicef.Term cupicef.prop C => B))) => + iff (or B A) (or C A)) + (Hl : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop C) => + Hr : (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop B) => + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (or B A) + (__ : cupicef.Term cupicef.prop (or B A) => or C A)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (or C A) + (__ : cupicef.Term cupicef.prop (or C A) => or B A)) + (H0 : cupicef.Term cupicef.prop (or B A) => + match____or cupicef.prop B A + (__ : cupicef.Term cupicef.prop (or B A) => or C A) + (H1 : cupicef.Term cupicef.prop B => or__introl C A (Hl H1)) + (H1 : cupicef.Term cupicef.prop A => or__intror C A H1) H0) + (H0 : cupicef.Term cupicef.prop (or C A) => + match____or cupicef.prop C A + (__ : cupicef.Term cupicef.prop (or C A) => or B A) + (H1 : cupicef.Term cupicef.prop C => or__introl B A (Hr H1)) + (H1 : cupicef.Term cupicef.prop A => or__intror B A H1) H0)) + H. + +thm imp__iff__compat__l : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (iff B C) -> + cupicef.Term cupicef.prop + (iff + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => C))) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (iff B C) => + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__ : cupicef.Term cupicef.prop C => B)) + (__ : + cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__ : cupicef.Term cupicef.prop C => B))) => + iff + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => C))) + (Hl : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop C) => + Hr : (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop B) => + conj + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => C))) + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => C)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop C) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B))) + (H0 : + (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H1 : cupicef.Term cupicef.prop A => + Hl (H0 H1)) + (H0 : + (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop C) => + H1 : cupicef.Term cupicef.prop A => + Hr (H0 H1))) + H. + +thm imp__iff__compat__r : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (iff B C) -> + cupicef.Term cupicef.prop + (iff + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => A)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__0 : cupicef.Term cupicef.prop C => A))) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (iff B C) => + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__ : cupicef.Term cupicef.prop C => B)) + (__ : + cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__ : cupicef.Term cupicef.prop C => B))) => + iff + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => A)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__0 : cupicef.Term cupicef.prop C => A))) + (Hl : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop C) => + Hr : (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop B) => + conj + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)) + (__ : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop A) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__0 : cupicef.Term cupicef.prop C => A))) + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__ : cupicef.Term cupicef.prop C => A)) + (__ : + (__ : cupicef.Term cupicef.prop C -> + cupicef.Term cupicef.prop A) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => A))) + (H0 : + (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + H1 : cupicef.Term cupicef.prop C => + H0 (Hr H1)) + (H0 : + (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop A) => + H1 : cupicef.Term cupicef.prop B => + H0 (Hl H1))) + H. + +thm not__iff__compat : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (iff A B) -> + cupicef.Term cupicef.prop (iff (not A) (not B)) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (iff A B) => + imp__iff__compat__r False A B H. + +thm neg__false : + A : cupicef.Univ cupicef.prop -> + cupicef.Term cupicef.prop (iff (not A) (iff A False)) + +:= A : cupicef.Univ cupicef.prop => + cupicef.cast cupicef.prop cupicef.prop + (and + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => False)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop False) => + iff A False)) + (cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (iff A False) + (__ : cupicef.Term cupicef.prop (iff A False) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => False)))) + (iff (not A) (iff A False)) cupicef.I + (conj + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => False)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop False) => + iff A False)) + (cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (iff A False) + (__ : cupicef.Term cupicef.prop (iff A False) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => False))) + (H : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop False) => + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => False)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I False + (__ : cupicef.Term cupicef.prop False => A)) + H (H1 : cupicef.Term cupicef.prop False => False__ind A H1)) + (H0 : cupicef.Term cupicef.prop (iff A False) => + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => False)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I False + (__ : cupicef.Term cupicef.prop False => A)) + (__ : + cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => False)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + False (__ : cupicef.Term cupicef.prop False => A))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => False)) + (H : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop False) => + __tmp : + (__ : cupicef.Term cupicef.prop False -> + cupicef.Term cupicef.prop A) => + H) + H0)). + +def let__H0 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + Hl : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + Hr : (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop A) -> + H : cupicef.Term cupicef.prop (iff (and A B) (and A C)) -> + HypL : + (__ : cupicef.Term cupicef.prop (and A B) -> + cupicef.Term cupicef.prop (and A C)) -> + HypR : + (__ : cupicef.Term cupicef.prop (and A C) -> + cupicef.Term cupicef.prop (and A B)) -> + H0 : cupicef.Term cupicef.prop B -> + __ : cupicef.Term cupicef.prop (and A B) -> + cupicef.Term cupicef.prop C + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + Hl : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + Hr : (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop A) => + H : cupicef.Term cupicef.prop (iff (and A B) (and A C)) => + HypL : + (__ : cupicef.Term cupicef.prop (and A B) -> + cupicef.Term cupicef.prop (and A C)) => + HypR : + (__ : cupicef.Term cupicef.prop (and A C) -> + cupicef.Term cupicef.prop (and A B)) => + H0 : cupicef.Term cupicef.prop B => + __ : cupicef.Term cupicef.prop (and A B) => + match____and cupicef.prop A C + (__0 : cupicef.Term cupicef.prop (and A C) => C) + (__0 : cupicef.Term cupicef.prop A => + __1 : cupicef.Term cupicef.prop C => + __1) + (HypL __). + +def let__H1 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + Hl : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + Hr : (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop A) -> + H : cupicef.Term cupicef.prop (iff (and A B) (and A C)) -> + HypL : + (__ : cupicef.Term cupicef.prop (and A B) -> + cupicef.Term cupicef.prop (and A C)) -> + HypR : + (__ : cupicef.Term cupicef.prop (and A C) -> + cupicef.Term cupicef.prop (and A B)) -> + H0 : cupicef.Term cupicef.prop C -> + __ : cupicef.Term cupicef.prop (and A C) -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + Hl : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + Hr : (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop A) => + H : cupicef.Term cupicef.prop (iff (and A B) (and A C)) => + HypL : + (__ : cupicef.Term cupicef.prop (and A B) -> + cupicef.Term cupicef.prop (and A C)) => + HypR : + (__ : cupicef.Term cupicef.prop (and A C) -> + cupicef.Term cupicef.prop (and A B)) => + H0 : cupicef.Term cupicef.prop C => + __ : cupicef.Term cupicef.prop (and A C) => + match____and cupicef.prop A B + (__0 : cupicef.Term cupicef.prop (and A B) => B) + (__0 : cupicef.Term cupicef.prop A => + __1 : cupicef.Term cupicef.prop B => + __1) + (HypR __). + +thm and__cancel__l : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + __ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + __0 : (__0 : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop (iff (iff (and A B) (and A C)) (iff B C)) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + Hl : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + Hr : (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop A) => + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (iff (and A B) (and A C)) + (__ : cupicef.Term cupicef.prop (iff (and A B) (and A C)) => iff B C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (iff B C) + (__ : cupicef.Term cupicef.prop (iff B C) => iff (and A B) (and A C))) + (H : cupicef.Term cupicef.prop (iff (and A B) (and A C)) => + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (and A B) + (__ : cupicef.Term cupicef.prop (and A B) => and A C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (and A C) + (__ : cupicef.Term cupicef.prop (and A C) => and A B)) + (__ : + cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (and A B) + (__ : cupicef.Term cupicef.prop (and A B) => and A C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (and A C) + (__ : cupicef.Term cupicef.prop (and A C) => and A B))) => + iff B C) + (HypL : + (__ : cupicef.Term cupicef.prop (and A B) -> + cupicef.Term cupicef.prop (and A C)) => + HypR : + (__ : cupicef.Term cupicef.prop (and A C) -> + cupicef.Term cupicef.prop (and A B)) => + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__ : cupicef.Term cupicef.prop C => B)) + (H0 : cupicef.Term cupicef.prop B => + let__H0 A B C Hl Hr H HypL HypR H0 (conj A B (Hl H0) H0)) + (H0 : cupicef.Term cupicef.prop C => + let__H1 A B C Hl Hr H HypL HypR H0 (conj A C (Hr H0) H0))) + H) + (and__iff__compat__l A B C). + +def let__H2 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + Hl : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + Hr : (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop A) -> + H : cupicef.Term cupicef.prop (iff (and B A) (and C A)) -> + HypL : + (__ : cupicef.Term cupicef.prop (and B A) -> + cupicef.Term cupicef.prop (and C A)) -> + HypR : + (__ : cupicef.Term cupicef.prop (and C A) -> + cupicef.Term cupicef.prop (and B A)) -> + H0 : cupicef.Term cupicef.prop B -> + __ : cupicef.Term cupicef.prop (and B A) -> + cupicef.Term cupicef.prop C + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + Hl : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + Hr : (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop A) => + H : cupicef.Term cupicef.prop (iff (and B A) (and C A)) => + HypL : + (__ : cupicef.Term cupicef.prop (and B A) -> + cupicef.Term cupicef.prop (and C A)) => + HypR : + (__ : cupicef.Term cupicef.prop (and C A) -> + cupicef.Term cupicef.prop (and B A)) => + H0 : cupicef.Term cupicef.prop B => + __ : cupicef.Term cupicef.prop (and B A) => + match____and cupicef.prop C A + (__0 : cupicef.Term cupicef.prop (and C A) => C) + (__0 : cupicef.Term cupicef.prop C => + __1 : cupicef.Term cupicef.prop A => + __0) + (HypL __). + +def let__H3 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + Hl : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + Hr : (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop A) -> + H : cupicef.Term cupicef.prop (iff (and B A) (and C A)) -> + HypL : + (__ : cupicef.Term cupicef.prop (and B A) -> + cupicef.Term cupicef.prop (and C A)) -> + HypR : + (__ : cupicef.Term cupicef.prop (and C A) -> + cupicef.Term cupicef.prop (and B A)) -> + H0 : cupicef.Term cupicef.prop C -> + __ : cupicef.Term cupicef.prop (and C A) -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + Hl : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + Hr : (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop A) => + H : cupicef.Term cupicef.prop (iff (and B A) (and C A)) => + HypL : + (__ : cupicef.Term cupicef.prop (and B A) -> + cupicef.Term cupicef.prop (and C A)) => + HypR : + (__ : cupicef.Term cupicef.prop (and C A) -> + cupicef.Term cupicef.prop (and B A)) => + H0 : cupicef.Term cupicef.prop C => + __ : cupicef.Term cupicef.prop (and C A) => + match____and cupicef.prop B A + (__0 : cupicef.Term cupicef.prop (and B A) => B) + (__0 : cupicef.Term cupicef.prop B => + __1 : cupicef.Term cupicef.prop A => + __0) + (HypR __). + +thm and__cancel__r : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + __ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + __0 : (__0 : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop (iff (iff (and B A) (and C A)) (iff B C)) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + Hl : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + Hr : (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop A) => + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (iff (and B A) (and C A)) + (__ : cupicef.Term cupicef.prop (iff (and B A) (and C A)) => iff B C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (iff B C) + (__ : cupicef.Term cupicef.prop (iff B C) => iff (and B A) (and C A))) + (H : cupicef.Term cupicef.prop (iff (and B A) (and C A)) => + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (and B A) + (__ : cupicef.Term cupicef.prop (and B A) => and C A)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (and C A) + (__ : cupicef.Term cupicef.prop (and C A) => and B A)) + (__ : + cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (and B A) + (__ : cupicef.Term cupicef.prop (and B A) => and C A)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (and C A) + (__ : cupicef.Term cupicef.prop (and C A) => and B A))) => + iff B C) + (HypL : + (__ : cupicef.Term cupicef.prop (and B A) -> + cupicef.Term cupicef.prop (and C A)) => + HypR : + (__ : cupicef.Term cupicef.prop (and C A) -> + cupicef.Term cupicef.prop (and B A)) => + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__ : cupicef.Term cupicef.prop C => B)) + (H0 : cupicef.Term cupicef.prop B => + let__H2 A B C Hl Hr H HypL HypR H0 (conj B A H0 (Hl H0))) + (H0 : cupicef.Term cupicef.prop C => + let__H3 A B C Hl Hr H HypL HypR H0 (conj C A H0 (Hr H0)))) + H) + (and__iff__compat__r A B C). + +thm and__comm : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + cupicef.Term cupicef.prop (iff (and A B) (and B A)) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (and A B) + (__ : cupicef.Term cupicef.prop (and A B) => and B A)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (and B A) + (__ : cupicef.Term cupicef.prop (and B A) => and A B)) + (H : cupicef.Term cupicef.prop (and A B) => + match____and cupicef.prop A B + (__ : cupicef.Term cupicef.prop (and A B) => and B A) + (H0 : cupicef.Term cupicef.prop A => + H1 : cupicef.Term cupicef.prop B => + conj B A H1 H0) + H) + (H : cupicef.Term cupicef.prop (and B A) => + match____and cupicef.prop B A + (__ : cupicef.Term cupicef.prop (and B A) => and A B) + (H0 : cupicef.Term cupicef.prop B => + H1 : cupicef.Term cupicef.prop A => + conj A B H1 H0) + H). + +thm and__assoc : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + cupicef.Term cupicef.prop (iff (and (and A B) C) (and A (and B C))) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (and (and A B) C) + (__ : cupicef.Term cupicef.prop (and (and A B) C) => and A (and B C))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (and A (and B C)) + (__ : cupicef.Term cupicef.prop (and A (and B C)) => and (and A B) C)) + (H : cupicef.Term cupicef.prop (and (and A B) C) => + match____and cupicef.prop (and A B) C + (__ : cupicef.Term cupicef.prop (and (and A B) C) => and A (and B C)) + (H0 : cupicef.Term cupicef.prop (and A B) => + match____and cupicef.prop A B + (__ : cupicef.Term cupicef.prop (and A B) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__0 : cupicef.Term cupicef.prop C => and A (and B C))) + (H1 : cupicef.Term cupicef.prop A => + H2 : cupicef.Term cupicef.prop B => + H3 : cupicef.Term cupicef.prop C => + conj A (and B C) H1 (conj B C H2 H3)) + H0) + H) + (H : cupicef.Term cupicef.prop (and A (and B C)) => + match____and cupicef.prop A (and B C) + (__ : cupicef.Term cupicef.prop (and A (and B C)) => and (and A B) C) + (H0 : cupicef.Term cupicef.prop A => + H1 : cupicef.Term cupicef.prop (and B C) => + match____and cupicef.prop B C + (__ : cupicef.Term cupicef.prop (and B C) => and (and A B) C) + (H2 : cupicef.Term cupicef.prop B => + H3 : cupicef.Term cupicef.prop C => + conj (and A B) C (conj A B H0 H2) H3) + H1) + H). + +def let__o : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + Fl : + (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop (not A)) -> + Fr : + (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop (not A)) -> + H : cupicef.Term cupicef.prop (iff (or A B) (or A C)) -> + Hl : + (__ : cupicef.Term cupicef.prop (or A B) -> + cupicef.Term cupicef.prop (or A C)) -> + Hr : + (__ : cupicef.Term cupicef.prop (or A C) -> + cupicef.Term cupicef.prop (or A B)) -> + H0 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop (or A C) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + Fl : + (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop (not A)) => + Fr : + (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop (not A)) => + H : cupicef.Term cupicef.prop (iff (or A B) (or A C)) => + Hl : + (__ : cupicef.Term cupicef.prop (or A B) -> + cupicef.Term cupicef.prop (or A C)) => + Hr : + (__ : cupicef.Term cupicef.prop (or A C) -> + cupicef.Term cupicef.prop (or A B)) => + H0 : cupicef.Term cupicef.prop B => + Hl (or__intror A B H0). + +def let__n : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + Fl : + (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop (not A)) -> + Fr : + (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop (not A)) -> + H : cupicef.Term cupicef.prop (iff (or A B) (or A C)) -> + Hl : + (__ : cupicef.Term cupicef.prop (or A B) -> + cupicef.Term cupicef.prop (or A C)) -> + Hr : + (__ : cupicef.Term cupicef.prop (or A C) -> + cupicef.Term cupicef.prop (or A B)) -> + H0 : cupicef.Term cupicef.prop B -> + H1 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + Fl : + (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop (not A)) => + Fr : + (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop (not A)) => + H : cupicef.Term cupicef.prop (iff (or A B) (or A C)) => + Hl : + (__ : cupicef.Term cupicef.prop (or A B) -> + cupicef.Term cupicef.prop (or A C)) => + Hr : + (__ : cupicef.Term cupicef.prop (or A C) -> + cupicef.Term cupicef.prop (or A B)) => + H0 : cupicef.Term cupicef.prop B => + H1 : cupicef.Term cupicef.prop A => + Fl H0 H1. + +def let__o0 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + Fl : + (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop (not A)) -> + Fr : + (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop (not A)) -> + H : cupicef.Term cupicef.prop (iff (or A B) (or A C)) -> + Hl : + (__ : cupicef.Term cupicef.prop (or A B) -> + cupicef.Term cupicef.prop (or A C)) -> + Hr : + (__ : cupicef.Term cupicef.prop (or A C) -> + cupicef.Term cupicef.prop (or A B)) -> + H0 : cupicef.Term cupicef.prop C -> + cupicef.Term cupicef.prop (or A B) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + Fl : + (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop (not A)) => + Fr : + (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop (not A)) => + H : cupicef.Term cupicef.prop (iff (or A B) (or A C)) => + Hl : + (__ : cupicef.Term cupicef.prop (or A B) -> + cupicef.Term cupicef.prop (or A C)) => + Hr : + (__ : cupicef.Term cupicef.prop (or A C) -> + cupicef.Term cupicef.prop (or A B)) => + H0 : cupicef.Term cupicef.prop C => + Hr (or__intror A C H0). + +def let__n0 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + Fl : + (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop (not A)) -> + Fr : + (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop (not A)) -> + H : cupicef.Term cupicef.prop (iff (or A B) (or A C)) -> + Hl : + (__ : cupicef.Term cupicef.prop (or A B) -> + cupicef.Term cupicef.prop (or A C)) -> + Hr : + (__ : cupicef.Term cupicef.prop (or A C) -> + cupicef.Term cupicef.prop (or A B)) -> + H0 : cupicef.Term cupicef.prop C -> + H1 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + Fl : + (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop (not A)) => + Fr : + (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop (not A)) => + H : cupicef.Term cupicef.prop (iff (or A B) (or A C)) => + Hl : + (__ : cupicef.Term cupicef.prop (or A B) -> + cupicef.Term cupicef.prop (or A C)) => + Hr : + (__ : cupicef.Term cupicef.prop (or A C) -> + cupicef.Term cupicef.prop (or A B)) => + H0 : cupicef.Term cupicef.prop C => + H1 : cupicef.Term cupicef.prop A => + Fr H0 H1. + +thm or__cancel__l : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + __ : + (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop (not A)) -> + __0 : + (__0 : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop (not A)) -> + cupicef.Term cupicef.prop (iff (iff (or A B) (or A C)) (iff B C)) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + Fl : + (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop (not A)) => + Fr : + (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop (not A)) => + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (iff (or A B) (or A C)) + (__ : cupicef.Term cupicef.prop (iff (or A B) (or A C)) => iff B C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (iff B C) + (__ : cupicef.Term cupicef.prop (iff B C) => iff (or A B) (or A C))) + (H : cupicef.Term cupicef.prop (iff (or A B) (or A C)) => + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (or A B) + (__ : cupicef.Term cupicef.prop (or A B) => or A C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (or A C) + (__ : cupicef.Term cupicef.prop (or A C) => or A B)) + (__ : + cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (or A B) + (__ : cupicef.Term cupicef.prop (or A B) => or A C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (or A C) + (__ : cupicef.Term cupicef.prop (or A C) => or A B))) => + iff B C) + (Hl : + (__ : cupicef.Term cupicef.prop (or A B) -> + cupicef.Term cupicef.prop (or A C)) => + Hr : + (__ : cupicef.Term cupicef.prop (or A C) -> + cupicef.Term cupicef.prop (or A B)) => + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__ : cupicef.Term cupicef.prop C => B)) + (H0 : cupicef.Term cupicef.prop B => + match____or cupicef.prop A C + (__ : cupicef.Term cupicef.prop (or A C) => C) + (H1 : cupicef.Term cupicef.prop A => + match____False cupicef.prop + (__ : cupicef.Term cupicef.prop False => C) + (let__n A B C Fl Fr H Hl Hr H0 H1)) + (H1 : cupicef.Term cupicef.prop C => H1) + (let__o A B C Fl Fr H Hl Hr H0)) + (H0 : cupicef.Term cupicef.prop C => + match____or cupicef.prop A B + (__ : cupicef.Term cupicef.prop (or A B) => B) + (H1 : cupicef.Term cupicef.prop A => + match____False cupicef.prop + (__ : cupicef.Term cupicef.prop False => B) + (let__n0 A B C Fl Fr H Hl Hr H0 H1)) + (H1 : cupicef.Term cupicef.prop B => H1) + (let__o0 A B C Fl Fr H Hl Hr H0))) + H) + (or__iff__compat__l A B C). + +def let__o1 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + Fl : + (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop (not A)) -> + Fr : + (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop (not A)) -> + H : cupicef.Term cupicef.prop (iff (or B A) (or C A)) -> + Hl : + (__ : cupicef.Term cupicef.prop (or B A) -> + cupicef.Term cupicef.prop (or C A)) -> + Hr : + (__ : cupicef.Term cupicef.prop (or C A) -> + cupicef.Term cupicef.prop (or B A)) -> + H0 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop (or C A) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + Fl : + (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop (not A)) => + Fr : + (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop (not A)) => + H : cupicef.Term cupicef.prop (iff (or B A) (or C A)) => + Hl : + (__ : cupicef.Term cupicef.prop (or B A) -> + cupicef.Term cupicef.prop (or C A)) => + Hr : + (__ : cupicef.Term cupicef.prop (or C A) -> + cupicef.Term cupicef.prop (or B A)) => + H0 : cupicef.Term cupicef.prop B => + Hl (or__introl B A H0). + +def let__n1 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + Fl : + (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop (not A)) -> + Fr : + (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop (not A)) -> + H : cupicef.Term cupicef.prop (iff (or B A) (or C A)) -> + Hl : + (__ : cupicef.Term cupicef.prop (or B A) -> + cupicef.Term cupicef.prop (or C A)) -> + Hr : + (__ : cupicef.Term cupicef.prop (or C A) -> + cupicef.Term cupicef.prop (or B A)) -> + H0 : cupicef.Term cupicef.prop B -> + H1 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + Fl : + (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop (not A)) => + Fr : + (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop (not A)) => + H : cupicef.Term cupicef.prop (iff (or B A) (or C A)) => + Hl : + (__ : cupicef.Term cupicef.prop (or B A) -> + cupicef.Term cupicef.prop (or C A)) => + Hr : + (__ : cupicef.Term cupicef.prop (or C A) -> + cupicef.Term cupicef.prop (or B A)) => + H0 : cupicef.Term cupicef.prop B => + H1 : cupicef.Term cupicef.prop A => + Fl H0 H1. + +def let__o2 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + Fl : + (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop (not A)) -> + Fr : + (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop (not A)) -> + H : cupicef.Term cupicef.prop (iff (or B A) (or C A)) -> + Hl : + (__ : cupicef.Term cupicef.prop (or B A) -> + cupicef.Term cupicef.prop (or C A)) -> + Hr : + (__ : cupicef.Term cupicef.prop (or C A) -> + cupicef.Term cupicef.prop (or B A)) -> + H0 : cupicef.Term cupicef.prop C -> + cupicef.Term cupicef.prop (or B A) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + Fl : + (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop (not A)) => + Fr : + (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop (not A)) => + H : cupicef.Term cupicef.prop (iff (or B A) (or C A)) => + Hl : + (__ : cupicef.Term cupicef.prop (or B A) -> + cupicef.Term cupicef.prop (or C A)) => + Hr : + (__ : cupicef.Term cupicef.prop (or C A) -> + cupicef.Term cupicef.prop (or B A)) => + H0 : cupicef.Term cupicef.prop C => + Hr (or__introl C A H0). + +def let__n2 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + Fl : + (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop (not A)) -> + Fr : + (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop (not A)) -> + H : cupicef.Term cupicef.prop (iff (or B A) (or C A)) -> + Hl : + (__ : cupicef.Term cupicef.prop (or B A) -> + cupicef.Term cupicef.prop (or C A)) -> + Hr : + (__ : cupicef.Term cupicef.prop (or C A) -> + cupicef.Term cupicef.prop (or B A)) -> + H0 : cupicef.Term cupicef.prop C -> + H1 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + Fl : + (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop (not A)) => + Fr : + (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop (not A)) => + H : cupicef.Term cupicef.prop (iff (or B A) (or C A)) => + Hl : + (__ : cupicef.Term cupicef.prop (or B A) -> + cupicef.Term cupicef.prop (or C A)) => + Hr : + (__ : cupicef.Term cupicef.prop (or C A) -> + cupicef.Term cupicef.prop (or B A)) => + H0 : cupicef.Term cupicef.prop C => + H1 : cupicef.Term cupicef.prop A => + Fr H0 H1. + +thm or__cancel__r : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + __ : + (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop (not A)) -> + __0 : + (__0 : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop (not A)) -> + cupicef.Term cupicef.prop (iff (iff (or B A) (or C A)) (iff B C)) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + Fl : + (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop (not A)) => + Fr : + (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop (not A)) => + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (iff (or B A) (or C A)) + (__ : cupicef.Term cupicef.prop (iff (or B A) (or C A)) => iff B C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (iff B C) + (__ : cupicef.Term cupicef.prop (iff B C) => iff (or B A) (or C A))) + (H : cupicef.Term cupicef.prop (iff (or B A) (or C A)) => + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (or B A) + (__ : cupicef.Term cupicef.prop (or B A) => or C A)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (or C A) + (__ : cupicef.Term cupicef.prop (or C A) => or B A)) + (__ : + cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (or B A) + (__ : cupicef.Term cupicef.prop (or B A) => or C A)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (or C A) + (__ : cupicef.Term cupicef.prop (or C A) => or B A))) => + iff B C) + (Hl : + (__ : cupicef.Term cupicef.prop (or B A) -> + cupicef.Term cupicef.prop (or C A)) => + Hr : + (__ : cupicef.Term cupicef.prop (or C A) -> + cupicef.Term cupicef.prop (or B A)) => + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__ : cupicef.Term cupicef.prop C => B)) + (H0 : cupicef.Term cupicef.prop B => + match____or cupicef.prop C A + (__ : cupicef.Term cupicef.prop (or C A) => C) + (H1 : cupicef.Term cupicef.prop C => H1) + (H1 : cupicef.Term cupicef.prop A => + match____False cupicef.prop + (__ : cupicef.Term cupicef.prop False => C) + (let__n1 A B C Fl Fr H Hl Hr H0 H1)) + (let__o1 A B C Fl Fr H Hl Hr H0)) + (H0 : cupicef.Term cupicef.prop C => + match____or cupicef.prop B A + (__ : cupicef.Term cupicef.prop (or B A) => B) + (H1 : cupicef.Term cupicef.prop B => H1) + (H1 : cupicef.Term cupicef.prop A => + match____False cupicef.prop + (__ : cupicef.Term cupicef.prop False => B) + (let__n2 A B C Fl Fr H Hl Hr H0 H1)) + (let__o2 A B C Fl Fr H Hl Hr H0))) + H) + (or__iff__compat__r A B C). + +thm or__comm : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + cupicef.Term cupicef.prop (iff (or A B) (or B A)) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (or A B) + (__ : cupicef.Term cupicef.prop (or A B) => or B A)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (or B A) + (__ : cupicef.Term cupicef.prop (or B A) => or A B)) + (H : cupicef.Term cupicef.prop (or A B) => + match____or cupicef.prop A B + (__ : cupicef.Term cupicef.prop (or A B) => or B A) + (H0 : cupicef.Term cupicef.prop A => or__intror B A H0) + (H0 : cupicef.Term cupicef.prop B => or__introl B A H0) H) + (H : cupicef.Term cupicef.prop (or B A) => + match____or cupicef.prop B A + (__ : cupicef.Term cupicef.prop (or B A) => or A B) + (H0 : cupicef.Term cupicef.prop B => or__intror A B H0) + (H0 : cupicef.Term cupicef.prop A => or__introl A B H0) H). + +thm or__assoc : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + cupicef.Term cupicef.prop (iff (or (or A B) C) (or A (or B C))) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (or (or A B) C) + (__ : cupicef.Term cupicef.prop (or (or A B) C) => or A (or B C))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (or A (or B C)) + (__ : cupicef.Term cupicef.prop (or A (or B C)) => or (or A B) C)) + (H : cupicef.Term cupicef.prop (or (or A B) C) => + match____or cupicef.prop (or A B) C + (__ : cupicef.Term cupicef.prop (or (or A B) C) => or A (or B C)) + (H0 : cupicef.Term cupicef.prop (or A B) => + match____or cupicef.prop A B + (__ : cupicef.Term cupicef.prop (or A B) => or A (or B C)) + (H1 : cupicef.Term cupicef.prop A => or__introl A (or B C) H1) + (H1 : cupicef.Term cupicef.prop B => + or__intror A (or B C) (or__introl B C H1)) + H0) + (H0 : cupicef.Term cupicef.prop C => + or__intror A (or B C) (or__intror B C H0)) + H) + (H : cupicef.Term cupicef.prop (or A (or B C)) => + match____or cupicef.prop A (or B C) + (__ : cupicef.Term cupicef.prop (or A (or B C)) => or (or A B) C) + (H0 : cupicef.Term cupicef.prop A => + or__introl (or A B) C (or__introl A B H0)) + (H0 : cupicef.Term cupicef.prop (or B C) => + match____or cupicef.prop B C + (__ : cupicef.Term cupicef.prop (or B C) => or (or A B) C) + (H1 : cupicef.Term cupicef.prop B => + or__introl (or A B) C (or__intror A B H1)) + (H1 : cupicef.Term cupicef.prop C => or__intror (or A B) C H1) H0) + H). + +thm iff__and : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (iff A B) -> + cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => A))) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (iff A B) => + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)) + (__ : + cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A))) => + and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => A))) + (H0 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H1 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)) + H0 H1) + H. + +thm iff__to__and : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + cupicef.Term cupicef.prop + (iff (iff A B) + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)))) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (iff A B) + (__ : cupicef.Term cupicef.prop (iff A B) => + and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => A)))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A))) + (__ : + cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A))) => + iff A B)) + (H : cupicef.Term cupicef.prop (iff A B) => + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)) + (__ : + cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A))) => + and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => A))) + (Hl : + (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + Hr : + (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)) + (H0 : cupicef.Term cupicef.prop A => Hl H0) + (H0 : cupicef.Term cupicef.prop B => Hr H0)) + H) + (H : + cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A))) => + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)) + (__ : + cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A))) => + iff A B) + (Hl : + (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + Hr : + (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)) + (H0 : cupicef.Term cupicef.prop A => Hl H0) + (H0 : cupicef.Term cupicef.prop B => Hr H0)) + H). + +def IF__then__else : + P : cupicef.Univ cupicef.prop -> + Q : cupicef.Univ cupicef.prop -> + R : cupicef.Univ cupicef.prop -> + cupicef.Univ cupicef.prop + +:= P : cupicef.Univ cupicef.prop => + Q : cupicef.Univ cupicef.prop => + R : cupicef.Univ cupicef.prop => + or (and P Q) (and (not P) R). + +def ex : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + cupicef.Univ cupicef.prop. + +def ex__intro : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (P x) -> + cupicef.Term cupicef.prop (ex A P). + +[ ] cupicef.code_guarded? _ (ex__intro _ _ _ _) --> cupicef.guarded. + +def match____ex : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + P0 : (cupicef.Term cupicef.prop (ex A P) -> cupicef.Univ s) -> + case__ex__intro : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (P x0) -> + cupicef.Term s (P0 (ex__intro A P x0 __))) -> + x : cupicef.Term cupicef.prop (ex A P) -> + cupicef.Term s (P0 x). + +[ s, A, P, P0, case__ex__intro, x0, __] + match____ex s A P P0 case__ex__intro (ex__intro A P x0 __) + --> + case__ex__intro x0 __. + +[ s, A, P, P0, s'] + match____ex s A P + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P0 x))) + --> + match____ex s' A P (x => P0 x). + +def ex__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + P0 : cupicef.Univ cupicef.prop -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (P x) -> + cupicef.Term cupicef.prop P0) -> + e : cupicef.Term cupicef.prop (ex A P) -> + cupicef.Term cupicef.prop P0 + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + P0 : cupicef.Univ cupicef.prop => + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (P x) -> + cupicef.Term cupicef.prop P0) => + e : cupicef.Term cupicef.prop (ex A P) => + match____ex cupicef.prop A P + (__ : cupicef.Term cupicef.prop (ex A P) => P0) f e. + +def ex2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + cupicef.Univ cupicef.prop. + +def ex__intro2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (P x) -> + __0 : cupicef.Term cupicef.prop (Q x) -> + cupicef.Term cupicef.prop (ex2 A P Q). + +[ ] cupicef.code_guarded? _ (ex__intro2 _ _ _ _ _ _) --> cupicef.guarded. + +def match____ex2 : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + P0 : (cupicef.Term cupicef.prop (ex2 A P Q) -> cupicef.Univ s) -> + case__ex__intro2 : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (P x0) -> + __0 : cupicef.Term cupicef.prop (Q x0) -> + cupicef.Term s (P0 (ex__intro2 A P Q x0 __ __0))) -> + x : cupicef.Term cupicef.prop (ex2 A P Q) -> + cupicef.Term s (P0 x). + +[ s, A, P, Q, P0, case__ex__intro2, x0, __, __0] + match____ex2 s A P Q P0 case__ex__intro2 (ex__intro2 A P Q x0 __ __0) + --> + case__ex__intro2 x0 __ __0. + +[ s, A, P, Q, P0, s'] + match____ex2 s A P Q + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P0 x))) + --> + match____ex2 s' A P Q (x => P0 x). + +def ex2__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + P0 : cupicef.Univ cupicef.prop -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (P x) -> + __0 : cupicef.Term cupicef.prop (Q x) -> + cupicef.Term cupicef.prop P0) -> + e : cupicef.Term cupicef.prop (ex2 A P Q) -> + cupicef.Term cupicef.prop P0 + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + P0 : cupicef.Univ cupicef.prop => + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (P x) -> + __0 : cupicef.Term cupicef.prop (Q x) -> + cupicef.Term cupicef.prop P0) => + e : cupicef.Term cupicef.prop (ex2 A P Q) => + match____ex2 cupicef.prop A P Q + (__ : cupicef.Term cupicef.prop (ex2 A P Q) => P0) f e. + +def all : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x). + +thm inst : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (all A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x0)) -> + cupicef.Term cupicef.prop (P x) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.prop)) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x0)) + (H : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (P x0)) => + P x))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (all A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x0)) + (__ : + cupicef.Term cupicef.prop + (all A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x0)) => + P x))) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (P x0)) => + H x). + +thm gen : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + B : cupicef.Univ cupicef.prop -> + f : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop (P y)) -> + __ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop (all A P) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => P y))) + (f : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop (P y)) => + cupicef.prod cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop)) + cupicef.I B + (H : cupicef.Term cupicef.prop B => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x))))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => P y))) + (f : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop (P y)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => all A P)))) + cupicef.I + (B : cupicef.Univ cupicef.prop => + f : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop (P y)) => + H : cupicef.Term cupicef.prop B => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + f x H). + +def eq : + A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) -> + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + __ : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + cupicef.Univ cupicef.prop. + +def eq__refl : + A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) -> + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + cupicef.Term cupicef.prop (eq A x x). + +[ ] cupicef.code_guarded? _ (eq__refl _ _) --> cupicef.guarded. + +def match____eq : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) -> + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + cupicef.Term cupicef.prop (eq A x __) -> + cupicef.Univ s) -> + case__eq__refl : cupicef.Term s (P x (eq__refl A x)) -> + __ : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + x0 : cupicef.Term cupicef.prop (eq A x __) -> + cupicef.Term s (P __ x0). + +[ s, A, x, P, case__eq__refl] + match____eq s A x P case__eq__refl {x} (eq__refl A x) + --> + case__eq__refl. + +[ s, A, x, P, s'] + match____eq s A x + (__ => x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P __ x))) + --> + match____eq s' A x (__ => x => P __ x). + +def eq__rect : + A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) -> + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) -> + y : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + e : cupicef.Term cupicef.prop (eq A x y) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) => + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + y : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + e : cupicef.Term cupicef.prop (eq A x y) => + match____eq (cupicef.type (cupicef.s cupicef.z)) A x + (y0 : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + __ : cupicef.Term cupicef.prop (eq A x y0) => + P y0) + f y e. + +def eq__ind : + A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) -> + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + cupicef.Univ cupicef.prop) -> + f : cupicef.Term cupicef.prop (P x) -> + y : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + e : cupicef.Term cupicef.prop (eq A x y) -> + cupicef.Term cupicef.prop (P y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) => + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + cupicef.Univ cupicef.prop) => + f : cupicef.Term cupicef.prop (P x) => + y : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + e : cupicef.Term cupicef.prop (eq A x y) => + match____eq cupicef.prop A x + (y0 : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + __ : cupicef.Term cupicef.prop (eq A x y0) => + P y0) + f y e. + +def eq__rec : + A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) -> + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + cupicef.Univ cupicef.set) -> + f : cupicef.Term cupicef.set (P x) -> + y : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + e : cupicef.Term cupicef.prop (eq A x y) -> + cupicef.Term cupicef.set (P y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) => + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + cupicef.Univ cupicef.set) => + eq__rect A x + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P). + +def let__f : + A : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + h1 : cupicef.Term cupicef.prop A -> + h2 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop False) -> + cupicef.Term cupicef.prop False + +:= A : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + h1 : cupicef.Term cupicef.prop A => + h2 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop False) => + h2 h1. + +thm absurd : + A : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop A -> + __0 : cupicef.Term cupicef.prop (not A) -> + cupicef.Term cupicef.prop C + +:= cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (C : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop)) + cupicef.I A + (h1 : cupicef.Term cupicef.prop A => + cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => False)) + (h2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop False) => + C))))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (C : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (__ : cupicef.Term cupicef.prop A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + not A) (__0 : cupicef.Term cupicef.prop (not A) => C))))) + cupicef.I + (A : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + h1 : cupicef.Term cupicef.prop A => + h2 : + (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop False) => + match____False cupicef.prop (__ : cupicef.Term cupicef.prop False => C) + (let__f A C h1 h2)). + +def eq__sym : + A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) -> + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + y : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + __ : cupicef.Term cupicef.prop (eq A x y) -> + cupicef.Term cupicef.prop (eq A y x) + +:= A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) => + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + y : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + H : cupicef.Term cupicef.prop (eq A x y) => + match____eq cupicef.prop A x + (y0 : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + __ : cupicef.Term cupicef.prop (eq A x y0) => + eq A y0 x) + (eq__refl A x) y H. + +def eq__trans : + A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) -> + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + y : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + z : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + __ : cupicef.Term cupicef.prop (eq A x y) -> + __0 : cupicef.Term cupicef.prop (eq A y z) -> + cupicef.Term cupicef.prop (eq A x z) + +:= A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) => + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + y : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + z : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + H : cupicef.Term cupicef.prop (eq A x y) => + H0 : cupicef.Term cupicef.prop (eq A y z) => + match____eq cupicef.prop A y + (y0 : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + __ : cupicef.Term cupicef.prop (eq A y y0) => + eq A x y0) + H z H0. + +def f__equal : + A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) -> + B : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) B) -> + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + y : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + __ : cupicef.Term cupicef.prop (eq A x y) -> + cupicef.Term cupicef.prop (eq B (f x) (f y)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) => + B : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) B) => + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + y : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + H : cupicef.Term cupicef.prop (eq A x y) => + match____eq cupicef.prop A x + (y0 : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + __ : cupicef.Term cupicef.prop (eq A x y0) => + eq B (f x) (f y0)) + (eq__refl B (f x)) y H. + +thm not__eq__sym : + A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) -> + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + y : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + __ : cupicef.Term cupicef.prop (not (eq A x y)) -> + cupicef.Term cupicef.prop (not (eq A y x)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) => + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + y : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + cupicef.cast + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (not (eq A x y)) + (h1 : cupicef.Term cupicef.prop (not (eq A x y)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eq A y x) + (h2 : cupicef.Term cupicef.prop (eq A y x) => False))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (not (eq A x y)) + (__ : cupicef.Term cupicef.prop (not (eq A x y)) => not (eq A y x))) + cupicef.I + (h1 : cupicef.Term cupicef.prop (not (eq A x y)) => + h2 : cupicef.Term cupicef.prop (eq A y x) => + h1 + (match____eq cupicef.prop A y + (y0 : + cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + __ : cupicef.Term cupicef.prop (eq A y y0) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (not (eq A y0 y)) + (h3 : cupicef.Term cupicef.prop (not (eq A y0 y)) => eq A y0 y)) + (h3 : cupicef.Term cupicef.prop (not (eq A y y)) => eq__refl A y) + x h2 h1)). + +def eq__ind__r : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + __ : cupicef.Term cupicef.prop (P x) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y x) -> + cupicef.Term cupicef.prop (P y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + H : cupicef.Term cupicef.prop (P x) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y x) => + eq__ind + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P y0) H y + (eq__sym + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y x H0). + +def eq__rec__r : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.set) -> + __ : cupicef.Term cupicef.set (P x) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y x) -> + cupicef.Term cupicef.set (P y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.set) => + H : cupicef.Term cupicef.set (P x) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y x) => + eq__rec + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P y0) H y + (eq__sym + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y x H0). + +def eq__rect__r : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + H : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y x) => + eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P y0) H y + (eq__sym + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y x H0). + +def rew__opp__r : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P y)) + (eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P (eq__rect__r A y P a x H) y H) + a) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I (P y0) + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y0) => + eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P y0)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P (eq__rect__r A y0 P a0 x e) y0 e) + a0)) + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + a0) + y H a. + +def rew__opp__l : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (eq__rect__r A y P + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P a y H) + x H) + a) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) => + eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (eq__rect__r A y0 P + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P a y0 e) + x e) + a) + (eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + a) + y H. + +thm f__equal2 : + A1 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + A2 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 -> + y1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 -> + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 -> + y2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 -> + __ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 y1) -> + __0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2) (f y1 y2)) + +:= A1 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + A2 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 => + y1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 => + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 => + y2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 => + H : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 y1) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 => + __ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 y) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) + (__0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) => + eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2) (f y y2))) + (H0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 => + __ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y) => + eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2) (f x1 y)) + (eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2)) + y2 H0) + y1 H. + +thm f__equal3 : + A1 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + A2 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + A3 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 -> + __1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A3 -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 -> + y1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 -> + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 -> + y2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 -> + x3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A3 -> + y3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A3 -> + __ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 y1) -> + __0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) -> + __1 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3) (f y1 y2 y3)) + +:= A1 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + A2 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + A3 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 -> + __1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A3 -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 => + y1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 => + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 => + y2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 => + x3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A3 => + y3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A3 => + H : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 y1) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 => + __ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 y) => + cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) + (__0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) + (__1 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) => + eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3) (f y y2 y3)))) + (H0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 => + __ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) + (__0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) => + eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3) (f x1 y y3))) + (H1 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A3 => + __ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y) => + eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3) (f x1 x2 y)) + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3)) + y3 H1) + y2 H0) + y1 H. + +thm f__equal4 : + A1 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + A2 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + A3 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + A4 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 -> + __1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A3 -> + __2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A4 -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 -> + y1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 -> + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 -> + y2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 -> + x3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A3 -> + y3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A3 -> + x4 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A4 -> + y4 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A4 -> + __ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 y1) -> + __0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) -> + __1 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) -> + __2 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3 x4) (f y1 y2 y3 y4)) + +:= A1 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + A2 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + A3 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + A4 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 -> + __1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A3 -> + __2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A4 -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 => + y1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 => + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 => + y2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 => + x3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A3 => + y3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A3 => + x4 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A4 => + y4 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A4 => + H : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 y1) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 => + __ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 y) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) + (__0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) => + cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) + (__1 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) + (__2 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) => + eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3 x4) (f y y2 y3 y4))))) + (H0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 => + __ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y) => + cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) + (__0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) + (__1 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) => + eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3 x4) (f x1 y y3 y4)))) + (H1 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A3 => + __ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) + (__0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) => + eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3 x4) (f x1 x2 y y4))) + (H2 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) => + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A4 => + __ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y) => + eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3 x4) (f x1 x2 x3 y)) + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3 x4)) + y4 H2) + y3 H1) + y2 H0) + y1 H. + +thm f__equal5 : + A1 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + A2 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + A3 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + A4 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + A5 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 -> + __1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A3 -> + __2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A4 -> + __3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A5 -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 -> + y1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 -> + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 -> + y2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 -> + x3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A3 -> + y3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A3 -> + x4 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A4 -> + y4 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A4 -> + x5 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A5 -> + y5 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A5 -> + __ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 y1) -> + __0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) -> + __1 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) -> + __2 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) -> + __3 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A5) + x5 y5) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3 x4 x5) (f y1 y2 y3 y4 y5)) + +:= A1 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + A2 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + A3 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + A4 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + A5 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 -> + __1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A3 -> + __2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A4 -> + __3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A5 -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 => + y1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 => + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 => + y2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 => + x3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A3 => + y3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A3 => + x4 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A4 => + y4 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A4 => + x5 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A5 => + y5 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A5 => + H : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 y1) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A1 => + __ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 y) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) + (__0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) + (__1 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) + (__2 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A5) + x5 y5) + (__3 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A5) + x5 y5) => + eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3 x4 x5) (f y y2 y3 y4 y5)))))) + (H0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A2 => + __ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) + (__0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) + (__1 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A5) + x5 y5) + (__2 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A5) + x5 y5) => + eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3 x4 x5) (f x1 y y3 y4 y5))))) + (H1 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A3 => + __ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) + (__0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A5) + x5 y5) + (__1 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A5) + x5 y5) => + eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3 x4 x5) (f x1 x2 y y4 y5)))) + (H2 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) => + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A4 => + __ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A5) + x5 y5) + (__0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A5) + x5 y5) => + eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3 x4 x5) (f x1 x2 x3 y y5))) + (H3 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A5) + x5 y5) => + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A5) + x5 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A5 => + __ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A5) + x5 y) => + eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3 x4 x5) (f x1 x2 x3 x4 y)) + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3 x4 x5)) + y5 H3) + y4 H2) + y3 H1) + y2 H0) + y1 H. + +def f__equal__compose : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + C : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + g : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C) -> + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a b) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + (g (f a)) (g (f b)))) + (f__equal + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + g (f a) (f b) + (f__equal + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f a b e)) + (f__equal + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + g (f a0)) + a b e)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + C : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + g : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C) => + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a b) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a y) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + (g (f a)) (g (f y)))) + (f__equal + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + g (f a) (f y) + (f__equal + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f a y e0)) + (f__equal + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + g (f a0)) + a y e0)) + (eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + (g (f a)) (g (f a)))) + (f__equal + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + g (f a0)) + a a + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a))) + b e. + +def eq__trans__refl__l : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (eq__trans + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x y + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x) + e) + e) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0)) + (eq__trans + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x y0 + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x) + e0) + e0) + (eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + y e. + +def eq__trans__refl__r : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (eq__trans + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y y e + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y)) + e) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0)) + (eq__trans + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0 y0 e0 + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0)) + e0) + (eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + y e. + +def eq__sym__involutive : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (eq__sym + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y x + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y e)) + e) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0)) + (eq__sym + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0 x + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0 e0)) + e0) + (eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + y e. + +def eq__trans__sym__inv__l : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y y)) + (eq__trans + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y x y + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y e) + e) + (eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0 y0)) + (eq__trans + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0 x y0 + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0 e0) + e0) + (eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0)) + (eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + y e. + +def eq__trans__sym__inv__r : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (eq__trans + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y x e + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y e)) + (eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (eq__trans + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0 x e0 + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0 e0)) + (eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + (eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + y e. + +def eq__trans__assoc : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + t : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + e_27_ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y z) -> + e_27__27_ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + z t) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x t)) + (eq__trans + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y t e + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y z t e_27_ e_27__27_)) + (eq__trans + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x z t + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y z e e_27_) + e_27__27_)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + t : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + e_27_ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y z) => + e_27__27_ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + z t) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + z + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + z y0) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0)) + (eq__trans + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y y0 e + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y z y0 e_27_ e0)) + (eq__trans + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x z y0 + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y z e e_27_) + e0)) + (eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x z)) + (eq__trans + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x z z + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y z e e_27_) + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + z))) + t e_27__27_. + +def let__e : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) -> + Hf : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a (f a))) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a (f a)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) => + Hf : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a (f a))) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hf a. + +def let__a0 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) -> + Hf : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a (f a))) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) => + Hf : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a (f a))) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + f a. + +def let__e0 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) -> + Hf : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a (f a))) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a (f a)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) => + Hf : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a (f a))) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hf a. + +def let__a1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) -> + Hf : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a (f a))) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) => + Hf : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a (f a))) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + f a. + +def eq__id__comm__l : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) -> + Hf : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a (f a))) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) (f (f a)))) + (f__equal + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + f a (f a) (Hf a)) + (Hf (f a))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) => + Hf : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a (f a))) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.cast cupicef.prop cupicef.prop + ((e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) (f a)) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) (f (f a)))) + (match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + __ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a y) => + eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) (f y)) + e (f a) (Hf a)) + (Hf (f a))) + (eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a))) + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) (f (f a)))) + (f__equal + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + f a (f a) (Hf a)) + (Hf (f a))) + cupicef.I + (eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) (f a))) + (eq__trans + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a (f a) + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a (f a) (Hf a)) + (Hf a)) + (e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) (f a)) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) (f (f a)))) + (match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + __ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a y) => + eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) (f y)) + e (f a) (Hf a)) + (Hf (f a))) + (match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a y) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y (f y))) + (match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + __ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a y0) => + eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y (f y0)) + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y a (f a) + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a y e0) + (Hf a)) + y e0) + (Hf y)) + (match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a y) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a y)) + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a a y + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a a + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a)) + e0) + e0) + (eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a a)) + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a)) + (f a) (let__e0 A f Hf a)) + (f a) (let__e A f Hf a)) + (eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a)) + (eq__trans__sym__inv__l A a (f a) (Hf a))). + +def let__Hfsymf : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) -> + Hf : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a0 (f a0)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) => + Hf : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a)) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq__sym + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a0) a0 (Hf a0). + +def let__e1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) -> + Hf : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f a)) (f (f (f a))))) + (f__equal + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + f (f a) (f (f a)) (let__Hfsymf A f Hf a (f a))) + (let__Hfsymf A f Hf a (f (f a)))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) => + Hf : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a)) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq__id__comm__l A f (let__Hfsymf A f Hf a) (f a). + +def let__e2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) -> + Hf : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f a)) (f (f (f a)))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) => + Hf : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a)) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + let__Hfsymf A f Hf a (f (f a)). + +def let__e3 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) -> + Hf : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) (f (f a)))) + (f__equal + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + f a (f a) (let__Hfsymf A f Hf a a)) + (let__Hfsymf A f Hf a (f a))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) => + Hf : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a)) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq__id__comm__l A f (let__Hfsymf A f Hf a) a. + +def let__e4 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) -> + Hf : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) (f (f a))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) => + Hf : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a)) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + let__Hfsymf A f Hf a (f a). + +def let__e5 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) -> + Hf : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) => + Hf : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a)) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hf a. + +def eq__id__comm__r : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) -> + Hf : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f a)) (f a))) + (f__equal + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + f (f a) a (Hf a)) + (Hf (f a))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) => + Hf : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + 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cupicef.I A) + (f (f (f a))) (f (f a)))) + (Hf (f (f a))) + (e1 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f (f a))) (f (f a))) => + eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f (f a))) (f (f a)))) + e1 (Hf (f (f a)))) + (eq__refl + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f (f a))) (f (f a)))) + (Hf (f (f a)))) + (eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f (f a))) (f (f (f a))) (f (f a)) + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f (f a)))) + (Hf (f (f a)))) + (eq__trans__refl__l A (f (f (f a))) (f (f a)) + (Hf (f (f a)))))) + a (let__e5 A f Hf a))) + (let__Hfsymf A f Hf a (f a)) (let__e3 A f Hf a)) + (let__Hfsymf A f Hf a (f (f a))) (let__e1 A f Hf a))) + (eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f a))) + (eq__trans__sym__inv__l A (f (f (f a))) (f (f a)) (Hf (f (f a))))). + +thm eq__refl__map__distr : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f x))) + (f__equal + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f x x + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + (eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f x))) + (eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x)). + +def eq__trans__map__distr : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + e_27_ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y z) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f z))) + (f__equal + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f x z + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y z e e_27_)) + (eq__trans + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y) (f z) + (f__equal + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f x y e) + (f__equal + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f y z e_27_))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + e_27_ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y z) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y y0) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y0))) + (f__equal + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f x y0 + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y y0 e e0)) + (eq__trans + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y) (f y0) + (f__equal + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f x y e) + (f__equal + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f y y0 e0))) + (eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y))) + (eq__trans + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y) (f y) + (f__equal + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f x y e) + (f__equal + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f y y + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y)))) + z e_27_. + +def eq__sym__map__distr : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f y) (f x))) + (eq__sym + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y) + (f__equal + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f x y e)) + (f__equal + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f y x + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y e))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f y0) (f x))) + (eq__sym + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y0) + (f__equal + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f x y0 e0)) + (f__equal + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f y0 x + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0 e0))) + (eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f x))) + (f__equal + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f x x + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)))) + y e. + +def eq__trans__sym__distr : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + e_27_ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y z) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + z x)) + (eq__sym + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x z + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y z e e_27_)) + (eq__trans + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + z y x + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y z e_27_) + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y e))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0 z) + (e_27_ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0 z) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + z x)) + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x z + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0 z e0 e_27_)) + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + z y0 x + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0 z e_27_) + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0 e0)))) + (e_27_ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x z) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0 x)) + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0 + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x y0 + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x) + e0)) + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0 x x + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0 e0) + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)))) + (eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x x + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)))) + z e_27_) + y e. + +thm eq__trans__rew__distr : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + e_27_ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y z) -> + k : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P z)) + (eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P k z + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y z e e_27_)) + (eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y P + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P k y e) + z e_27_)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) => + cupicef.prod cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop)) + cupicef.I + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0 z) + (e_27_ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0 z) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I (P x) + (k : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P z)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P k z + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0 z e0 e_27_)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0 P + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P k y0 e0) + z e_27_)))) + (e_27_ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x z) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I (P x) + (k : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P y0)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P k y0 + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x y0 + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x) + e0)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P k x + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + y0 e0))) + (k : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P k x + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + x + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x))) + z e_27_) + y e. + +thm rew__const : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + k : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) P -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I P) + (eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P) k + y e) + k) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I P + (k : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) P => + eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I P) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P) + k y0 e0) + k)) + (k : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) P => + eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I P) + k) + y e. + +def subrelation : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) -> + R_27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) => + R_27_ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I B + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x y) + (__ : cupicef.Term cupicef.prop (R x y) => R_27_ x y))). + +def unique : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (P x_27_) + (__ : cupicef.Term cupicef.prop (P x_27_) => + eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_))). + +def uniqueness : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (P x) + (__ : cupicef.Term cupicef.prop (P x) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (P y) + (__0 : cupicef.Term cupicef.prop (P y) => + eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))). + +thm unique__existence : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + cupicef.Term cupicef.prop + (iff + (and + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x)) + (uniqueness A P)) + (ex A + (unique A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x)))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (and + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x)) + (uniqueness A P)) + (__ : + cupicef.Term cupicef.prop + (and + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x)) + (uniqueness A P)) => + ex A + (unique A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x)))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (ex A + (unique A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x))) + (__ : + cupicef.Term cupicef.prop + (ex A + (unique A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x))) => + and + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x)) + (uniqueness A P))) + (H : + cupicef.Term cupicef.prop + (and + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x)) + (uniqueness A P)) => + match____and cupicef.prop + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x)) + (uniqueness A P) + (__ : + cupicef.Term cupicef.prop + (and + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x)) + (uniqueness A P)) => + ex A + (unique A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x))) + (H0 : + cupicef.Term cupicef.prop + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x)) => + match____ex cupicef.prop A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x) + (__ : + cupicef.Term cupicef.prop + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (uniqueness A P) + (__0 : cupicef.Term cupicef.prop (uniqueness A P) => + ex A + (unique A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x)))) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hx : cupicef.Term cupicef.prop (P x) => + Huni : cupicef.Term cupicef.prop (uniqueness A P) => + ex__intro A + (unique A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x0)) + x + (cupicef.cast cupicef.prop cupicef.prop + (and (P x) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x_27_ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P x_27_) + (__ : cupicef.Term cupicef.prop (P x_27_) => + eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_)))) + (unique A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x0) + x) + cupicef.I + (conj (P x) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x_27_ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P x_27_) + (__ : cupicef.Term cupicef.prop (P x_27_) => + eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_))) + Hx + (x_27_ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H1 : cupicef.Term cupicef.prop (P x_27_) => + Huni x x_27_ Hx H1)))) + H0) + H) + (H : + cupicef.Term cupicef.prop + (ex A + (unique A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x))) => + match____ex cupicef.prop A + (unique A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x)) + (__ : + cupicef.Term cupicef.prop + (ex A + (unique A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x))) => + and + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x)) + (uniqueness A P)) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (unique A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x0) + x) => + match____and cupicef.prop + ((x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x0) + x) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + ((x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x0) x_27_) + (__ : + cupicef.Term cupicef.prop + ((x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x0) x_27_) => + eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_))) + (__ : + cupicef.Term cupicef.prop + (and (P x) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x_27_ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P x_27_) + (__ : cupicef.Term cupicef.prop (P x_27_) => + eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_)))) => + and + (ex A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x0)) + (uniqueness A P)) + (Hx : cupicef.Term cupicef.prop (P x) => + Huni : + (x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (P x_27_) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_)) => + conj + (ex A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x0)) + (uniqueness A P) + (ex__intro A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x0) + x Hx) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I A + (x_27_ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x_27__27_ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (P x_27_) + (Hx_27_ : cupicef.Term cupicef.prop (P x_27_) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P x_27__27_) + (Hx_27__27_ : + cupicef.Term cupicef.prop (P x_27__27_) => + eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x_27_ x_27__27_))))) + (uniqueness A P) cupicef.I + (x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x_27__27_ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hx_27_ : cupicef.Term cupicef.prop (P x_27_) => + Hx_27__27_ : cupicef.Term cupicef.prop (P x_27__27_) => + eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x_27_ x x_27__27_ + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_ (Huni x_27_ Hx_27_)) + (Huni x_27__27_ Hx_27__27_)))) + H0) + H). + +def let__H : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + H : + cupicef.Term cupicef.prop + (ex A + (unique A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x))) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H0 : + cupicef.Term cupicef.prop + (unique A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x0) x) -> + Hp : cupicef.Term cupicef.prop (P x) -> + Huniq : + (x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (P x_27_) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_)) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + H1 : + cupicef.Term cupicef.prop + (ex A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x0) (Q x0))) -> + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H2 : cupicef.Term cupicef.prop (and (P x0) (Q x0)) -> + HPx0 : cupicef.Term cupicef.prop (P x0) -> + HQx0 : cupicef.Term cupicef.prop (Q x0) -> + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + HPx1 : cupicef.Term cupicef.prop (P x1) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 x1) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + H : + cupicef.Term cupicef.prop + (ex A + (unique A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x))) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (unique A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x0) + x) => + Hp : cupicef.Term cupicef.prop (P x) => + Huniq : + (x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (P x_27_) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_)) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + H1 : + cupicef.Term cupicef.prop + (ex A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x0) (Q x0))) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H2 : cupicef.Term cupicef.prop (and (P x0) (Q x0)) => + HPx0 : cupicef.Term cupicef.prop (P x0) => + HQx0 : cupicef.Term cupicef.prop (Q x0) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + HPx1 : cupicef.Term cupicef.prop (P x1) => + eq__trans + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 x x1 + (eq__sym + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x0 (Huniq x0 HPx0)) + (Huniq x1 HPx1). + +thm forall__exists__unique__domain__coincide : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + __ : + cupicef.Term cupicef.prop + (ex A + (unique A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x))) -> + Q : + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + cupicef.Term cupicef.prop + (iff + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (P x) + (__0 : cupicef.Term cupicef.prop (P x) => Q x))) + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x) (Q x)))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + H : + cupicef.Term cupicef.prop + (ex A + (unique A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x))) => + match____ex cupicef.prop A + (unique A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x)) + (__ : + cupicef.Term cupicef.prop + (ex A + (unique A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x))) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Q : + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + iff + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + P x) (__0 : cupicef.Term cupicef.prop (P x) => Q x))) + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x) (Q x))))) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (unique A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x0) + x) => + match____and cupicef.prop + ((x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x0) x) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + ((x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x0) x_27_) + (__ : + cupicef.Term cupicef.prop + ((x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x0) x_27_) => + eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_))) + (__ : + cupicef.Term cupicef.prop + (and (P x) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x_27_ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P x_27_) + (__ : cupicef.Term cupicef.prop (P x_27_) => + eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_)))) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (Q : + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + iff + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + P x0) (__0 : cupicef.Term cupicef.prop (P x0) => Q x0))) + (ex A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x0) (Q x0))))) + (Hp : cupicef.Term cupicef.prop (P x) => + Huniq : + (x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (P x_27_) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_)) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + conj + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + P x0) (__ : cupicef.Term cupicef.prop (P x0) => Q x0))) + (__ : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (P x0) -> + cupicef.Term cupicef.prop (Q x0)) => + ex A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x0) (Q x0)))) + (cupicef.prod cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (ex A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x0) (Q x0))) + (__ : + cupicef.Term cupicef.prop + (ex A + (x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x0) (Q x0))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + P x0) (__0 : cupicef.Term cupicef.prop (P x0) => Q x0)))) + (H1 : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (P x0) -> + cupicef.Term cupicef.prop (Q x0)) => + ex__intro A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x0) (Q x0)) + x (conj (P x) (Q x) Hp (H1 x Hp))) + (H1 : + cupicef.Term cupicef.prop + (ex A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x0) (Q x0))) => + match____ex cupicef.prop A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x0) (Q x0)) + (__ : + cupicef.Term cupicef.prop + (ex A + (x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x0) (Q x0))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + P x0) (__0 : cupicef.Term cupicef.prop (P x0) => Q x0))) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H2 : cupicef.Term cupicef.prop (and (P x0) (Q x0)) => + match____and cupicef.prop (P x0) (Q x0) + (__ : cupicef.Term cupicef.prop (and (P x0) (Q x0)) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P x1) + (__0 : cupicef.Term cupicef.prop (P x1) => Q x1))) + (HPx0 : cupicef.Term cupicef.prop (P x0) => + HQx0 : cupicef.Term cupicef.prop (Q x0) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + HPx1 : cupicef.Term cupicef.prop (P x1) => + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + __ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P y) (HPx2 : cupicef.Term cupicef.prop (P y) => Q y)) + (HPx2 : cupicef.Term cupicef.prop (P x0) => HQx0) x1 + (let__H A P H x H0 Hp Huniq Q H1 x0 H2 HPx0 HQx0 x1 HPx1) + HPx1) + H2) + H1)) + H0) + H. + +def let__i : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + H : + (Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + cupicef.Term cupicef.prop + (iff + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (P x) + (__ : cupicef.Term cupicef.prop (P x) => Q x))) + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x) (Q x))))) -> + cupicef.Term cupicef.prop + (iff + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (P x) + (__ : cupicef.Term cupicef.prop (P x) => P x))) + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x) (P x)))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + H : + (Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + cupicef.Term cupicef.prop + (iff + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + P x) (__ : cupicef.Term cupicef.prop (P x) => Q x))) + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x) (Q x))))) => + H P. + +def let__i0 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + H : + (Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + cupicef.Term cupicef.prop + (iff + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (P x) + (__ : cupicef.Term cupicef.prop (P x) => Q x))) + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x) (Q x))))) -> + H0 : + (__ : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (P x) -> + cupicef.Term cupicef.prop (P x)) -> + cupicef.Term cupicef.prop + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x) (P x)))) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H1 : cupicef.Term cupicef.prop (and (P x) (P x)) -> + Hx : cupicef.Term cupicef.prop (P x) -> + __tmp : cupicef.Term cupicef.prop (P x) -> + __tmp0 : + (__ : + cupicef.Term cupicef.prop + (ex A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x0) (P x0))) -> + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term cupicef.prop (P x0) -> + cupicef.Term cupicef.prop (P x0)) -> + cupicef.Term cupicef.prop + (iff + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (P x0) + (__ : cupicef.Term cupicef.prop (P x0) => + (x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_) + x0))) + (ex A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x0) + ((x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_) + x0)))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + H : + (Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + cupicef.Term cupicef.prop + (iff + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + P x) (__ : cupicef.Term cupicef.prop (P x) => Q x))) + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x) (Q x))))) => + H0 : + (__ : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (P x) -> + cupicef.Term cupicef.prop (P x)) -> + cupicef.Term cupicef.prop + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x) (P x)))) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H1 : cupicef.Term cupicef.prop (and (P x) (P x)) => + Hx : cupicef.Term cupicef.prop (P x) => + __tmp : cupicef.Term cupicef.prop (P x) => + __tmp0 : + (__ : + cupicef.Term cupicef.prop + (ex A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x0) (P x0))) -> + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term cupicef.prop (P x0) -> + cupicef.Term cupicef.prop (P x0)) => + H + (x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_). + +thm forall__exists__coincide__unique__domain : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + __ : + (Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + cupicef.Term cupicef.prop + (iff + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (P x) + (__ : cupicef.Term cupicef.prop (P x) => Q x))) + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x) (Q x))))) -> + cupicef.Term cupicef.prop + (ex A + (unique A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + H : + (Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + cupicef.Term cupicef.prop + (iff + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + P x) (__ : cupicef.Term cupicef.prop (P x) => Q x))) + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x) (Q x))))) => + match____and cupicef.prop + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (P x) + (__ : cupicef.Term cupicef.prop (P x) => P x))) + (__ : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (P x) -> + cupicef.Term cupicef.prop (P x)) => + ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x) (P x)))) + (cupicef.prod cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x) (P x))) + (__ : + cupicef.Term cupicef.prop + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x) (P x))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (P x) + (__0 : cupicef.Term cupicef.prop (P x) => P x)))) + (__ : + cupicef.Term cupicef.prop + (and + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P x) (__ : cupicef.Term cupicef.prop (P x) => P x))) + (__ : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (P x) -> + cupicef.Term cupicef.prop (P x)) => + ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x) (P x)))) + (cupicef.prod cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x) (P x))) + (__ : + cupicef.Term cupicef.prop + (ex A + (x : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + and (P x) (P x))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P x) (__0 : cupicef.Term cupicef.prop (P x) => P x))))) => + ex A + (unique A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x))) + (H0 : + (__ : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (P x) -> + cupicef.Term cupicef.prop (P x)) -> + cupicef.Term cupicef.prop + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x) (P x)))) => + match____ex cupicef.prop A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x) (P x)) + (__ : + cupicef.Term cupicef.prop + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x) (P x))) => + cupicef.prod + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.rule + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x) (P x))) + (__0 : + cupicef.Term cupicef.prop + (ex A + (x : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x) (P x))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + P x) (__1 : cupicef.Term cupicef.prop (P x) => P x)))) + (__0 : + (__0 : + cupicef.Term cupicef.prop + (ex A + (x : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x) (P x))) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __1 : cupicef.Term cupicef.prop (P x) -> + cupicef.Term cupicef.prop (P x)) => + ex A + (unique A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x)))) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H1 : cupicef.Term cupicef.prop (and (P x) (P x)) => + match____and cupicef.prop (P x) (P x) + (__ : cupicef.Term cupicef.prop (and (P x) (P x)) => + cupicef.prod + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.rule + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (ex A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x0) (P x0))) + (__0 : + cupicef.Term cupicef.prop + (ex A + (x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + and (P x0) (P x0))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P x0) + (__1 : cupicef.Term cupicef.prop (P x0) => P x0)))) + (__0 : + (__0 : + cupicef.Term cupicef.prop + (ex A + (x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + and (P x0) (P x0))) -> + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __1 : cupicef.Term cupicef.prop (P x0) -> + cupicef.Term cupicef.prop (P x0)) => + ex A + (unique A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x0)))) + (Hx : cupicef.Term cupicef.prop (P x) => + __tmp : cupicef.Term cupicef.prop (P x) => + __tmp0 : + (__ : + cupicef.Term cupicef.prop + (ex A + (x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + and (P x0) (P x0))) -> + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term cupicef.prop (P x0) -> + cupicef.Term cupicef.prop (P x0)) => + ex__intro A + (unique A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x0)) + x + (conj (P x) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x_27_ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P x_27_) + (__ : cupicef.Term cupicef.prop (P x_27_) => + eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_))) + Hx + (match____and cupicef.prop + (cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P x0) + (__ : cupicef.Term cupicef.prop (P x0) => + (x_27_ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_) + x0))) + (__ : + (x0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (P x0) -> + cupicef.Term cupicef.prop + ((x_27_ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_) + x0)) => + ex A + (x0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + and (P x0) + ((x_27_ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_) + x0)))) + (cupicef.prod cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (ex A + (x0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + and (P x0) + ((x_27_ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_) + x0))) + (__ : + cupicef.Term cupicef.prop + (ex A + (x0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + and (P x0) + ((x_27_ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + eq + (cupicef.cast + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_) + x0))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P x0) + (__0 : cupicef.Term cupicef.prop (P x0) => + (x_27_ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_) + x0)))) + (__ : + cupicef.Term cupicef.prop + (and + (cupicef.prod + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.prop + (cupicef.rule + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.prop) + cupicef.I + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P x0) + (__ : cupicef.Term cupicef.prop (P x0) => + eq + (cupicef.cast + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x0))) + (__ : + (x0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (P x0) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x0)) => + ex A + (x0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + and (P x0) + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x0)))) + (cupicef.prod cupicef.prop + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (ex A + (x0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + and (P x0) + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x0))) + (__ : + cupicef.Term cupicef.prop + (ex A + (x0 : + cupicef.Term + (cupicef.type + (cupicef.s cupicef.z)) + A => + and (P x0) + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A) + x x0))) => + cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (P x0) + (__0 : cupicef.Term cupicef.prop (P x0) => + eq + (cupicef.cast + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x0))))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x_27_ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P x_27_) + (__0 : cupicef.Term cupicef.prop (P x_27_) => + eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_))) + (__tmp1 : + (__ : + (x0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (P x0) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x0)) -> + cupicef.Term cupicef.prop + (ex A + (x0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + and (P x0) + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x0)))) => + Huniq : + (__ : + cupicef.Term cupicef.prop + (ex A + (x0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + and (P x0) + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x0))) -> + x0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term cupicef.prop (P x0) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x0)) => + Huniq + (ex__intro A + (x0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + and (P x0) + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x0)) + x + (conj (P x) + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) + Hx + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)))) + (let__i0 A P H H0 x H1 Hx __tmp __tmp0)))) + H1) + (H0 + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H1 : cupicef.Term cupicef.prop (P x) => + H1))) + (let__i A P H). + +def inhabited : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Univ cupicef.prop. + +def inhabits : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (inhabited A). + +[ ] cupicef.code_guarded? _ (inhabits _ _) --> cupicef.guarded. + +def match____inhabited : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : (cupicef.Term cupicef.prop (inhabited A) -> cupicef.Univ s) -> + case__inhabits : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term s (P (inhabits A __))) -> + x : cupicef.Term cupicef.prop (inhabited A) -> + cupicef.Term s (P x). + +[ s, A, P, case__inhabits, __] + match____inhabited s A P case__inhabits (inhabits A __) + --> + case__inhabits __. + +[ s, A, P, s'] + match____inhabited s A + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____inhabited s' A (x => P x). + +def inhabited__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : cupicef.Univ cupicef.prop -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop P) -> + i : cupicef.Term cupicef.prop (inhabited A) -> + cupicef.Term cupicef.prop P + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : cupicef.Univ cupicef.prop => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop P) => + i : cupicef.Term cupicef.prop (inhabited A) => + match____inhabited cupicef.prop A + (__ : cupicef.Term cupicef.prop (inhabited A) => P) f i. + +thm exists__inhabited : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + __ : + cupicef.Term cupicef.prop + (ex A (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x)) -> + cupicef.Term cupicef.prop (inhabited A) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + H : + cupicef.Term cupicef.prop + (ex A (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x)) => + match____ex cupicef.prop A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x) + (__ : + cupicef.Term cupicef.prop + (ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x)) => + inhabited A) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (P x) => + inhabits A x) + H. + +thm inhabited__covariant : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + __ : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + __0 : cupicef.Term cupicef.prop (inhabited A) -> + cupicef.Term cupicef.prop (inhabited B) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + H : cupicef.Term cupicef.prop (inhabited A) => + match____inhabited cupicef.prop A + (__ : cupicef.Term cupicef.prop (inhabited A) => inhabited B) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + inhabits B (f x)) + H. + +thm eq__stepl : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + __0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x z) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + z y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H1 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + H2 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x z) => + eq__ind + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (z0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + z0 y) + H1 z H2. + +thm iff__stepl : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (iff A B) -> + __0 : cupicef.Term cupicef.prop (iff A C) -> + cupicef.Term cupicef.prop (iff C B) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + H : cupicef.Term cupicef.prop (iff A B) => + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)) + (__ : + cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (iff A C) + (__0 : cupicef.Term cupicef.prop (iff A C) => iff C B)) + (H0 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H1 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + H2 : cupicef.Term cupicef.prop (iff A C) => + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__ : cupicef.Term cupicef.prop C => A)) + (__ : + cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__ : cupicef.Term cupicef.prop C => A))) => + iff C B) + (H3 : + (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop C) => + H4 : + (__ : cupicef.Term cupicef.prop C -> cupicef.Term cupicef.prop A) => + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I C + (__ : cupicef.Term cupicef.prop C => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => C)) + (H5 : cupicef.Term cupicef.prop C => H0 (H1 (H0 (H4 H5)))) + (H5 : cupicef.Term cupicef.prop B => H3 (H1 H5))) + H2) + H. + +thm eq__ex__uncurried : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + u1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + v1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + u2 : cupicef.Term cupicef.prop (P u1) -> + v2 : cupicef.Term cupicef.prop (P v1) -> + pq : + cupicef.Term cupicef.prop + (ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1)) + (p : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2)) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex A P)) + (ex__intro A P u1 u2) (ex__intro A P v1 v2)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + u1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + v1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + u2 : cupicef.Term cupicef.prop (P u1) => + v2 : cupicef.Term cupicef.prop (P v1) => + pq : + cupicef.Term cupicef.prop + (ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1)) + (p : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2)) => + match____ex cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1)) + (p : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) + (__ : + cupicef.Term cupicef.prop + (ex + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1)) + (p : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2)) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex A P)) + (ex__intro A P u1 u2) (ex__intro A P v1 v2)) + (p : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + q : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) => + match____eq cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + (y : cupicef.Term cupicef.prop (P v1) => + __ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + y) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex A P)) + (ex__intro A P u1 u2) (ex__intro A P v1 y)) + (cupicef.cast cupicef.prop cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex A P)) + (ex__intro A P u1 u2) + (ex__intro A P v1 + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p))) + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex A P)) + (ex__intro A P u1 u2) + (ex__intro A P v1 + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p))) + cupicef.I + (match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 y) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex A P)) + (ex__intro A P u1 u2) + (ex__intro A P y + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 y e))) + (eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex A P)) + (ex__intro A P u1 + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 u1 + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1)))) + v1 p)) + v2 q) + pq. + +def eq__ex : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + u1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + v1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + u2 : cupicef.Term cupicef.prop (P u1) -> + v2 : cupicef.Term cupicef.prop (P v1) -> + p : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) -> + q : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex A P)) + (ex__intro A P u1 u2) (ex__intro A P v1 v2)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + u1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + v1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + u2 : cupicef.Term cupicef.prop (P u1) => + v2 : cupicef.Term cupicef.prop (P v1) => + p : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + q : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) => + eq__ex__uncurried A P u1 v1 u2 v2 + (ex__intro + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1)) + (p0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p0) + v2) + p q). + +def eq__ex__hprop : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + P__hprop : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (P x) -> + q : cupicef.Term cupicef.prop (P x) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + p q)) -> + u1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + v1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + u2 : cupicef.Term cupicef.prop (P u1) -> + v2 : cupicef.Term cupicef.prop (P v1) -> + p : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex A P)) + (ex__intro A P u1 u2) (ex__intro A P v1 v2)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + P__hprop : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (P x) -> + q : cupicef.Term cupicef.prop (P x) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + p q)) => + u1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + v1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + u2 : cupicef.Term cupicef.prop (P u1) => + v2 : cupicef.Term cupicef.prop (P v1) => + p : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + eq__ex A P u1 v1 u2 v2 p + (P__hprop v1 + (eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2). + +thm rew__ex : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) -> + cupicef.Univ cupicef.prop) -> + u : + cupicef.Term cupicef.prop + (ex (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p)) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (ex (P y) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y) => + Q y p))) + (eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + ex (P a) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + Q a p))) + u y H) + (match____ex cupicef.prop (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (u0 : + cupicef.Term cupicef.prop + (ex (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p)) => + ex (P y) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y) => + Q y p)) + (u1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + u2 : cupicef.Term cupicef.prop (Q x u1) => + ex__intro (P y) (Q y) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P u1 y H) + (match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x a) => + Q a + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P u1 a x0)) + u2 y H)) + u)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) -> + cupicef.Univ cupicef.prop) => + u : + cupicef.Term cupicef.prop + (ex (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p)) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (ex (P y0) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y0) => + Q y0 p))) + (eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + ex (P a) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + Q a p))) + u y0 e) + (match____ex cupicef.prop (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (u0 : + cupicef.Term cupicef.prop + (ex (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p)) => + ex (P y0) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y0) => + Q y0 p)) + (u1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + u2 : cupicef.Term cupicef.prop (Q x u1) => + ex__intro (P y0) (Q y0) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P u1 y0 e) + (match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x a) => + Q a + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P u1 a x0)) + u2 y0 e)) + u)) + (match____ex cupicef.prop (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => Q x p) + (e : + cupicef.Term cupicef.prop + (ex (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p)) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (ex (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p))) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + ex (P a) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P a) => + Q a p))) + e x + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + (match____ex cupicef.prop (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (u0 : + cupicef.Term cupicef.prop + (ex (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) => + ex (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p)) + (u1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + u2 : cupicef.Term cupicef.prop (Q x u1) => + ex__intro (P x) (Q x) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P u1 x + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + u2) + e)) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + q : cupicef.Term cupicef.prop (Q x x0) => + eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (ex (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p))) + (ex__intro (P x) (Q x) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P x0 x + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + q)) + u) + y H. + +thm eq__ex2__uncurried : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + u1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + v1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + u2 : cupicef.Term cupicef.prop (P u1) -> + v2 : cupicef.Term cupicef.prop (P v1) -> + u3 : cupicef.Term cupicef.prop (Q u1) -> + v3 : cupicef.Term cupicef.prop (Q v1) -> + pq : + cupicef.Term cupicef.prop + (ex2 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1)) + (p : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) + (p : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + v3)) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex2 A P Q)) + (ex__intro2 A P Q u1 u2 u3) (ex__intro2 A P Q v1 v2 v3)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + u1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + v1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + u2 : cupicef.Term cupicef.prop (P u1) => + v2 : cupicef.Term cupicef.prop (P v1) => + u3 : cupicef.Term cupicef.prop (Q u1) => + v3 : cupicef.Term cupicef.prop (Q v1) => + pq : + cupicef.Term cupicef.prop + (ex2 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1)) + (p : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) + (p : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + v3)) => + match____ex2 cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1)) + (p : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) + (p : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + v3) + (__ : + cupicef.Term cupicef.prop + (ex2 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1)) + (p : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) + (p : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + v3)) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex2 A P Q)) + (ex__intro2 A P Q u1 u2 u3) (ex__intro2 A P Q v1 v2 v3)) + (p : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + q : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) => + r : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + v3) => + match____eq cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + (y : cupicef.Term cupicef.prop (Q v1) => + __ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + y) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex2 A P Q)) + (ex__intro2 A P Q u1 u2 u3) (ex__intro2 A P Q v1 v2 y)) + (match____eq cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + (y : cupicef.Term cupicef.prop (P v1) => + __ : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + y) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex2 A P Q)) + (ex__intro2 A P Q u1 u2 u3) + (ex__intro2 A P Q v1 y + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p))) + (match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 y) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex2 A P Q)) + (ex__intro2 A P Q u1 u2 u3) + (ex__intro2 A P Q y + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 y e) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 y e))) + (cupicef.cast cupicef.prop cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex2 A P Q)) + (ex__intro2 A P Q u1 u2 u3) (ex__intro2 A P Q u1 u2 u3)) + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex2 A P Q)) + (ex__intro2 A P Q u1 u2 u3) + (ex__intro2 A P Q u1 + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 u1 + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 u1 + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1)))) + cupicef.I + (eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex2 A P Q)) + (ex__intro2 A P Q u1 u2 u3))) + v1 p) + v2 q) + v3 r) + pq. + +def eq__ex2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + u1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + v1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + u2 : cupicef.Term cupicef.prop (P u1) -> + v2 : cupicef.Term cupicef.prop (P v1) -> + u3 : cupicef.Term cupicef.prop (Q u1) -> + v3 : cupicef.Term cupicef.prop (Q v1) -> + p : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) -> + q : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) -> + r : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + v3) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex2 A P Q)) + (ex__intro2 A P Q u1 u2 u3) (ex__intro2 A P Q v1 v2 v3)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + u1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + v1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + u2 : cupicef.Term cupicef.prop (P u1) => + v2 : cupicef.Term cupicef.prop (P v1) => + u3 : cupicef.Term cupicef.prop (Q u1) => + v3 : cupicef.Term cupicef.prop (Q v1) => + p : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + q : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) => + r : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + v3) => + eq__ex2__uncurried A P Q u1 v1 u2 v2 u3 v3 + (ex__intro2 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1)) + (p0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p0) + v2) + (p0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p0) + v3) + p q r). + +def eq__ex2__hprop : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + P__hprop : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (P x) -> + q : cupicef.Term cupicef.prop (P x) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + p q)) -> + Q__hprop : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (Q x) -> + q : cupicef.Term cupicef.prop (Q x) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q x)) + p q)) -> + u1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + v1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + u2 : cupicef.Term cupicef.prop (P u1) -> + v2 : cupicef.Term cupicef.prop (P v1) -> + u3 : cupicef.Term cupicef.prop (Q u1) -> + v3 : cupicef.Term cupicef.prop (Q v1) -> + p : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex2 A P Q)) + (ex__intro2 A P Q u1 u2 u3) (ex__intro2 A P Q v1 v2 v3)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + P__hprop : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (P x) -> + q : cupicef.Term cupicef.prop (P x) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + p q)) => + Q__hprop : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (Q x) -> + q : cupicef.Term cupicef.prop (Q x) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q x)) + p q)) => + u1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + v1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + u2 : cupicef.Term cupicef.prop (P u1) => + v2 : cupicef.Term cupicef.prop (P v1) => + u3 : cupicef.Term cupicef.prop (Q u1) => + v3 : cupicef.Term cupicef.prop (Q v1) => + p : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + eq__ex2 A P Q u1 v1 u2 v2 u3 v3 p + (P__hprop v1 + (eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) + (Q__hprop v1 + (eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + v3). + +thm rew__ex2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) -> + cupicef.Univ cupicef.prop) -> + R : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) -> + cupicef.Univ cupicef.prop) -> + u : + cupicef.Term cupicef.prop + (ex2 (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p)) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (ex2 (P y) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y) => + Q y p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y) => + R y p))) + (eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + ex2 (P a) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + Q a p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + R a p))) + u y H) + (match____ex2 cupicef.prop (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p) + (u0 : + cupicef.Term cupicef.prop + (ex2 (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p)) => + ex2 (P y) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y) => + Q y p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y) => + R y p)) + (u1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + u2 : cupicef.Term cupicef.prop (Q x u1) => + u3 : cupicef.Term cupicef.prop (R x u1) => + ex__intro2 (P y) (Q y) (R y) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P u1 y H) + (match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x a) => + Q a + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P u1 a x0)) + u2 y H) + (match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x a) => + R a + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P u1 a x0)) + u3 y H)) + u)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) -> + cupicef.Univ cupicef.prop) => + R : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) -> + cupicef.Univ cupicef.prop) => + u : + cupicef.Term cupicef.prop + (ex2 (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p)) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (ex2 (P y0) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y0) => + Q y0 p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y0) => + R y0 p))) + (eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + ex2 (P a) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + Q a p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + R a p))) + u y0 e) + (match____ex2 cupicef.prop (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p) + (u0 : + cupicef.Term cupicef.prop + (ex2 (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p)) => + ex2 (P y0) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y0) => + Q y0 p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y0) => + R y0 p)) + (u1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + u2 : cupicef.Term cupicef.prop (Q x u1) => + u3 : cupicef.Term cupicef.prop (R x u1) => + ex__intro2 (P y0) (Q y0) (R y0) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P u1 y0 e) + (match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x a) => + Q a + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P u1 a x0)) + u2 y0 e) + (match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x0 : + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x a) => + R a + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P u1 a x0)) + u3 y0 e)) + u)) + (match____ex2 cupicef.prop (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => Q x p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => R x p) + (e : + cupicef.Term cupicef.prop + (ex2 (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p)) => + eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (ex2 (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p))) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + ex2 (P a) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P a) => + Q a p) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P a) => + R a p))) + e x + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + (match____ex2 cupicef.prop (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p) + (u0 : + cupicef.Term cupicef.prop + (ex2 (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + R x p)) => + ex2 (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p)) + (u1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + u2 : cupicef.Term cupicef.prop (Q x u1) => + u3 : cupicef.Term cupicef.prop (R x u1) => + ex__intro2 (P x) (Q x) (R x) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P u1 x + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + u2 u3) + e)) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + q : cupicef.Term cupicef.prop (Q x x0) => + r : cupicef.Term cupicef.prop (R x x0) => + eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (ex2 (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p))) + (ex__intro2 (P x) (Q x) (R x) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P x0 x + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + q r)) + u) + y H. + +(; End of translation. ;) diff --git a/out/Coq__Init__Logic__Type.dk b/out/Coq__Init__Logic__Type.dk new file mode 100644 index 000000000..0df0e4c20 --- /dev/null +++ b/out/Coq__Init__Logic__Type.dk @@ -0,0 +1,398 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def notT : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.False). + +def identity__sym : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) A x + y) -> + cupicef.Term cupicef.prop + (Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) A y x) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) A + x y) => + Coq__Init__Datatypes.match____identity cupicef.prop A x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + i : + cupicef.Term cupicef.prop + (Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) + A x y0) => + Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) A y0 + x) + (Coq__Init__Datatypes.identity__refl A x) y H. + +def identity__trans : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) A x + y) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) A y + z) -> + cupicef.Term cupicef.prop + (Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) A x z) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) A + x y) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) A + y z) => + Coq__Init__Datatypes.match____identity cupicef.prop A y + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + i : + cupicef.Term cupicef.prop + (Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) + A y y0) => + Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) A x + y0) + H z H0. + +def identity__congr : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) A x + y) -> + cupicef.Term cupicef.prop + (Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) B + (f x) (f y)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) A + x y) => + Coq__Init__Datatypes.match____identity cupicef.prop A x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + i : + cupicef.Term cupicef.prop + (Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) + A x y0) => + Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) B + (f x) (f y0)) + (Coq__Init__Datatypes.identity__refl B (f x)) y H. + +thm not__identity__sym : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (notT + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (Coq__Init__Datatypes.identity + (cupicef.type (cupicef.s cupicef.z)) A x y))) -> + cupicef.Term cupicef.prop + (notT + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) + A y x))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.cast + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (notT + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Init__Datatypes.identity + (cupicef.type (cupicef.s cupicef.z)) A x y))) + (H : + cupicef.Term cupicef.prop + (notT + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Init__Datatypes.identity + (cupicef.type (cupicef.s cupicef.z)) A x y))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Datatypes.identity + (cupicef.type (cupicef.s cupicef.z)) A y x) + (H_27_ : + cupicef.Term cupicef.prop + (Coq__Init__Datatypes.identity + (cupicef.type (cupicef.s cupicef.z)) A y x) => + Coq__Init__Logic.False))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (notT + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Init__Datatypes.identity + (cupicef.type (cupicef.s cupicef.z)) A x y))) + (__ : + cupicef.Term cupicef.prop + (notT + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Init__Datatypes.identity + (cupicef.type (cupicef.s cupicef.z)) A x y))) => + notT + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Init__Datatypes.identity + (cupicef.type (cupicef.s cupicef.z)) A y x)))) + cupicef.I + (H : + cupicef.Term cupicef.prop + (notT + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Init__Datatypes.identity + (cupicef.type (cupicef.s cupicef.z)) A x y))) => + H_27_ : + cupicef.Term cupicef.prop + (Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) + A y x) => + H + (Coq__Init__Datatypes.match____identity cupicef.prop A y + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + i : + cupicef.Term cupicef.prop + (Coq__Init__Datatypes.identity + (cupicef.type (cupicef.s cupicef.z)) A y y0) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (notT + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Init__Datatypes.identity + (cupicef.type (cupicef.s cupicef.z)) A y0 y))) + (H0 : + cupicef.Term cupicef.prop + (notT + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Init__Datatypes.identity + (cupicef.type (cupicef.s cupicef.z)) A y0 y))) => + Coq__Init__Datatypes.identity + (cupicef.type (cupicef.s cupicef.z)) A y0 y)) + (H0 : + cupicef.Term cupicef.prop + (notT + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Init__Datatypes.identity + (cupicef.type (cupicef.s cupicef.z)) A y y))) => + Coq__Init__Datatypes.identity__refl A y) + x H_27_ H)). + +def identity__ind__r : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + __ : cupicef.Term cupicef.prop (P a) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : + cupicef.Term cupicef.prop + (Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) A y + a) -> + cupicef.Term cupicef.prop (P y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + H : cupicef.Term cupicef.prop (P x) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) A + y x) => + Coq__Init__Datatypes.match____identity cupicef.prop A x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + i : + cupicef.Term cupicef.prop + (Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) + A x y0) => + P y0) + H y (identity__sym A y x H0). + +def identity__rec__r : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.set) -> + __ : cupicef.Term cupicef.set (P a) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : + cupicef.Term cupicef.prop + (Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) A y + a) -> + cupicef.Term cupicef.set (P y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.set) => + H : cupicef.Term cupicef.set (P x) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) A + y x) => + Coq__Init__Datatypes.match____identity cupicef.set A x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + i : + cupicef.Term cupicef.prop + (Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) + A x y0) => + P y0) + H y (identity__sym A y x H0). + +def identity__rect__r : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : + cupicef.Term cupicef.prop + (Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) A y + a) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + H : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) A + y x) => + Coq__Init__Datatypes.match____identity + (cupicef.type (cupicef.s cupicef.z)) A x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + i : + cupicef.Term cupicef.prop + (Coq__Init__Datatypes.identity (cupicef.type (cupicef.s cupicef.z)) + A x y0) => + P y0) + H y (identity__sym A y x H0). + +(; End of translation. ;) diff --git a/out/Coq__Init__Nat.dk b/out/Coq__Init__Nat.dk new file mode 100644 index 000000000..0ad139f6c --- /dev/null +++ b/out/Coq__Init__Nat.dk @@ -0,0 +1,1457 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def t : + cupicef.Univ cupicef.set + +:= Coq__Init__Datatypes.nat. + +def zero : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= Coq__Init__Datatypes.O. + +def one : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= Coq__Init__Datatypes.S Coq__Init__Datatypes.O. + +def two : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= Coq__Init__Datatypes.S (Coq__Init__Datatypes.S Coq__Init__Datatypes.O). + +def succ : + __ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= Coq__Init__Datatypes.S. + +def pred : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + n (u : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => u) n. + +def add : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat))))) + (c => + c + (add0 => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + m + (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.S (add0 p m)) + n)) + cupicef.0. + +def double : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => add n n. + +def mul : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat))))) + (c => + c + (mul0 => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O + (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + add m (mul0 p m)) + n)) + cupicef.0. + +def sub : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat))))) + (c => + c + (sub0 => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + n + (k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + n + (l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + sub0 k l) + m) + n)) + cupicef.0. + +def eqb : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.bool))))) + (c => + c + (eqb0 => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.bool) + (Coq__Init__Datatypes.match____nat cupicef.set + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.false) + m) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (m_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + eqb0 n_27_ m_27_) + m) + n)) + cupicef.0. + +def leb : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.bool))))) + (c => + c + (leb0 => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (m_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + leb0 n_27_ m_27_) + m) + n)) + cupicef.0. + +def ltb : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + leb (Coq__Init__Datatypes.S n) m. + +def compare : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.comparison + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.comparison))))) + (c => + c + (compare0 => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.comparison) + (Coq__Init__Datatypes.match____nat cupicef.set + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.comparison) + Coq__Init__Datatypes.Eq + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.Lt) + m) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.comparison) + Coq__Init__Datatypes.Gt + (m_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + compare0 n_27_ m_27_) + m) + n)) + cupicef.0. + +def max : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat))))) + (c => + c + (max0 => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + m + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + n + (m_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.S (max0 n_27_ m_27_)) + m) + n)) + cupicef.0. + +def min : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat))))) + (c => + c + (min0 => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O + (m_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.S (min0 n_27_ m_27_)) + m) + n)) + cupicef.0. + +def even : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.bool)))) + (c => + c + (even0 => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + even0 n_27_) + n0) + n)) + cupicef.0. + +def odd : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.negb (even n). + +def pow : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 cupicef.set + (cupicef.prod cupicef.set (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat))))) + (c => + c + (pow0 => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O) + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + mul n (pow0 n m0)) + m)) + cupicef.0. + +def tail__add : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat))))) + (c => + c + (tail__add0 => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + m + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + tail__add0 n0 (Coq__Init__Datatypes.S m)) + n)) + cupicef.0. + +def tail__addmul : + r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 cupicef.set + (cupicef.prod cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set))) + cupicef.I Coq__Init__Datatypes.nat + (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set + (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat)))))) + (c => + c + (tail__addmul0 => + r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + r + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + tail__addmul0 (tail__add m r) n0 m) + n)) + cupicef.0. + +def tail__mul : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + tail__addmul Coq__Init__Datatypes.O n m. + +def of__uint__acc : + d : cupicef.Term cupicef.set Coq__Init__Decimal.uint -> + acc : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + cupicef.I Coq__Init__Decimal.uint + (d : cupicef.Term cupicef.set Coq__Init__Decimal.uint => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (acc : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat))))) + (c => + c + (of__uint__acc0 => + d : cupicef.Term cupicef.set Coq__Init__Decimal.uint => + acc : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Decimal.match____uint cupicef.set + (d0 : cupicef.Term cupicef.set Coq__Init__Decimal.uint => + Coq__Init__Datatypes.nat) + acc + (d0 : cupicef.Term cupicef.set Coq__Init__Decimal.uint => + of__uint__acc0 d0 + (tail__mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + Coq__Init__Datatypes.O)))))))))) + acc)) + (d0 : cupicef.Term cupicef.set Coq__Init__Decimal.uint => + of__uint__acc0 d0 + (Coq__Init__Datatypes.S + (tail__mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + Coq__Init__Datatypes.O)))))))))) + acc))) + (d0 : cupicef.Term cupicef.set Coq__Init__Decimal.uint => + of__uint__acc0 d0 + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (tail__mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + Coq__Init__Datatypes.O)))))))))) + acc)))) + (d0 : cupicef.Term cupicef.set Coq__Init__Decimal.uint => + of__uint__acc0 d0 + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (tail__mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + Coq__Init__Datatypes.O)))))))))) + acc))))) + (d0 : cupicef.Term cupicef.set Coq__Init__Decimal.uint => + of__uint__acc0 d0 + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (tail__mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + Coq__Init__Datatypes.O)))))))))) + acc)))))) + (d0 : cupicef.Term cupicef.set Coq__Init__Decimal.uint => + of__uint__acc0 d0 + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (tail__mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + Coq__Init__Datatypes.O)))))))))) + acc))))))) + (d0 : cupicef.Term cupicef.set Coq__Init__Decimal.uint => + of__uint__acc0 d0 + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (tail__mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + Coq__Init__Datatypes.O)))))))))) + acc)))))))) + (d0 : cupicef.Term cupicef.set Coq__Init__Decimal.uint => + of__uint__acc0 d0 + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (tail__mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + Coq__Init__Datatypes.O)))))))))) + acc))))))))) + (d0 : cupicef.Term cupicef.set Coq__Init__Decimal.uint => + of__uint__acc0 d0 + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (tail__mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + ( + Coq__Init__Datatypes.S + Coq__Init__Datatypes.O)))))))))) + acc)))))))))) + (d0 : cupicef.Term cupicef.set Coq__Init__Decimal.uint => + of__uint__acc0 d0 + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (tail__mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + ( + Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S + Coq__Init__Datatypes.O)))))))))) + acc))))))))))) + d)) + cupicef.0. + +def of__uint : + d : cupicef.Term cupicef.set Coq__Init__Decimal.uint -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= d : cupicef.Term cupicef.set Coq__Init__Decimal.uint => + of__uint__acc d Coq__Init__Datatypes.O. + +def to__little__uint : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + acc : cupicef.Term cupicef.set Coq__Init__Decimal.uint -> + cupicef.Term cupicef.set Coq__Init__Decimal.uint + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Decimal.uint + (acc : cupicef.Term cupicef.set Coq__Init__Decimal.uint => + Coq__Init__Decimal.uint))))) + (c => + c + (to__little__uint0 => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + acc : cupicef.Term cupicef.set Coq__Init__Decimal.uint => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Decimal.uint) + acc + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + to__little__uint0 n0 (Coq__Init__Decimal.Little__succ acc)) + n)) + cupicef.0. + +def to__uint : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Decimal.uint + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Decimal.rev + (to__little__uint n (Coq__Init__Decimal.D0 Coq__Init__Decimal.Nil)). + +def of__int : + d : cupicef.Term cupicef.set Coq__Init__Decimal.int -> + cupicef.Term cupicef.set + (Coq__Init__Datatypes.option cupicef.set Coq__Init__Datatypes.nat) + +:= d : cupicef.Term cupicef.set Coq__Init__Decimal.int => + Coq__Init__Decimal.match____int cupicef.set + (x : cupicef.Term cupicef.set Coq__Init__Decimal.int => + Coq__Init__Datatypes.option cupicef.set Coq__Init__Datatypes.nat) + (u : cupicef.Term cupicef.set Coq__Init__Decimal.uint => + Coq__Init__Datatypes.Some + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (of__uint u)) + (d0 : cupicef.Term cupicef.set Coq__Init__Decimal.uint => + Coq__Init__Datatypes.None + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat)) + (Coq__Init__Decimal.norm d). + +def to__int : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Decimal.int + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Decimal.Pos (to__uint n). + +def divmod : + x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + u : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set + (Coq__Init__Datatypes.prod cupicef.set cupicef.set + Coq__Init__Datatypes.nat Coq__Init__Datatypes.nat) + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set))) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)))) + cupicef.I Coq__Init__Datatypes.nat + (x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set))) + cupicef.I Coq__Init__Datatypes.nat + (y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set + (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + cupicef.I Coq__Init__Datatypes.nat + (q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (u : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.prod cupicef.set cupicef.set + Coq__Init__Datatypes.nat Coq__Init__Datatypes.nat))))))) + (c => + c + (divmod0 => + x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + u : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (x0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.prod cupicef.set cupicef.set + Coq__Init__Datatypes.nat Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.pair + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + q u) + (x_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (u0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.prod cupicef.set cupicef.set + Coq__Init__Datatypes.nat Coq__Init__Datatypes.nat) + (divmod0 x_27_ y (Coq__Init__Datatypes.S q) y) + (u_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + divmod0 x_27_ y q u_27_) + u) + x)) + cupicef.0. + +def div : + x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (y0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + y + (y_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.fst + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (divmod x y_27_ Coq__Init__Datatypes.O y_27_)) + y. + +def modulo : + x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (y0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + y + (y_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + sub y_27_ + (Coq__Init__Datatypes.snd + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (divmod x y_27_ Coq__Init__Datatypes.O y_27_))) + y. + +def gcd : + a : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + cupicef.I Coq__Init__Datatypes.nat + (a : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat))))) + (c => + c + (gcd0 => + a : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (a0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + b + (a_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + gcd0 (modulo b (Coq__Init__Datatypes.S a_27_)) + (Coq__Init__Datatypes.S a_27_)) + a)) + cupicef.0. + +def square : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => mul n n. + +def sqrt__iter : + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set))) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)))) + cupicef.I Coq__Init__Datatypes.nat + (k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set))) + cupicef.I Coq__Init__Datatypes.nat + (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set + (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + cupicef.I Coq__Init__Datatypes.nat + (q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat))))))) + (c => + c + (sqrt__iter0 => + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (k0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + p + (k_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (r0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + (sqrt__iter0 k_27_ (Coq__Init__Datatypes.S p) + (Coq__Init__Datatypes.S (Coq__Init__Datatypes.S q)) + (Coq__Init__Datatypes.S (Coq__Init__Datatypes.S q))) + (r_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + sqrt__iter0 k_27_ p q r_27_) + r) + k)) + cupicef.0. + +def sqrt : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + sqrt__iter n Coq__Init__Datatypes.O Coq__Init__Datatypes.O + Coq__Init__Datatypes.O. + +def log2__iter : + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set))) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)))) + cupicef.I Coq__Init__Datatypes.nat + (k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set))) + cupicef.I Coq__Init__Datatypes.nat + (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set + (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + cupicef.I Coq__Init__Datatypes.nat + (q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat))))))) + (c => + c + (log2__iter0 => + k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (k0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + p + (k_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (r0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + (log2__iter0 k_27_ (Coq__Init__Datatypes.S p) + (Coq__Init__Datatypes.S q) q) + (r_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + log2__iter0 k_27_ p (Coq__Init__Datatypes.S q) r_27_) + r) + k)) + cupicef.0. + +def log2 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + log2__iter (pred n) Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O) Coq__Init__Datatypes.O. + +def iter : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Datatypes.nat__rect + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) x + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) n. + +def div2 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat)))) + (c => + c + (div3 => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.S (div3 n_27_)) + n0) + n)) + cupicef.0. + +def testbit : + a : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 cupicef.set + (cupicef.prod cupicef.set (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + cupicef.I Coq__Init__Datatypes.nat + (a : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.bool))))) + (c => + c + (testbit0 => + a : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.bool) + (odd a) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + testbit0 (div2 a) n0) + n)) + cupicef.0. + +def shiftl : + a : + cupicef.Term cupicef.set + ((__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) Coq__Init__Datatypes.O) -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set + ((__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) n) + +:= a : + cupicef.Term cupicef.set + ((__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) Coq__Init__Datatypes.O) => + Coq__Init__Datatypes.nat__rect + (cupicef.cast + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.set (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Coq__Init__Datatypes.nat + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat)) + a (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => double). + +def shiftr : + a : + cupicef.Term cupicef.set + ((__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) Coq__Init__Datatypes.O) -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set + ((__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) n) + +:= a : + cupicef.Term cupicef.set + ((__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) Coq__Init__Datatypes.O) => + Coq__Init__Datatypes.nat__rect + (cupicef.cast + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.set (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Coq__Init__Datatypes.nat + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat)) + a (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => div2). + +def bitwise : + op : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool) -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + a : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 cupicef.set + (cupicef.prod + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set))) + (cupicef.rule + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)))) + cupicef.I + (cupicef.prod cupicef.set + (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + cupicef.I Coq__Init__Datatypes.bool + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.bool + (__0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.bool))) + (op : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool) => + cupicef.prod cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set))) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set + (cupicef.rule cupicef.set cupicef.set) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set cupicef.set)) + cupicef.I Coq__Init__Datatypes.nat + (a : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat))))))) + (c => + c + (bitwise0 => + op : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool) => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + a : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + add + (Coq__Init__Datatypes.match____bool cupicef.set + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O) + Coq__Init__Datatypes.O (op (odd a) (odd b))) + (mul + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (bitwise0 op n_27_ (div2 a) (div2 b)))) + n)) + cupicef.0. + +def land : + a : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= a : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + bitwise Coq__Init__Datatypes.andb a a b. + +def lor : + a : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= a : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + bitwise Coq__Init__Datatypes.orb (max a b) a b. + +def ldiff : + a : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= a : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + bitwise + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + b_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + Coq__Init__Datatypes.andb b0 (Coq__Init__Datatypes.negb b_27_)) + a a b. + +def lxor : + a : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + b : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= a : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + b : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + bitwise Coq__Init__Datatypes.xorb (max a b) a b. + +(; End of translation. ;) diff --git a/out/Coq__Init__Notations.dk b/out/Coq__Init__Notations.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Init__Notations.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Init__Peano.dk b/out/Coq__Init__Peano.dk new file mode 100644 index 000000000..aea633de0 --- /dev/null +++ b/out/Coq__Init__Peano.dk @@ -0,0 +1,5988 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def eq__S : + x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S x) (Coq__Init__Datatypes.S y)) + +:= Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.S. + +def f__equal__nat : + B : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) -> + f : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) B) -> + x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + x y) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.eq B (f x) (f y)) + +:= Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat). + +def f__equal__pred : + x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.pred x) (Coq__Init__Nat.pred y)) + +:= Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Nat.pred. + +thm pred__Sn : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n (Coq__Init__Nat.pred (Coq__Init__Datatypes.S n))) + +:= cupicef.cast (cupicef.rule cupicef.set cupicef.prop) + (cupicef.rule cupicef.set cupicef.prop) + (cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I + Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n n)) + (cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I + Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n (Coq__Init__Nat.pred (Coq__Init__Datatypes.S n)))) + cupicef.I + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n). + +def eq__add__S : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n) (Coq__Init__Datatypes.S m)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n) (Coq__Init__Datatypes.S m)) => + Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Nat.pred (Coq__Init__Datatypes.S n) + (Coq__Init__Datatypes.S m) H. + +thm not__eq__S : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n) (Coq__Init__Datatypes.S m))) + +:= cupicef.cast + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n) (Coq__Init__Datatypes.S m)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n) + (Coq__Init__Datatypes.S m)) => + Coq__Init__Logic.False))))) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n) (Coq__Init__Datatypes.S m)))))) + cupicef.I + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n m)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n) (Coq__Init__Datatypes.S m)) => + H (eq__add__S n m H0)). + +def IsSucc : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat (cupicef.type (cupicef.s cupicef.z)) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False + (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.True) + n. + +def let__H0 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n)) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n)) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat (cupicef.type (cupicef.s cupicef.z)) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.True + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.False) + e) + Coq__Init__Logic.I (Coq__Init__Datatypes.S n) H. + +thm O__S : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n)) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n))) + cupicef.I + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n)) => + Coq__Init__Logic.False__ind Coq__Init__Logic.False (let__H0 n H)). + +thm n__Sn : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n (Coq__Init__Datatypes.S n))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__ind + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 (Coq__Init__Datatypes.S n0))) + (O__S Coq__Init__Datatypes.O) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 (Coq__Init__Datatypes.S n0))) => + not__eq__S n0 (Coq__Init__Datatypes.S n0) IHn) + n. + +def f__equal2__plus : + x1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + y1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + x2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + y2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + x1 y1) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + x2 y2) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add x1 x2) (Coq__Init__Nat.add y1 y2)) + +:= Coq__Init__Logic.f__equal2 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Nat.add. + +def f__equal2__nat : + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + x1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + y1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + x2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + y2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + x1 y1) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + x2 y2) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2) (f y1 y2)) + +:= Coq__Init__Logic.f__equal2 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat). + +thm plus__n__O : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n (Coq__Init__Nat.add n Coq__Init__Datatypes.O)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__ind + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 (Coq__Init__Nat.add n0 Coq__Init__Datatypes.O)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O + (Coq__Init__Nat.add Coq__Init__Datatypes.O Coq__Init__Datatypes.O)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O)) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 (Coq__Init__Nat.add n0 Coq__Init__Datatypes.O)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S + (Coq__Init__Nat.add n0 Coq__Init__Datatypes.O))) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) + Coq__Init__Datatypes.O)) + cupicef.I + (f__equal__nat + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.S n0 + (Coq__Init__Nat.add n0 Coq__Init__Datatypes.O) IHn)) + n. + +thm plus__O__n : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add Coq__Init__Datatypes.O n) n) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n. + +thm plus__n__Sm : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S (Coq__Init__Nat.add n m)) + (Coq__Init__Nat.add n (Coq__Init__Datatypes.S m))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__ind + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S (Coq__Init__Nat.add n0 m)) + (Coq__Init__Nat.add n0 (Coq__Init__Datatypes.S m))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m) (Coq__Init__Datatypes.S m)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S + (Coq__Init__Nat.add Coq__Init__Datatypes.O m)) + (Coq__Init__Nat.add Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S m))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m))) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S (Coq__Init__Nat.add n0 m)) + (Coq__Init__Nat.add n0 (Coq__Init__Datatypes.S m))) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S (Coq__Init__Nat.add n0 m))) + (Coq__Init__Datatypes.S + (Coq__Init__Nat.add n0 (Coq__Init__Datatypes.S m)))) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m)) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m))) + cupicef.I + (f__equal__nat + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.S + (Coq__Init__Datatypes.S (Coq__Init__Nat.add n0 m)) + (Coq__Init__Nat.add n0 (Coq__Init__Datatypes.S m)) IHn)) + n. + +thm plus__Sn__m : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n) m) + (Coq__Init__Datatypes.S (Coq__Init__Nat.add n m))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S (Coq__Init__Nat.add n m)). + +def f__equal2__mult : + x1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + y1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + x2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + y2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + x1 y1) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + x2 y2) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.mul x1 x2) (Coq__Init__Nat.mul y1 y2)) + +:= Coq__Init__Logic.f__equal2 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Nat.mul. + +thm mult__n__O : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Nat.mul n Coq__Init__Datatypes.O)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__ind + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Nat.mul n0 Coq__Init__Datatypes.O)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O + (Coq__Init__Nat.mul Coq__Init__Datatypes.O Coq__Init__Datatypes.O)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O)) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O + (Coq__Init__Nat.mul n0 Coq__Init__Datatypes.O)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O + (Coq__Init__Nat.mul n0 Coq__Init__Datatypes.O)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O + (Coq__Init__Nat.mul (Coq__Init__Datatypes.S n0) + Coq__Init__Datatypes.O)) + cupicef.I IHn) + n. + +def let__n : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Nat.mul p m) p) + (Coq__Init__Nat.mul p (Coq__Init__Datatypes.S m))) -> + cupicef.Term cupicef.set Coq__Init__Datatypes.nat + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Nat.mul p m) p) + (Coq__Init__Nat.mul p (Coq__Init__Datatypes.S m))) => + Coq__Init__Nat.mul p (Coq__Init__Datatypes.S m). + +thm mult__n__Sm : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Nat.mul n m) n) + (Coq__Init__Nat.mul n (Coq__Init__Datatypes.S m))) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__ind + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Nat.mul n0 m) n0) + (Coq__Init__Nat.mul n0 (Coq__Init__Datatypes.S m))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Nat.mul Coq__Init__Datatypes.O m) + Coq__Init__Datatypes.O) + (Coq__Init__Nat.mul Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S m))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O)) + (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Nat.mul p m) p) + (Coq__Init__Nat.mul p (Coq__Init__Datatypes.S m))) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add + (Coq__Init__Nat.add m (Coq__Init__Nat.mul p m)) + (Coq__Init__Datatypes.S p)) + (Coq__Init__Datatypes.S + (Coq__Init__Nat.add m + (Coq__Init__Nat.mul p (Coq__Init__Datatypes.S m))))) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add + (Coq__Init__Nat.mul (Coq__Init__Datatypes.S p) m) + (Coq__Init__Datatypes.S p)) + (Coq__Init__Nat.mul (Coq__Init__Datatypes.S p) + (Coq__Init__Datatypes.S m))) + cupicef.I + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Nat.mul p m) p) + (y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Nat.mul p m) p) y) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add + (Coq__Init__Nat.add m (Coq__Init__Nat.mul p m)) + (Coq__Init__Datatypes.S p)) + (Coq__Init__Datatypes.S (Coq__Init__Nat.add m y))) + (Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S + (Coq__Init__Nat.add + (Coq__Init__Nat.add m (Coq__Init__Nat.mul p m)) p)) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 + (Coq__Init__Datatypes.S + (Coq__Init__Nat.add m + (Coq__Init__Nat.add (Coq__Init__Nat.mul p m) p)))) + (eq__S + (Coq__Init__Nat.add + (Coq__Init__Nat.add m (Coq__Init__Nat.mul p m)) p) + (Coq__Init__Nat.add m + (Coq__Init__Nat.add (Coq__Init__Nat.mul p m) p)) + (cupicef.cast cupicef.prop cupicef.prop + ((n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add + (Coq__Init__Nat.add n0 (Coq__Init__Nat.mul p m)) p) + (Coq__Init__Nat.add n0 + (Coq__Init__Nat.add (Coq__Init__Nat.mul p m) p))) + m) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add + (Coq__Init__Nat.add m (Coq__Init__Nat.mul p m)) p) + (Coq__Init__Nat.add m + (Coq__Init__Nat.add (Coq__Init__Nat.mul p m) p))) + cupicef.I + (Coq__Init__Datatypes.nat__ind + (n0 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add + (Coq__Init__Nat.add n0 (Coq__Init__Nat.mul p m)) + p) + (Coq__Init__Nat.add n0 + (Coq__Init__Nat.add (Coq__Init__Nat.mul p m) p))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Nat.mul p m) p) + (Coq__Init__Nat.add (Coq__Init__Nat.mul p m) p)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add + (Coq__Init__Nat.add Coq__Init__Datatypes.O + (Coq__Init__Nat.mul p m)) + p) + (Coq__Init__Nat.add Coq__Init__Datatypes.O + (Coq__Init__Nat.add (Coq__Init__Nat.mul p m) + p))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add (Coq__Init__Nat.mul p m) p))) + (cupicef.cast + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n0 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add + (Coq__Init__Nat.add n0 + (Coq__Init__Nat.mul p m)) + p) + (Coq__Init__Nat.add n0 + (Coq__Init__Nat.add + (Coq__Init__Nat.mul p m) p))) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add + (Coq__Init__Nat.add n0 + (Coq__Init__Nat.mul p m)) + p) + (Coq__Init__Nat.add n0 + (Coq__Init__Nat.add + (Coq__Init__Nat.mul p m) p))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S + (Coq__Init__Nat.add + (Coq__Init__Nat.add n0 + (Coq__Init__Nat.mul p m)) + p)) + (Coq__Init__Datatypes.S + (Coq__Init__Nat.add n0 + (Coq__Init__Nat.add + (Coq__Init__Nat.mul p m) p)))))) + (cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n0 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add + (Coq__Init__Nat.add n0 + (Coq__Init__Nat.mul p m)) + p) + (Coq__Init__Nat.add n0 + (Coq__Init__Nat.add + (Coq__Init__Nat.mul p m) p))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type + (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add + (Coq__Init__Nat.add n0 + (Coq__Init__Nat.mul p m)) + p) + (Coq__Init__Nat.add n0 + (Coq__Init__Nat.add + (Coq__Init__Nat.mul p m) p))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add + (Coq__Init__Nat.add + (Coq__Init__Datatypes.S n0) + (Coq__Init__Nat.mul p m)) + p) + (Coq__Init__Nat.add + (Coq__Init__Datatypes.S n0) + (Coq__Init__Nat.add + (Coq__Init__Nat.mul p m) p))))) + cupicef.I + (n0 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.add + (Coq__Init__Nat.add n0 + (Coq__Init__Nat.mul p m)) + p) + (Coq__Init__Nat.add n0 + (Coq__Init__Nat.add + (Coq__Init__Nat.mul p m) p))) => + f__equal__nat + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.S + (Coq__Init__Nat.add + (Coq__Init__Nat.add n0 + (Coq__Init__Nat.mul p m)) + p) + (Coq__Init__Nat.add n0 + (Coq__Init__Nat.add (Coq__Init__Nat.mul p m) + p)) + H0)) + m))) + (Coq__Init__Nat.add + (Coq__Init__Nat.add m (Coq__Init__Nat.mul p m)) + (Coq__Init__Datatypes.S p)) + (plus__n__Sm (Coq__Init__Nat.add m (Coq__Init__Nat.mul p m)) p)) + (Coq__Init__Nat.mul p (Coq__Init__Datatypes.S m)) H)) + n. + +def le : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop. + +def le__n : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (le n n). + +def le__S : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (le n m) -> + cupicef.Term cupicef.prop (le n (Coq__Init__Datatypes.S m)). + +[ ] cupicef.code_guarded? _ (le__n _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (le__S _ _ _) --> cupicef.guarded. + +def match____le : + s : cupicef.Sort -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (le n __) -> + cupicef.Univ s) -> + case__le__n : cupicef.Term s (P n (le__n n)) -> + case__le__S : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (le n m) -> + cupicef.Term s (P (Coq__Init__Datatypes.S m) (le__S n m __))) -> + __ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + x : cupicef.Term cupicef.prop (le n __) -> + cupicef.Term s (P __ x). + +[ s, n, P, case__le__n, case__le__S] + match____le s n P case__le__n case__le__S {n} (le__n n) + --> + case__le__n. + +[ s, n, P, case__le__n, case__le__S, m, __] + match____le s n P case__le__n case__le__S {Coq__Init__Datatypes.S m} + (le__S n m __) + --> + case__le__S m __. + +[ s, n, P, s'] + match____le s n + (__ => x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P __ x))) + --> + match____le s' n (__ => x => P __ x). + +def le__ind : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + f : cupicef.Term cupicef.prop (P n) -> + f0 : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (le n m) -> + __0 : cupicef.Term cupicef.prop (P m) -> + cupicef.Term cupicef.prop (P (Coq__Init__Datatypes.S m))) -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + l : cupicef.Term cupicef.prop (le n n0) -> + cupicef.Term cupicef.prop (P n0) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + f : cupicef.Term cupicef.prop (P n) => + f0 : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (le n m) -> + __0 : cupicef.Term cupicef.prop (P m) -> + cupicef.Term cupicef.prop (P (Coq__Init__Datatypes.S m))) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 cupicef.prop + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + le n n0) (l : cupicef.Term cupicef.prop (le n n0) => P n0))))) + (c => + c + (F => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l : cupicef.Term cupicef.prop (le n n0) => + match____le cupicef.prop n + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + __ : cupicef.Term cupicef.prop (le n n1) => + P n1) + f + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + l0 : cupicef.Term cupicef.prop (le n m) => + f0 m l0 (F m l0)) + n0 l)) + cupicef.0. + +def lt : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le (Coq__Init__Datatypes.S n) m. + +def ge : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le m n. + +def gt : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + lt m n. + +thm le__pred : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (le n m) -> + cupicef.Term cupicef.prop + (le (Coq__Init__Nat.pred n) (Coq__Init__Nat.pred m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (le n m) => + le__ind n + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le (Coq__Init__Nat.pred n) (Coq__Init__Nat.pred m0)) + (le__n (Coq__Init__Nat.pred n)) + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (le n m0) => + IHle : + cupicef.Term cupicef.prop + (le (Coq__Init__Nat.pred n) (Coq__Init__Nat.pred m0)) => + Coq__Init__Datatypes.match____nat cupicef.prop + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (le n n0) + (H1 : cupicef.Term cupicef.prop (le n n0) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le (Coq__Init__Nat.pred n) (Coq__Init__Nat.pred n0)) + (IHle0 : + cupicef.Term cupicef.prop + (le (Coq__Init__Nat.pred n) (Coq__Init__Nat.pred n0)) => + le (Coq__Init__Nat.pred n) + (Coq__Init__Nat.pred (Coq__Init__Datatypes.S n0))))) + (H1 : cupicef.Term cupicef.prop (le n Coq__Init__Datatypes.O) => + IHle0 : + cupicef.Term cupicef.prop + (le (Coq__Init__Nat.pred n) + (Coq__Init__Nat.pred Coq__Init__Datatypes.O)) => + cupicef.cast cupicef.prop cupicef.prop + (le (Coq__Init__Nat.pred n) + (Coq__Init__Nat.pred Coq__Init__Datatypes.O)) + (le (Coq__Init__Nat.pred n) + (Coq__Init__Nat.pred + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) + cupicef.I IHle0) + (m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H1 : cupicef.Term cupicef.prop (le n (Coq__Init__Datatypes.S m1)) => + IHle0 : + cupicef.Term cupicef.prop + (le (Coq__Init__Nat.pred n) + (Coq__Init__Nat.pred (Coq__Init__Datatypes.S m1))) => + cupicef.cast cupicef.prop cupicef.prop + (le (Coq__Init__Nat.pred n) (Coq__Init__Datatypes.S m1)) + (le (Coq__Init__Nat.pred n) + (Coq__Init__Nat.pred + (Coq__Init__Datatypes.S (Coq__Init__Datatypes.S m1)))) + cupicef.I (le__S (Coq__Init__Nat.pred n) m1 IHle0)) + m0 H0 IHle) + m H. + +thm le__S__n : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n) (Coq__Init__Datatypes.S m)) -> + cupicef.Term cupicef.prop (le n m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le__pred (Coq__Init__Datatypes.S n) (Coq__Init__Datatypes.S m). + +thm le__0__n : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (le Coq__Init__Datatypes.O n) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__ind + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le Coq__Init__Datatypes.O n0) + (le__n Coq__Init__Datatypes.O) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : cupicef.Term cupicef.prop (le Coq__Init__Datatypes.O n0) => + le__S Coq__Init__Datatypes.O n0 IHn) + n. + +thm le__n__S : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (le n m) -> + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n) (Coq__Init__Datatypes.S m)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : cupicef.Term cupicef.prop (le n m) => + le__ind n + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + le (Coq__Init__Datatypes.S n) (Coq__Init__Datatypes.S m0)) + (le__n (Coq__Init__Datatypes.S n)) + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (le n m0) => + IHle : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n) (Coq__Init__Datatypes.S m0)) => + le__S (Coq__Init__Datatypes.S n) (Coq__Init__Datatypes.S m0) IHle) + m H. + +thm nat__case : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + __ : cupicef.Term cupicef.prop (P Coq__Init__Datatypes.O) -> + __0 : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (P (Coq__Init__Datatypes.S m))) -> + cupicef.Term cupicef.prop (P n) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__ind + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.set cupicef.prop) cupicef.prop)) + (cupicef.rule + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.set cupicef.prop) + cupicef.prop))) + cupicef.I + (cupicef.prod cupicef.set (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + cupicef.I Coq__Init__Datatypes.nat + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + cupicef.prod cupicef.prop + (cupicef.rule (cupicef.rule cupicef.set cupicef.prop) cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.set cupicef.prop) + cupicef.prop)) + cupicef.I (P Coq__Init__Datatypes.O) + (__ : cupicef.Term cupicef.prop (P Coq__Init__Datatypes.O) => + cupicef.prod (cupicef.rule cupicef.set cupicef.prop) cupicef.prop + (cupicef.rule (cupicef.rule cupicef.set cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I + Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + P (Coq__Init__Datatypes.S m))) + (__0 : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (P (Coq__Init__Datatypes.S m))) => + P n0)))) + (P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + H : cupicef.Term cupicef.prop (P Coq__Init__Datatypes.O) => + H0 : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (P (Coq__Init__Datatypes.S m))) => + H) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + __ : cupicef.Term cupicef.prop (P Coq__Init__Datatypes.O) -> + __0 : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (P (Coq__Init__Datatypes.S m))) -> + cupicef.Term cupicef.prop (P n0)) => + P : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + H : cupicef.Term cupicef.prop (P Coq__Init__Datatypes.O) => + H0 : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (P (Coq__Init__Datatypes.S m))) => + H0 n0) + n. + +thm nat__double__ind : + R : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) -> + __ : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (R Coq__Init__Datatypes.O n)) -> + __0 : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (R (Coq__Init__Datatypes.S n) Coq__Init__Datatypes.O)) -> + __1 : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __1 : cupicef.Term cupicef.prop (R n m) -> + cupicef.Term cupicef.prop + (R (Coq__Init__Datatypes.S n) (Coq__Init__Datatypes.S m))) -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (R n m) + +:= R : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Univ cupicef.prop) => + H : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (R Coq__Init__Datatypes.O n)) => + H0 : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (R (Coq__Init__Datatypes.S n) Coq__Init__Datatypes.O)) => + H1 : + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (R n m) -> + cupicef.Term cupicef.prop + (R (Coq__Init__Datatypes.S n) (Coq__Init__Datatypes.S m))) => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__ind + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I + Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => R n0 m)) + H + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop (R n0 m)) => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.prop + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + R (Coq__Init__Datatypes.S n0) n1) + (H0 n0) + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H1 n0 m0 (IHn m0)) + m) + n. + +def let__H2 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat (cupicef.type (cupicef.s cupicef.z)) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.True) + e) + Coq__Init__Logic.I Coq__Init__Datatypes.O H1. + +def let__H3 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) -> + m2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H0 : cupicef.Term cupicef.prop (le (Coq__Init__Datatypes.S m1) m2) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m2) Coq__Init__Datatypes.O) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m2) Coq__Init__Datatypes.O) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + m2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (le (Coq__Init__Datatypes.S m1) m2) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m2) Coq__Init__Datatypes.O) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m2) Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m2) + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat (cupicef.type (cupicef.s cupicef.z)) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.True) + e) + Coq__Init__Logic.I Coq__Init__Datatypes.O H2. + +def let__H1 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + match____le cupicef.prop (Coq__Init__Datatypes.S m1) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + __ : cupicef.Term cupicef.prop (le (Coq__Init__Datatypes.S m1) n0) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) + (let__H2 n m m0 m1 H H0 H1)) + H0) + (m2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (le (Coq__Init__Datatypes.S m1) m2) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m2) Coq__Init__Datatypes.O) => + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m2) Coq__Init__Datatypes.O) => + Coq__Init__Logic.False__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le (Coq__Init__Datatypes.S m1) m2) + (H3 : + cupicef.Term cupicef.prop (le (Coq__Init__Datatypes.S m1) m2) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O)) + (let__H3 n m m0 m1 H m2 H0 H1 H2)) + H1 H0) + Coq__Init__Datatypes.O H. + +thm max__l : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (le m n) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.max n m) n) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__ind + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (le m0 n0) + (__ : cupicef.Term cupicef.prop (le m0 n0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.max n0 m0) n0))) + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.prop + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le n0 Coq__Init__Datatypes.O) + (__ : cupicef.Term cupicef.prop (le n0 Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.max Coq__Init__Datatypes.O n0) + Coq__Init__Datatypes.O)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le Coq__Init__Datatypes.O Coq__Init__Datatypes.O) + (H : + cupicef.Term cupicef.prop + (le Coq__Init__Datatypes.O Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le Coq__Init__Datatypes.O Coq__Init__Datatypes.O) + (__ : + cupicef.Term cupicef.prop + (le Coq__Init__Datatypes.O Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.max Coq__Init__Datatypes.O + Coq__Init__Datatypes.O) + Coq__Init__Datatypes.O)) + cupicef.I + (H : + cupicef.Term cupicef.prop + (le Coq__Init__Datatypes.O Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O)) + (m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) + (H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) + (__ : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.max Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S m1)) + Coq__Init__Datatypes.O)) + cupicef.I + (H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + let__H1 n m m0 m1 H + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O))) + m0) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (le m0 n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.max n0 m0) n0)) => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.prop + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le n1 (Coq__Init__Datatypes.S n0)) + (__ : + cupicef.Term cupicef.prop (le n1 (Coq__Init__Datatypes.S n0)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.max (Coq__Init__Datatypes.S n0) n1) + (Coq__Init__Datatypes.S n0))) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n0)) + (H : + cupicef.Term cupicef.prop + (le Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n0)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) (Coq__Init__Datatypes.S n0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n0)) + (__ : + cupicef.Term cupicef.prop + (le Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n0)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.max (Coq__Init__Datatypes.S n0) + Coq__Init__Datatypes.O) + (Coq__Init__Datatypes.S n0))) + cupicef.I + (H : + cupicef.Term cupicef.prop + (le Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n0)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0))) + (m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S n0)) + (H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S m1) + (Coq__Init__Datatypes.S n0)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S (Coq__Init__Nat.max n0 m1)) + (Coq__Init__Datatypes.S n0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S n0)) + (__ : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S m1) + (Coq__Init__Datatypes.S n0)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.max (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)) + (Coq__Init__Datatypes.S n0))) + cupicef.I + (H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S n0)) => + Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.S (Coq__Init__Nat.max n0 m1) n0 + (IHn m1 (le__S__n m1 n0 H)))) + m0) + n m. + +def let__H5 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn : + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.max n0 m0) m0)) -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.max n0 m0) m0)) => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat (cupicef.type (cupicef.s cupicef.z)) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.True) + e) + Coq__Init__Logic.I Coq__Init__Datatypes.O H1. + +def let__H6 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn : + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.max n0 m0) m0)) -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) -> + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H0 : cupicef.Term cupicef.prop (le (Coq__Init__Datatypes.S n0) m1) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.max n0 m0) m0)) => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) => + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (le (Coq__Init__Datatypes.S n0) m1) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat (cupicef.type (cupicef.s cupicef.z)) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.True) + e) + Coq__Init__Logic.I Coq__Init__Datatypes.O H2. + +def let__H4 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn : + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.max n0 m0) m0)) -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.max n0 m0) m0)) => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) => + match____le cupicef.prop (Coq__Init__Datatypes.S n0) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + __ : cupicef.Term cupicef.prop (le (Coq__Init__Datatypes.S n0) n1) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 Coq__Init__Datatypes.O) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) => + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) + (let__H5 n m n0 IHn m0 H H0 H1)) + H0) + (m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (le (Coq__Init__Datatypes.S n0) m1) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + Coq__Init__Logic.False__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le (Coq__Init__Datatypes.S n0) m1) + (H3 : + cupicef.Term cupicef.prop (le (Coq__Init__Datatypes.S n0) m1) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O)) + (let__H6 n m n0 IHn m0 H m1 H0 H1 H2)) + H1 H0) + Coq__Init__Datatypes.O H. + +thm max__r : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (le n m) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.max n m) m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__ind + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (le n0 m0) + (__ : cupicef.Term cupicef.prop (le n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.max n0 m0) m0))) + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.prop + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le Coq__Init__Datatypes.O n0) + (__ : cupicef.Term cupicef.prop (le Coq__Init__Datatypes.O n0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.max Coq__Init__Datatypes.O n0) n0)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le Coq__Init__Datatypes.O Coq__Init__Datatypes.O) + (H : + cupicef.Term cupicef.prop + (le Coq__Init__Datatypes.O Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le Coq__Init__Datatypes.O Coq__Init__Datatypes.O) + (__ : + cupicef.Term cupicef.prop + (le Coq__Init__Datatypes.O Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.max Coq__Init__Datatypes.O + Coq__Init__Datatypes.O) + Coq__Init__Datatypes.O)) + cupicef.I + (H : + cupicef.Term cupicef.prop + (le Coq__Init__Datatypes.O Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O)) + (m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1)) + (H : + cupicef.Term cupicef.prop + (le Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S m1))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1)) + (__ : + cupicef.Term cupicef.prop + (le Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.max Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S m1)) + (Coq__Init__Datatypes.S m1))) + cupicef.I + (H : + cupicef.Term cupicef.prop + (le Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1))) + m0) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.max n0 m0) m0)) => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.prop + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le (Coq__Init__Datatypes.S n0) n1) + (__ : + cupicef.Term cupicef.prop (le (Coq__Init__Datatypes.S n0) n1) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.max (Coq__Init__Datatypes.S n0) n1) n1)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) + (H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) + (__ : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.max (Coq__Init__Datatypes.S n0) + Coq__Init__Datatypes.O) + Coq__Init__Datatypes.O)) + cupicef.I + (H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) => + let__H4 n m n0 IHn m0 H + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O))) + (m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le (Coq__Init__Datatypes.S n0) (Coq__Init__Datatypes.S m1)) + (H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S (Coq__Init__Nat.max n0 m1)) + (Coq__Init__Datatypes.S m1))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le (Coq__Init__Datatypes.S n0) (Coq__Init__Datatypes.S m1)) + (__ : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.max (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)) + (Coq__Init__Datatypes.S m1))) + cupicef.I + (H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n0) (Coq__Init__Datatypes.S m1)) => + Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.S (Coq__Init__Nat.max n0 m1) m1 + (IHn m1 (le__S__n n0 m1 H)))) + m0) + n m. + +def let__H8 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn : + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.min n0 m0) n0)) -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.min n0 m0) n0)) => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat (cupicef.type (cupicef.s cupicef.z)) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.True) + e) + Coq__Init__Logic.I Coq__Init__Datatypes.O H1. + +def let__H9 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn : + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.min n0 m0) n0)) -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) -> + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H0 : cupicef.Term cupicef.prop (le (Coq__Init__Datatypes.S n0) m1) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.min n0 m0) n0)) => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) => + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (le (Coq__Init__Datatypes.S n0) m1) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat (cupicef.type (cupicef.s cupicef.z)) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.True) + e) + Coq__Init__Logic.I Coq__Init__Datatypes.O H2. + +def let__H7 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn : + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.min n0 m0) n0)) -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n0)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.min n0 m0) n0)) => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) => + match____le cupicef.prop (Coq__Init__Datatypes.S n0) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + __ : cupicef.Term cupicef.prop (le (Coq__Init__Datatypes.S n0) n1) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 Coq__Init__Datatypes.O) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n1 Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n0))) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) => + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n0)) + (let__H8 n m n0 IHn m0 H H0 H1)) + H0) + (m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (le (Coq__Init__Datatypes.S n0) m1) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + Coq__Init__Logic.False__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le (Coq__Init__Datatypes.S n0) m1) + (H3 : + cupicef.Term cupicef.prop (le (Coq__Init__Datatypes.S n0) m1) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n0))) + (let__H9 n m n0 IHn m0 H m1 H0 H1 H2)) + H1 H0) + Coq__Init__Datatypes.O H. + +thm min__l : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (le n m) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.min n m) n) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__ind + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (le n0 m0) + (__ : cupicef.Term cupicef.prop (le n0 m0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.min n0 m0) n0))) + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.prop + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le Coq__Init__Datatypes.O n0) + (__ : cupicef.Term cupicef.prop (le Coq__Init__Datatypes.O n0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.min Coq__Init__Datatypes.O n0) + Coq__Init__Datatypes.O)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le Coq__Init__Datatypes.O Coq__Init__Datatypes.O) + (H : + cupicef.Term cupicef.prop + (le Coq__Init__Datatypes.O Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le Coq__Init__Datatypes.O Coq__Init__Datatypes.O) + (__ : + cupicef.Term cupicef.prop + (le Coq__Init__Datatypes.O Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.min Coq__Init__Datatypes.O + Coq__Init__Datatypes.O) + Coq__Init__Datatypes.O)) + cupicef.I + (H : + cupicef.Term cupicef.prop + (le Coq__Init__Datatypes.O Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O)) + (m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1)) + (H : + cupicef.Term cupicef.prop + (le Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1)) + (__ : + cupicef.Term cupicef.prop + (le Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.min Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S m1)) + Coq__Init__Datatypes.O)) + cupicef.I + (H : + cupicef.Term cupicef.prop + (le Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O)) + m0) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (le n0 m0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.min n0 m0) n0)) => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.prop + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le (Coq__Init__Datatypes.S n0) n1) + (__ : + cupicef.Term cupicef.prop (le (Coq__Init__Datatypes.S n0) n1) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.min (Coq__Init__Datatypes.S n0) n1) + (Coq__Init__Datatypes.S n0))) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) + (H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) + (__ : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.min (Coq__Init__Datatypes.S n0) + Coq__Init__Datatypes.O) + (Coq__Init__Datatypes.S n0))) + cupicef.I + (H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) => + let__H7 n m n0 IHn m0 H + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O))) + (m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le (Coq__Init__Datatypes.S n0) (Coq__Init__Datatypes.S m1)) + (H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S (Coq__Init__Nat.min n0 m1)) + (Coq__Init__Datatypes.S n0))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le (Coq__Init__Datatypes.S n0) (Coq__Init__Datatypes.S m1)) + (__ : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.min (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)) + (Coq__Init__Datatypes.S n0))) + cupicef.I + (H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S n0) (Coq__Init__Datatypes.S m1)) => + Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.S (Coq__Init__Nat.min n0 m1) n0 + (IHn m1 (le__S__n n0 m1 H)))) + m0) + n m. + +def let__H11 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat (cupicef.type (cupicef.s cupicef.z)) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.True) + e) + Coq__Init__Logic.I Coq__Init__Datatypes.O H1. + +def let__H12 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) -> + m2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H0 : cupicef.Term cupicef.prop (le (Coq__Init__Datatypes.S m1) m2) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m2) Coq__Init__Datatypes.O) -> + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m2) Coq__Init__Datatypes.O) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + m2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (le (Coq__Init__Datatypes.S m1) m2) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m2) Coq__Init__Datatypes.O) => + H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m2) Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m2) + (e : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat (cupicef.type (cupicef.s cupicef.z)) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + Coq__Init__Logic.False + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.True) + e) + Coq__Init__Logic.I Coq__Init__Datatypes.O H2. + +def let__H10 : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + match____le cupicef.prop (Coq__Init__Datatypes.S m1) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + __ : cupicef.Term cupicef.prop (le (Coq__Init__Datatypes.S m1) n0) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1))) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1)) + (let__H11 n m m0 m1 H H0 H1)) + H0) + (m2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + H0 : cupicef.Term cupicef.prop (le (Coq__Init__Datatypes.S m1) m2) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m2) Coq__Init__Datatypes.O) => + (H2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S m2) Coq__Init__Datatypes.O) => + Coq__Init__Logic.False__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le (Coq__Init__Datatypes.S m1) m2) + (H3 : + cupicef.Term cupicef.prop (le (Coq__Init__Datatypes.S m1) m2) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1))) + (let__H12 n m m0 m1 H m2 H0 H1 H2)) + H1 H0) + Coq__Init__Datatypes.O H. + +thm min__r : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (le m n) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.min n m) m) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__ind + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (le m0 n0) + (__ : cupicef.Term cupicef.prop (le m0 n0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.min n0 m0) m0))) + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.prop + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le n0 Coq__Init__Datatypes.O) + (__ : cupicef.Term cupicef.prop (le n0 Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.min Coq__Init__Datatypes.O n0) n0)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le Coq__Init__Datatypes.O Coq__Init__Datatypes.O) + (H : + cupicef.Term cupicef.prop + (le Coq__Init__Datatypes.O Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le Coq__Init__Datatypes.O Coq__Init__Datatypes.O) + (__ : + cupicef.Term cupicef.prop + (le Coq__Init__Datatypes.O Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.min Coq__Init__Datatypes.O + Coq__Init__Datatypes.O) + Coq__Init__Datatypes.O)) + cupicef.I + (H : + cupicef.Term cupicef.prop + (le Coq__Init__Datatypes.O Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O)) + (m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) + (H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) + (__ : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.min Coq__Init__Datatypes.O + (Coq__Init__Datatypes.S m1)) + (Coq__Init__Datatypes.S m1))) + cupicef.I + (H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S m1) Coq__Init__Datatypes.O) => + let__H10 n m m0 m1 H + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O))) + m0) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + __ : cupicef.Term cupicef.prop (le m0 n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.min n0 m0) m0)) => + m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.prop + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le n1 (Coq__Init__Datatypes.S n0)) + (__ : + cupicef.Term cupicef.prop (le n1 (Coq__Init__Datatypes.S n0)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.min (Coq__Init__Datatypes.S n0) n1) n1)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n0)) + (H : + cupicef.Term cupicef.prop + (le Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n0)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n0)) + (__ : + cupicef.Term cupicef.prop + (le Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n0)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.min (Coq__Init__Datatypes.S n0) + Coq__Init__Datatypes.O) + Coq__Init__Datatypes.O)) + cupicef.I + (H : + cupicef.Term cupicef.prop + (le Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n0)) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O)) + (m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S n0)) + (H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S m1) + (Coq__Init__Datatypes.S n0)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S (Coq__Init__Nat.min n0 m1)) + (Coq__Init__Datatypes.S m1))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (le (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S n0)) + (__ : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S m1) + (Coq__Init__Datatypes.S n0)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Nat.min (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S m1)) + (Coq__Init__Datatypes.S m1))) + cupicef.I + (H : + cupicef.Term cupicef.prop + (le (Coq__Init__Datatypes.S m1) (Coq__Init__Datatypes.S n0)) => + Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.S (Coq__Init__Nat.min n0 m1) m1 + (IHn m1 (le__S__n m1 n0 H)))) + m0) + n m. + +thm nat__rect__succ__r : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.nat__rect + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) x + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) + (Coq__Init__Datatypes.S n)) + (Coq__Init__Datatypes.nat__rect + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) ( + f x) (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) + n)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__ind + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.nat__rect + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) x + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) + (Coq__Init__Datatypes.S n0)) + (Coq__Init__Datatypes.nat__rect + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) + (f x) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) n0)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f x) (f x)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.nat__rect + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) x + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) + (Coq__Init__Datatypes.nat__rect + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) + (f x) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) + Coq__Init__Datatypes.O)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f x))) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.nat__rect + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) + x + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) + (Coq__Init__Datatypes.S n0)) + (Coq__Init__Datatypes.nat__rect + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) + (f x) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) + n0)) => + cupicef.cast cupicef.prop cupicef.prop + ((a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f + (f + (Coq__Init__Datatypes.nat__rect + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + A) + x + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + f) + n0))) + (f a)) + (Coq__Init__Datatypes.nat__rect + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) + (f x) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) + n0)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.nat__rect + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) x + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) + (Coq__Init__Datatypes.S (Coq__Init__Datatypes.S n0))) + (Coq__Init__Datatypes.nat__rect + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) + (f x) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) + (Coq__Init__Datatypes.S n0))) + cupicef.I + (Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.nat__rect + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) x + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) + (Coq__Init__Datatypes.S n0)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f + (f + (Coq__Init__Datatypes.nat__rect + (n1 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + A) + x + (n1 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + f) + n0))) + (f a)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f + (Coq__Init__Datatypes.nat__rect + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + A) + x + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + f) + (Coq__Init__Datatypes.S n0)))) + (Coq__Init__Datatypes.nat__rect + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) + (f x) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) + n0) + IHn)) + n. + +thm nat__rect__plus : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.nat__rect + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) x + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) + (Coq__Init__Nat.add n m)) + (Coq__Init__Datatypes.nat__rect + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) + (Coq__Init__Datatypes.nat__rect + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) x + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) m) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) n)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__ind + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.set + (cupicef.rule (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop))) + (cupicef.rule cupicef.set + (cupicef.rule (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop)))) + cupicef.I Coq__Init__Datatypes.nat + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop)) + (cupicef.rule (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop))) + cupicef.I + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop)) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + A)) + (f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.nat__rect + (n1 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + A) + x + (n1 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + f) + (Coq__Init__Nat.add n0 m)) + (Coq__Init__Datatypes.nat__rect + (n1 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + A) + (Coq__Init__Datatypes.nat__rect + (n1 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + A) + x + (n1 : + cupicef.Term cupicef.set + Coq__Init__Datatypes.nat => + f) + m) + (n1 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + f) + n0)))))) + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.nat__rect + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) x + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) m) + (Coq__Init__Datatypes.nat__rect + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) x + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) m)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.nat__rect + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) x + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) + (Coq__Init__Nat.add Coq__Init__Datatypes.O m)) + (Coq__Init__Datatypes.nat__rect + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) + (Coq__Init__Datatypes.nat__rect + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) + x + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) + m) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) + Coq__Init__Datatypes.O)) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.nat__rect + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) x + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) m))) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.nat__rect + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) + x + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) + (Coq__Init__Nat.add n0 m)) + (Coq__Init__Datatypes.nat__rect + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) + (Coq__Init__Datatypes.nat__rect + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + A) + x + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + f) + m) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) + n0))) => + m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.cast cupicef.prop cupicef.prop + ((a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) + (f + (Coq__Init__Datatypes.nat__rect + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + A) + (Coq__Init__Datatypes.nat__rect + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + A) + x + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + f) + m) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + f) + n0))) + (Coq__Init__Datatypes.nat__rect + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) x + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) + (Coq__Init__Nat.add n0 m))) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.nat__rect + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) x + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) + (Coq__Init__Nat.add (Coq__Init__Datatypes.S n0) m)) + (Coq__Init__Datatypes.nat__rect + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) + (Coq__Init__Datatypes.nat__rect + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) + x + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) + m) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) + (Coq__Init__Datatypes.S n0))) + cupicef.I + (Coq__Init__Logic.eq__ind__r A + (Coq__Init__Datatypes.nat__rect + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) + (Coq__Init__Datatypes.nat__rect + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) + x + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) + m) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) + n0) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) + (f + (Coq__Init__Datatypes.nat__rect + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + A) + (Coq__Init__Datatypes.nat__rect + (n1 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + A) + x + (n1 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + f) + m) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + f) + n0))) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f + (Coq__Init__Datatypes.nat__rect + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + A) + (Coq__Init__Datatypes.nat__rect + (n1 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + A) + x + (n1 : + cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + f) + m) + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + f) + n0))) + (Coq__Init__Datatypes.nat__rect + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => A) x + (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => f) + (Coq__Init__Nat.add n0 m)) + (IHn m A f x))) + n. + +(; End of translation. ;) diff --git a/out/Coq__Init__Prelude.dk b/out/Coq__Init__Prelude.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Init__Prelude.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Init__Specif.dk b/out/Coq__Init__Specif.dk new file mode 100644 index 000000000..2fb197a60 --- /dev/null +++ b/out/Coq__Init__Specif.dk @@ -0,0 +1,48062 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def sig : + Coq__Init__Specif__1 : cupicef.Sort -> + A : cupicef.Univ Coq__Init__Specif__1 -> + P : (__ : cupicef.Term Coq__Init__Specif__1 A -> cupicef.Univ cupicef.prop) -> + cupicef.Univ Coq__Init__Specif__1. + +sig' : + A : cupicef.Univ cupicef.sinf -> + P : (__ : cupicef.Term cupicef.sinf A -> cupicef.Univ cupicef.prop) -> + cupicef.Univ cupicef.sinf. + +[ Coq__Init__Specif__1, A, P] + cupicef.code (cupicef.cU _) (sig Coq__Init__Specif__1 A P) + --> + sig' (cupicef.code (cupicef.cU Coq__Init__Specif__1) A) P. + +def Coq__Init__Specif__1 : + cupicef.Nat + +:= cupicef.s cupicef.z. + +def exist : + A : cupicef.Univ (cupicef.type Coq__Init__Specif__1) -> + P : + (__ : cupicef.Term (cupicef.type Coq__Init__Specif__1) A -> + cupicef.Univ cupicef.prop) -> + x : cupicef.Term (cupicef.type Coq__Init__Specif__1) A -> + __ : cupicef.Term cupicef.prop (P x) -> + cupicef.Term (cupicef.type Coq__Init__Specif__1) + (sig (cupicef.type Coq__Init__Specif__1) A P). + +[ ] cupicef.code_guarded? _ (exist _ _ _ _) --> cupicef.guarded. + +def match____sig : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type Coq__Init__Specif__1) -> + P : + (__ : cupicef.Term (cupicef.type Coq__Init__Specif__1) A -> + cupicef.Univ cupicef.prop) -> + P0 : + (cupicef.Term (cupicef.type Coq__Init__Specif__1) + (sig (cupicef.type Coq__Init__Specif__1) A P) -> + cupicef.Univ s) -> + case__exist : + (x0 : cupicef.Term (cupicef.type Coq__Init__Specif__1) A -> + __ : cupicef.Term cupicef.prop (P x0) -> + cupicef.Term s (P0 (exist A P x0 __))) -> + x : + cupicef.Term (cupicef.type Coq__Init__Specif__1) + (sig (cupicef.type Coq__Init__Specif__1) A P) -> + cupicef.Term s (P0 x). + +[ s, A, P, P0, case__exist, x0, __] + match____sig s A P P0 case__exist (exist A P x0 __) + --> + case__exist x0 __. + +[ s, A, P, P0, s'] + match____sig s A P + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P0 x))) + --> + match____sig s' A P (x => P0 x). + +def sig__rect : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + P0 : + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A P) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (P x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P0 (exist A P x p))) -> + s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A P) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P0 s) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + P0 : + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A P) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (P x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P0 (exist A P x p))) => + s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A P) => + match____sig (cupicef.type (cupicef.s cupicef.z)) A P + (s0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A P) => + P0 s0) + f s. + +def sig__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + P0 : + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A P) -> + cupicef.Univ cupicef.prop) -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (P x) -> + cupicef.Term cupicef.prop (P0 (exist A P x p))) -> + s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A P) -> + cupicef.Term cupicef.prop (P0 s) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + P0 : + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A P) -> + cupicef.Univ cupicef.prop) => + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (P x) -> + cupicef.Term cupicef.prop (P0 (exist A P x p))) => + s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A P) => + match____sig cupicef.prop A P + (s0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A P) => + P0 s0) + f s. + +def sig__rec : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + P0 : + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A P) -> + cupicef.Univ cupicef.set) -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (P x) -> + cupicef.Term cupicef.set (P0 (exist A P x p))) -> + s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A P) -> + cupicef.Term cupicef.set (P0 s) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + P0 : + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A P) -> + cupicef.Univ cupicef.set) => + sig__rect A P + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (sig (cupicef.type (cupicef.s cupicef.z)) A P) + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A P) => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (sig (cupicef.type (cupicef.s cupicef.z)) A P) + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A P) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P0). + +def sig2 : + Coq__Init__Specif__4 : cupicef.Sort -> + A : cupicef.Univ Coq__Init__Specif__4 -> + P : (__ : cupicef.Term Coq__Init__Specif__4 A -> cupicef.Univ cupicef.prop) -> + Q : (__ : cupicef.Term Coq__Init__Specif__4 A -> cupicef.Univ cupicef.prop) -> + cupicef.Univ Coq__Init__Specif__4. + +sig2' : + A : cupicef.Univ cupicef.sinf -> + P : (__ : cupicef.Term cupicef.sinf A -> cupicef.Univ cupicef.prop) -> + Q : (__ : cupicef.Term cupicef.sinf A -> cupicef.Univ cupicef.prop) -> + cupicef.Univ cupicef.sinf. + +[ Coq__Init__Specif__4, A, P, Q] + cupicef.code (cupicef.cU _) (sig2 Coq__Init__Specif__4 A P Q) + --> + sig2' (cupicef.code (cupicef.cU Coq__Init__Specif__4) A) P Q. + +def Coq__Init__Specif__4 : + cupicef.Nat + +:= cupicef.s cupicef.z. + +def exist2 : + A : cupicef.Univ (cupicef.type Coq__Init__Specif__4) -> + P : + (__ : cupicef.Term (cupicef.type Coq__Init__Specif__4) A -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term (cupicef.type Coq__Init__Specif__4) A -> + cupicef.Univ cupicef.prop) -> + x : cupicef.Term (cupicef.type Coq__Init__Specif__4) A -> + __ : cupicef.Term cupicef.prop (P x) -> + __0 : cupicef.Term cupicef.prop (Q x) -> + cupicef.Term (cupicef.type Coq__Init__Specif__4) + (sig2 (cupicef.type Coq__Init__Specif__4) A P Q). + +[ ] cupicef.code_guarded? _ (exist2 _ _ _ _ _ _) --> cupicef.guarded. + +def match____sig2 : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type Coq__Init__Specif__4) -> + P : + (__ : cupicef.Term (cupicef.type Coq__Init__Specif__4) A -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term (cupicef.type Coq__Init__Specif__4) A -> + cupicef.Univ cupicef.prop) -> + P0 : + (cupicef.Term (cupicef.type Coq__Init__Specif__4) + (sig2 (cupicef.type Coq__Init__Specif__4) A P Q) -> + cupicef.Univ s) -> + case__exist2 : + (x0 : cupicef.Term (cupicef.type Coq__Init__Specif__4) A -> + __ : cupicef.Term cupicef.prop (P x0) -> + __0 : cupicef.Term cupicef.prop (Q x0) -> + cupicef.Term s (P0 (exist2 A P Q x0 __ __0))) -> + x : + cupicef.Term (cupicef.type Coq__Init__Specif__4) + (sig2 (cupicef.type Coq__Init__Specif__4) A P Q) -> + cupicef.Term s (P0 x). + +[ s, A, P, Q, P0, case__exist2, x0, __, __0] + match____sig2 s A P Q P0 case__exist2 (exist2 A P Q x0 __ __0) + --> + case__exist2 x0 __ __0. + +[ s, A, P, Q, P0, s'] + match____sig2 s A P Q + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P0 x))) + --> + match____sig2 s' A P Q (x => P0 x). + +def sig2__rect : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + P0 : + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (P x) -> + q : cupicef.Term cupicef.prop (Q x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P0 (exist2 A P Q x p q))) -> + s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P0 s) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + P0 : + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (P x) -> + q : cupicef.Term cupicef.prop (Q x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P0 (exist2 A P Q x p q))) => + s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) => + match____sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q + (s0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) => + P0 s0) + f s. + +def sig2__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + P0 : + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) -> + cupicef.Univ cupicef.prop) -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (P x) -> + q : cupicef.Term cupicef.prop (Q x) -> + cupicef.Term cupicef.prop (P0 (exist2 A P Q x p q))) -> + s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) -> + cupicef.Term cupicef.prop (P0 s) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + P0 : + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) -> + cupicef.Univ cupicef.prop) => + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (P x) -> + q : cupicef.Term cupicef.prop (Q x) -> + cupicef.Term cupicef.prop (P0 (exist2 A P Q x p q))) => + s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) => + match____sig2 cupicef.prop A P Q + (s0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) => + P0 s0) + f s. + +def sig2__rec : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + P0 : + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) -> + cupicef.Univ cupicef.set) -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (P x) -> + q : cupicef.Term cupicef.prop (Q x) -> + cupicef.Term cupicef.set (P0 (exist2 A P Q x p q))) -> + s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) -> + cupicef.Term cupicef.set (P0 s) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + P0 : + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) -> + cupicef.Univ cupicef.set) => + sig2__rect A P Q + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P0). + +def sigT : + Coq__Init__Specif__7 : cupicef.Sort -> + Coq__Init__Specif__8 : cupicef.Sort -> + A : cupicef.Univ Coq__Init__Specif__7 -> + P : + (__ : cupicef.Term Coq__Init__Specif__7 A -> + cupicef.Univ Coq__Init__Specif__8) -> + cupicef.Univ (cupicef.sup Coq__Init__Specif__7 Coq__Init__Specif__8). + +sigT' : + A : cupicef.Univ cupicef.sinf -> + P : (__ : cupicef.Term cupicef.sinf A -> cupicef.Univ cupicef.sinf) -> + cupicef.Univ cupicef.sinf. + +[ Coq__Init__Specif__7, Coq__Init__Specif__8, A, P] + cupicef.code (cupicef.cU _) + (sigT Coq__Init__Specif__7 Coq__Init__Specif__8 A P) + --> + sigT' (cupicef.code (cupicef.cU Coq__Init__Specif__7) A) + (__ => cupicef.code (cupicef.cU Coq__Init__Specif__8) (P __)). + +def Coq__Init__Specif__7 : + cupicef.Nat + +:= cupicef.s cupicef.z. + +def Coq__Init__Specif__8 : + cupicef.Nat + +:= cupicef.s cupicef.z. + +def existT : + A : cupicef.Univ (cupicef.type Coq__Init__Specif__7) -> + P : + (__ : cupicef.Term (cupicef.type Coq__Init__Specif__7) A -> + cupicef.Univ (cupicef.type Coq__Init__Specif__8)) -> + x : cupicef.Term (cupicef.type Coq__Init__Specif__7) A -> + __ : cupicef.Term (cupicef.type Coq__Init__Specif__8) (P x) -> + cupicef.Term + (cupicef.sup (cupicef.type Coq__Init__Specif__7) + (cupicef.type Coq__Init__Specif__8)) + (sigT (cupicef.type Coq__Init__Specif__7) + (cupicef.type Coq__Init__Specif__8) A P). + +[ ] cupicef.code_guarded? _ (existT _ _ _ _) --> cupicef.guarded. + +def match____sigT : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type Coq__Init__Specif__7) -> + P : + (__ : cupicef.Term (cupicef.type Coq__Init__Specif__7) A -> + cupicef.Univ (cupicef.type Coq__Init__Specif__8)) -> + P0 : + (cupicef.Term + (cupicef.sup (cupicef.type Coq__Init__Specif__7) + (cupicef.type Coq__Init__Specif__8)) + (sigT (cupicef.type Coq__Init__Specif__7) + (cupicef.type Coq__Init__Specif__8) A P) -> + cupicef.Univ s) -> + case__existT : + (x0 : cupicef.Term (cupicef.type Coq__Init__Specif__7) A -> + __ : cupicef.Term (cupicef.type Coq__Init__Specif__8) (P x0) -> + cupicef.Term s (P0 (existT A P x0 __))) -> + x : + cupicef.Term + (cupicef.sup (cupicef.type Coq__Init__Specif__7) + (cupicef.type Coq__Init__Specif__8)) + (sigT (cupicef.type Coq__Init__Specif__7) + (cupicef.type Coq__Init__Specif__8) A P) -> + cupicef.Term s (P0 x). + +[ s, A, P, P0, case__existT, x0, __] + match____sigT s A P P0 case__existT (existT A P x0 __) + --> + case__existT x0 __. + +[ s, A, P, P0, s'] + match____sigT s A P + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P0 x))) + --> + match____sigT s' A P (x => P0 x). + +def sigT__rect : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + P0 : + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P0 (existT A P x p))) -> + s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P0 s) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + P0 : + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P0 (existT A P x p))) => + s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P) => + match____sigT (cupicef.type (cupicef.s cupicef.z)) A P + (s0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P) => + P0 s0) + f s. + +def sigT__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + P0 : + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P) -> + cupicef.Univ cupicef.prop) -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) -> + cupicef.Term cupicef.prop (P0 (existT A P x p))) -> + s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P) -> + cupicef.Term cupicef.prop (P0 s) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + P0 : + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P) -> + cupicef.Univ cupicef.prop) => + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) -> + cupicef.Term cupicef.prop (P0 (existT A P x p))) => + s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P) => + match____sigT cupicef.prop A P + (s0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P) => + P0 s0) + f s. + +def sigT__rec : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + P0 : + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P) -> + cupicef.Univ cupicef.set) -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) -> + cupicef.Term cupicef.set (P0 (existT A P x p))) -> + s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P) -> + cupicef.Term cupicef.set (P0 s) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + P0 : + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P) -> + cupicef.Univ cupicef.set) => + sigT__rect A P + (cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P) + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P) => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P) + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P0). + +def sigT2 : + Coq__Init__Specif__11 : cupicef.Sort -> + Coq__Init__Specif__12 : cupicef.Sort -> + Coq__Init__Specif__13 : cupicef.Sort -> + A : cupicef.Univ Coq__Init__Specif__11 -> + P : + (__ : cupicef.Term Coq__Init__Specif__11 A -> + cupicef.Univ Coq__Init__Specif__12) -> + Q : + (__ : cupicef.Term Coq__Init__Specif__11 A -> + cupicef.Univ Coq__Init__Specif__13) -> + cupicef.Univ + (cupicef.sup Coq__Init__Specif__11 + (cupicef.sup Coq__Init__Specif__12 Coq__Init__Specif__13)). + +sigT2' : + A : cupicef.Univ cupicef.sinf -> + P : (__ : cupicef.Term cupicef.sinf A -> cupicef.Univ cupicef.sinf) -> + Q : (__ : cupicef.Term cupicef.sinf A -> cupicef.Univ cupicef.sinf) -> + cupicef.Univ cupicef.sinf. + +[ Coq__Init__Specif__11, Coq__Init__Specif__12, Coq__Init__Specif__13, A, P, Q] + cupicef.code (cupicef.cU _) + (sigT2 Coq__Init__Specif__11 Coq__Init__Specif__12 Coq__Init__Specif__13 + A P Q) + --> + sigT2' (cupicef.code (cupicef.cU Coq__Init__Specif__11) A) + (__ => cupicef.code (cupicef.cU Coq__Init__Specif__12) (P __)) + (__ => cupicef.code (cupicef.cU Coq__Init__Specif__13) (Q __)). + +def Coq__Init__Specif__11 : + cupicef.Nat + +:= cupicef.s cupicef.z. + +def Coq__Init__Specif__12 : + cupicef.Nat + +:= cupicef.s cupicef.z. + +def Coq__Init__Specif__13 : + cupicef.Nat + +:= cupicef.s cupicef.z. + +def existT2 : + A : cupicef.Univ (cupicef.type Coq__Init__Specif__11) -> + P : + (__ : cupicef.Term (cupicef.type Coq__Init__Specif__11) A -> + cupicef.Univ (cupicef.type Coq__Init__Specif__12)) -> + Q : + (__ : cupicef.Term (cupicef.type Coq__Init__Specif__11) A -> + cupicef.Univ (cupicef.type Coq__Init__Specif__13)) -> + x : cupicef.Term (cupicef.type Coq__Init__Specif__11) A -> + __ : cupicef.Term (cupicef.type Coq__Init__Specif__12) (P x) -> + __0 : cupicef.Term (cupicef.type Coq__Init__Specif__13) (Q x) -> + cupicef.Term + (cupicef.sup (cupicef.type Coq__Init__Specif__11) + (cupicef.sup (cupicef.type Coq__Init__Specif__12) + (cupicef.type Coq__Init__Specif__13))) + (sigT2 (cupicef.type Coq__Init__Specif__11) + (cupicef.type Coq__Init__Specif__12) + (cupicef.type Coq__Init__Specif__13) A P Q). + +[ ] cupicef.code_guarded? _ (existT2 _ _ _ _ _ _) --> cupicef.guarded. + +def match____sigT2 : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type Coq__Init__Specif__11) -> + P : + (__ : cupicef.Term (cupicef.type Coq__Init__Specif__11) A -> + cupicef.Univ (cupicef.type Coq__Init__Specif__12)) -> + Q : + (__ : cupicef.Term (cupicef.type Coq__Init__Specif__11) A -> + cupicef.Univ (cupicef.type Coq__Init__Specif__13)) -> + P0 : + (cupicef.Term + (cupicef.sup (cupicef.type Coq__Init__Specif__11) + (cupicef.sup (cupicef.type Coq__Init__Specif__12) + (cupicef.type Coq__Init__Specif__13))) + (sigT2 (cupicef.type Coq__Init__Specif__11) + (cupicef.type Coq__Init__Specif__12) + (cupicef.type Coq__Init__Specif__13) A P Q) -> + cupicef.Univ s) -> + case__existT2 : + (x0 : cupicef.Term (cupicef.type Coq__Init__Specif__11) A -> + __ : cupicef.Term (cupicef.type Coq__Init__Specif__12) (P x0) -> + __0 : cupicef.Term (cupicef.type Coq__Init__Specif__13) (Q x0) -> + cupicef.Term s (P0 (existT2 A P Q x0 __ __0))) -> + x : + cupicef.Term + (cupicef.sup (cupicef.type Coq__Init__Specif__11) + (cupicef.sup (cupicef.type Coq__Init__Specif__12) + (cupicef.type Coq__Init__Specif__13))) + (sigT2 (cupicef.type Coq__Init__Specif__11) + (cupicef.type Coq__Init__Specif__12) + (cupicef.type Coq__Init__Specif__13) A P Q) -> + cupicef.Term s (P0 x). + +[ s, A, P, Q, P0, case__existT2, x0, __, __0] + match____sigT2 s A P Q P0 case__existT2 (existT2 A P Q x0 __ __0) + --> + case__existT2 x0 __ __0. + +[ s, A, P, Q, P0, s'] + match____sigT2 s A P Q + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P0 x))) + --> + match____sigT2 s' A P Q (x => P0 x). + +def sigT2__rect : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + P0 : + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P0 (existT2 A P Q x p q))) -> + s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P0 s) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + P0 : + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P0 (existT2 A P Q x p q))) => + s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q) => + match____sigT2 (cupicef.type (cupicef.s cupicef.z)) A P Q + (s0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q) => + P0 s0) + f s. + +def sigT2__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + P0 : + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q) -> + cupicef.Univ cupicef.prop) -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q x) -> + cupicef.Term cupicef.prop (P0 (existT2 A P Q x p q))) -> + s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q) -> + cupicef.Term cupicef.prop (P0 s) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + P0 : + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q) -> + cupicef.Univ cupicef.prop) => + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q x) -> + cupicef.Term cupicef.prop (P0 (existT2 A P Q x p q))) => + s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q) => + match____sigT2 cupicef.prop A P Q + (s0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q) => + P0 s0) + f s. + +def sigT2__rec : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + P0 : + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q) -> + cupicef.Univ cupicef.set) -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q x) -> + cupicef.Term cupicef.set (P0 (existT2 A P Q x p q))) -> + s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q) -> + cupicef.Term cupicef.set (P0 s) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + P0 : + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q) -> + cupicef.Univ cupicef.set) => + sigT2__rect A P Q + (cupicef.cast + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q) + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q) => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q) + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P0). + +def proj1__sig : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + e : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A P) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + e : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A P) => + match____sig (cupicef.type (cupicef.s cupicef.z)) A P + (e0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A P) => + A) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + b : cupicef.Term cupicef.prop (P a) => + a) + e. + +def proj2__sig : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + e : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A P) -> + cupicef.Term cupicef.prop (P (proj1__sig A P e)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + e : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A P) => + match____sig cupicef.prop A P + (e0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A P) => + P (proj1__sig A P e0)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + b : cupicef.Term cupicef.prop (P a) => + b) + e. + +def sig__of__sig2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + X : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A P) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + X : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) => + exist A P + (match____sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q + (X0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) => + A) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + __ : cupicef.Term cupicef.prop (P a) => + __0 : cupicef.Term cupicef.prop (Q a) => + a) + X) + (match____sig2 cupicef.prop A P Q + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) => + P + (match____sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q + (s0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) => + A) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + __ : cupicef.Term cupicef.prop (P a) => + __0 : cupicef.Term cupicef.prop (Q a) => + a) + s)) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p : cupicef.Term cupicef.prop (P x) => + __ : cupicef.Term cupicef.prop (Q x) => + p) + X). + +def proj3__sig : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + e : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) -> + cupicef.Term cupicef.prop (Q (proj1__sig A P (sig__of__sig2 A P Q e))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + e : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) => + match____sig2 cupicef.prop A P Q + (e0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) => + Q (proj1__sig A P (sig__of__sig2 A P Q e0))) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + b : cupicef.Term cupicef.prop (P a) => + c : cupicef.Term cupicef.prop (Q a) => + c) + e. + +def projT1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + x : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + x : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P) => + match____sigT (cupicef.type (cupicef.s cupicef.z)) A P + (x0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P) => + A) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + a) + x. + +def projT2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + x : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (projT1 A P x)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + x : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P) => + match____sigT (cupicef.type (cupicef.s cupicef.z)) A P + (x0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P) => + P (projT1 A P x0)) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + h : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x0) => + h) + x. + +def sigT__of__sigT2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + X : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q) -> + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + X : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q) => + existT A P + (match____sigT2 (cupicef.type (cupicef.s cupicef.z)) A P Q + (X0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q) => + A) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q a) => + a) + X) + (match____sigT2 (cupicef.type (cupicef.s cupicef.z)) A P Q + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q) => + P + (match____sigT2 (cupicef.type (cupicef.s cupicef.z)) A P Q + (s0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q) => + A) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q a) => + a) + s)) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q x) => + p) + X). + +def projT3 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + e : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Q (projT1 A P (sigT__of__sigT2 A P Q e))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + e : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q) => + match____sigT2 (cupicef.type (cupicef.s cupicef.z)) A P Q + (e0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q) => + Q (projT1 A P (sigT__of__sigT2 A P Q e0))) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + c : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q a) => + c) + e. + +def sig__of__sigT : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + X : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sigT (cupicef.type (cupicef.s cupicef.z)) cupicef.prop A P) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A P) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + X : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sigT (cupicef.type (cupicef.s cupicef.z)) cupicef.prop A P) => + exist A P + (projT1 A + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + X) + (projT2 A + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + X). + +def sigT__of__sig : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + X : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A P) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sigT (cupicef.type (cupicef.s cupicef.z)) cupicef.prop A P) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + X : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A P) => + existT A + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P) + (proj1__sig A P X) (proj2__sig A P X). + +def sig2__of__sigT2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + X : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) cupicef.prop cupicef.prop A + P Q) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + X : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) cupicef.prop cupicef.prop + A P Q) => + exist2 A P Q + (projT1 A + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + (sigT__of__sigT2 A + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + X)) + (projT2 A + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + (sigT__of__sigT2 A + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + X)) + (projT3 A + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + X). + +def sigT2__of__sig2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + X : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) cupicef.prop cupicef.prop A P + Q) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + X : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q) => + existT2 A + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I Q) + (proj1__sig A P (sig__of__sig2 A P Q X)) + (proj2__sig A P (sig__of__sig2 A P Q X)) (proj3__sig A P Q X). + +def sigT__eta : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + p + (existT A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + p))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + match____sigT cupicef.prop A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + s + (existT A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + s) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + s))) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + (existT A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (existT A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0)) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (existT A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0)))) + p. + +def sig__eta : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + p + (exist A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + p))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + match____sig cupicef.prop A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + s + (exist A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + s) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + s))) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p0 : cupicef.Term cupicef.prop (P x) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + (exist A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (exist A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (exist A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0)))) + p. + +def sigT2__eta : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a))) + p + (existT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + p)) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + p)) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + p))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + match____sigT2 cupicef.prop A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a))) + s + (existT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s)) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s)) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + s))) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q x) => + Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a))) + (existT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (existT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q))) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (existT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q))) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + (existT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q)))) + p. + +def sig2__eta : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a))) + p + (exist2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + p)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + p)) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + p))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + match____sig2 cupicef.prop A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a))) + s + (exist2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s)) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + s))) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p0 : cupicef.Term cupicef.prop (P x) => + q : cupicef.Term cupicef.prop (Q x) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a))) + (exist2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (exist2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q))) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (exist2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q))) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + (exist2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q)))) + p. + +thm exists__to__inhabited__sig : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.inhabited + (sig (cupicef.type (cupicef.s cupicef.z)) A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x)) => + Coq__Init__Logic.match____ex cupicef.prop A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x)) => + Coq__Init__Logic.inhabited + (sig (cupicef.type (cupicef.s cupicef.z)) A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x))) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term cupicef.prop (P x) => + Coq__Init__Logic.inhabits + (sig (cupicef.type (cupicef.s cupicef.z)) A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x0)) + (exist A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x0) + x y)) + H. + +thm inhabited__sig__to__exists : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.inhabited + (sig (cupicef.type (cupicef.s cupicef.z)) A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.inhabited + (sig (cupicef.type (cupicef.s cupicef.z)) A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x))) => + Coq__Init__Logic.match____inhabited cupicef.prop + (sig (cupicef.type (cupicef.s cupicef.z)) A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.inhabited + (sig (cupicef.type (cupicef.s cupicef.z)) A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x))) => + Coq__Init__Logic.ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x)) + (X : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x)) => + match____sig cupicef.prop A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x) + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P x)) => + Coq__Init__Logic.ex A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x)) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term cupicef.prop (P x) => + Coq__Init__Logic.ex__intro A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x0) + x y) + X) + H. + +def projT1__eq : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) v)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) => + Coq__Init__Logic.f__equal + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) + u v p. + +def projT2__eq : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + v) + (projT1__eq A P u v p)) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) v)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + u + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + x : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u s) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + s))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + s) + (projT1__eq A P u s x)) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) s)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (projT1__eq A P u u + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u)))) + v p. + +def eq__existT__uncurried : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + u1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + v1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + u2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P u1) -> + v2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P v1) -> + pq : + cupicef.Term cupicef.prop + (sigT cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 P u2 v1 p) + v2)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P)) + (existT A P u1 u2) (existT A P v1 v2)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + u1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + v1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + u2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P u1) => + v2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P v1) => + pq : + cupicef.Term cupicef.prop + (sigT cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 P u2 v1 p) + v2)) => + match____sigT cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1)) + (cupicef.cast + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 P u2 v1 p) + v2)) + (s : + cupicef.Term cupicef.prop + (sigT cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 P u2 v1 p) + v2)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P)) + (existT A P u1 u2) (existT A P v1 v2)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 P u2 v1 p) + v2) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 P u2 v1 p) + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P v1) => + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 P u2 v1 p) + y) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P)) + (existT A P u1 u2) (existT A P v1 y)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P)) + (existT A P u1 u2) + (existT A P v1 + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 P u2 v1 p))) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P)) + (existT A P u1 u2) + (existT A P v1 + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 P u2 v1 p))) + cupicef.I + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 y) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P)) + (existT A P u1 u2) + (existT A P y + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 P u2 y e))) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P)) + (existT A P u1 + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 P u2 u1 + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1)))) + v1 p)) + v2 q) + pq. + +def eq__sigT__uncurried : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + pq : + cupicef.Term cupicef.prop + (sigT cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + u v) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + pq : + cupicef.Term cupicef.prop + (sigT cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) => + match____sigT cupicef.prop A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (sigT cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + 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cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + s) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + s) + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term 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cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (existT A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + u1 u2)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (existT A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + u1 u2)) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) => + match____sigT cupicef.prop A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (sigT cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + 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(Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (existT A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u1 u2)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (existT A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u1 u2)) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + s) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + s))) + (pq1 : + cupicef.Term cupicef.prop + (sigT cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (existT A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u1 u2)) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + s)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (existT A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u1 u2)) + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + s)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + s))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (existT A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u1 u2)) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (projT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (existT A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u1 u2)) + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + s) + p) + (projT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + s))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (existT A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u1 u2) + s)) + (v1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + v2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P v1) => + pq1 : + cupicef.Term cupicef.prop + (sigT cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (existT A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + u1 u2)) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (existT A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + v1 v2))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (existT A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u1 u2)) + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (existT A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v1 v2))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (existT A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v1 v2)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (existT A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u1 u2)) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (existT A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u1 u2)) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (existT A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v1 v2)) + p) + (projT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (existT A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + v1 v2)))) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (existT A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u1 u2) + (existT A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v1 v2)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (existT A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u1 u2) + (existT A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v1 v2)) + cupicef.I + (eq__existT__uncurried A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u1 v1 u2 v2 pq1)) + v pq0) + u pq. + +def eq__sigT : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + v)) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + v)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + u v) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + v)) => + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + v)) => + eq__sigT__uncurried A P u v + (existT + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (cupicef.cast + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) + (p0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + u) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + v)) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + u) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + v)) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p0) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + p q). + +def eq__sigT__hprop : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + P__hprop : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + p q)) -> + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + v)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + u v) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + P__hprop : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + p q)) => + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + v)) => + eq__sigT A P u v p + (P__hprop + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) v) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + v) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) v)). + +def eq__sigT__uncurried__iff : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) + (sigT cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) => + sigT cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (sigT cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (__ : + cupicef.Term cupicef.prop + (sigT cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + u) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + u) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + v) + p) + (projT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) => + Coq__Init__Logic.eq__ind__r + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + v + (u0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) => + sigT cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u0) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u0) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u0) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u0) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (existT + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (cupicef.cast + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + v) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + v)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + u H) + (eq__sigT__uncurried A P u v). + +def let__f : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + Q : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Q (eq__sigT A P u v p q))) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Q (eq__sigT A P u v (projT1__eq A P u v p) (projT2__eq A P u v p))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + Q : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Q (eq__sigT A P u v p q))) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) => + f (projT1__eq A P u v p) (projT2__eq A P u v p). + +def eq__sigT__rect : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + Q : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Q (eq__sigT A P u v p q))) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q p) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + Q : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Q (eq__sigT A P u v p q))) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) => + match____sigT (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + cupicef.prod + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I + (cupicef.prod cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + s v) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a))) + s v) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + (Q0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a))) + s v) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + cupicef.prod cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + s v) + (p0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a))) + s v) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Q0 + (eq__sigT A P s v (projT1__eq A P s v p0) + (projT2__eq A P s v p0))) + (f0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Q0 + (eq__sigT A P s v (projT1__eq A P s v p0) + (projT2__eq A P s v p0))) => + Q0 p0)))) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + 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+ v p1))) => + Coq__Init__Logic.match____eq + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + (existT A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) x + p0) + (y : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (existT A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0) + y) => + cupicef.prod + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.prod cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s 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(cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (existT A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + x p0) + y) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + (Q1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (existT A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + x p0) + y) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Q1 + (eq__sigT A P + (existT A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0) + y + (projT1__eq A P + (existT A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0) + y e) + (projT2__eq A P + (existT A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0) + y e))) + (f1 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Q1 + (eq__sigT A P + (existT A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0) + y + (projT1__eq A P + (existT A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0) + y e) + (projT2__eq A P + (existT A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0) + y e))) => + Q1 e))) + (Q1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a))) + (existT A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0) + (existT A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0)) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f1 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Q1 + (eq__sigT A P + (existT A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0) + (existT A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0) + (projT1__eq A P + (existT A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + x p0) + (existT A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + x p0) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (existT A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0))) + (projT2__eq A P + (existT A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + x p0) + (existT A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + x p0) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (existT A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0))))) => + f1) + v p1 Q0 f0) + u Q p (let__f A P u v Q f p). + +def eq__sigT__rec : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + Q : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Univ cupicef.set) -> + f : + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) -> + cupicef.Term cupicef.set (Q (eq__sigT A P u v p q))) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Term cupicef.set (Q p) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + Q : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Univ cupicef.set) => + eq__sigT__rect A P u v + (cupicef.cast + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a))) + u v) => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a))) + u v) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I Q). + +def eq__sigT__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + Q : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Univ cupicef.prop) -> + f : + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) -> + cupicef.Term cupicef.prop (Q (eq__sigT A P u v p q))) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Term cupicef.prop (Q p) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + Q : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Univ cupicef.prop) => + eq__sigT__rec A P u v + (cupicef.cast + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + (cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a))) + u v) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a))) + u v) => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + cupicef.I Q). + +def eq__sigT__hprop__iff : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + P__hprop : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + p q)) -> + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + v))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + P__hprop : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + p q)) => + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) + (x : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + v))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + v)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) => + Coq__Init__Logic.f__equal + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) + u v p) + (eq__sigT__hprop A P P__hprop u v). + +def eq__sigT__nondep : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) -> + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) v)) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) u) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) v)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B))) + u v) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) => + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) v)) => + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) u) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) v)) => + eq__sigT A (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + u v p + (Coq__Init__Logic.eq__trans + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + a) + u) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + a) + v) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) u) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) v) + (Coq__Init__Logic.rew__const A B + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + a) + v) + p + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + u)) + q). + +def rew__sigT : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) -> + cupicef.Univ cupicef.prop) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sigT (cupicef.type (cupicef.s cupicef.z)) cupicef.prop (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p)) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) cupicef.prop (P y) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y) => + Q y p))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + sigT (cupicef.type (cupicef.s cupicef.z)) cupicef.prop (P a) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + Q a p)) + u y H) + (existT (P y) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P y) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P y) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I (Q y)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (projT1 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p)) + u) + y H) + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x a) => + Q a + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (projT1 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) + u) + a x0)) + (projT2 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p)) + u) + y H))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) -> + cupicef.Univ cupicef.prop) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sigT (cupicef.type (cupicef.s cupicef.z)) cupicef.prop (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p)) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) cupicef.prop (P y0) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y0) => + Q y0 p))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + sigT (cupicef.type (cupicef.s cupicef.z)) cupicef.prop (P a) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + Q a p)) + u y0 e) + (existT (P y0) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P y0) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y0) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P y0) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y0) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I (Q y0)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (projT1 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) + u) + y0 e) + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x a) => + Q a + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (projT1 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + Q x p)) + u) + a x0)) + (projT2 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) + u) + y0 e))) + (match____sigT cupicef.prop (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p)) + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sigT (cupicef.type (cupicef.s cupicef.z)) cupicef.prop ( + P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) cupicef.prop ( + P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + sigT (cupicef.type (cupicef.s cupicef.z)) cupicef.prop ( + P a) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + Q a p)) + s x + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + (existT (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I (Q x)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (projT1 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) + s) + x + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + (projT2 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) + s))) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + q : cupicef.Term cupicef.prop (Q x x0) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) cupicef.prop ( + P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p))) + (existT (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I (Q x)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (projT1 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) + (existT (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + Q x p)) + x0 q)) + x + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + (projT2 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) + (existT (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) + x0 q)))) + u) + y H. + +def proj1__sig__eq : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) v)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) => + Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) + u v p. + +def proj2__sig__eq : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + v) + (proj1__sig__eq A P u v p)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) v)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + u + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + x : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u s) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + s))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + s) + (proj1__sig__eq A P u s x)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) s)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (proj1__sig__eq A P u u + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u)))) + v p. + +def eq__exist__uncurried : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + u1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + v1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + u2 : cupicef.Term cupicef.prop (P u1) -> + v2 : cupicef.Term cupicef.prop (P v1) -> + pq : + cupicef.Term cupicef.prop + (sig cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (sig (cupicef.type (cupicef.s cupicef.z)) A P)) + (exist A P u1 u2) (exist A P v1 v2)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + u1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + v1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + u2 : cupicef.Term cupicef.prop (P u1) => + v2 : cupicef.Term cupicef.prop (P v1) => + pq : + cupicef.Term cupicef.prop + (sig cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2)) => + match____sig cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) + (s : + cupicef.Term cupicef.prop + (sig cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (sig (cupicef.type (cupicef.s cupicef.z)) A P)) + (exist A P u1 u2) (exist A P v1 v2)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + (y : cupicef.Term cupicef.prop (P v1) => + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + y) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (sig (cupicef.type (cupicef.s cupicef.z)) A P)) + (exist A P u1 u2) (exist A P v1 y)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (sig (cupicef.type (cupicef.s cupicef.z)) A P)) + (exist A P u1 u2) + (exist A P v1 + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p))) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (sig (cupicef.type (cupicef.s cupicef.z)) A P)) + (exist A P u1 u2) + (exist A P v1 + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p))) + cupicef.I + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 y) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (sig (cupicef.type (cupicef.s cupicef.z)) A P)) + (exist A P u1 u2) + (exist A P y + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 y e))) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (sig (cupicef.type (cupicef.s cupicef.z)) A P)) + (exist A P u1 + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 u1 + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1)))) + v1 p)) + v2 q) + pq. + +def eq__sig__uncurried : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + pq : + cupicef.Term cupicef.prop + (sig cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + u v) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + pq : + cupicef.Term cupicef.prop + (sig cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) => + match____sig cupicef.prop A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (sig cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + s) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + s) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + s) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + s) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (pq0 : + cupicef.Term cupicef.prop + (sig cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + s) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + s) + (proj1__sig A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + s) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + 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(cupicef.type (cupicef.s cupicef.z)) A => + P a) + (exist A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u1 u2)) + (cupicef.cast + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (exist A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u1 u2)) + (proj1__sig A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + s) + p) + (proj2__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + s))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (exist A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u1 u2) + s)) + (v1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + v2 : cupicef.Term cupicef.prop (P v1) => + pq1 : + cupicef.Term cupicef.prop + (sig cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (exist A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + u1 u2)) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (exist A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + v1 v2))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (exist A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u1 u2)) + (proj1__sig A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (exist A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v1 v2))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (exist A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v1 v2)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (exist A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u1 u2)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a)) + (proj2__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (exist A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u1 u2)) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (exist A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v1 v2)) + p) + (proj2__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (exist A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + v1 v2)))) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (exist A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u1 u2) + (exist A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v1 v2)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (exist A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u1 u2) + (exist A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v1 v2)) + cupicef.I + (eq__exist__uncurried A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u1 v1 u2 v2 pq1)) + v pq0) + u pq. + +def eq__sig : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + v)) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + v)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + u v) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + v)) => + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + v)) => + eq__sig__uncurried A P u v + (exist + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (p0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p0) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + v)) + p q). + +def let__f0 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + Q : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Q (eq__sig A P u v p q))) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Q + (eq__sig A P u v (proj1__sig__eq A P u v p) (proj2__sig__eq A P u v p))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + Q : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Q (eq__sig A P u v p q))) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) => + f (proj1__sig__eq A P u v p) (proj2__sig__eq A P u v p). + +def eq__sig__rect : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + Q : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Q (eq__sig A P u v p q))) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q p) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + Q : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Q (eq__sig A P u v p q))) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) => + match____sig (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + cupicef.prod + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I + (cupicef.prod cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + s v) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a))) + s v) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + (Q0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a))) + s v) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + cupicef.prod cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + s v) + (p0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a))) + s v) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Q0 + (eq__sig A P s v (proj1__sig__eq A P s v p0) + (proj2__sig__eq A P s v p0))) + (f0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Q0 + (eq__sig A P s v (proj1__sig__eq A P s v p0) + (proj2__sig__eq A P s v p0))) => + Q0 p0)))) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p0 : cupicef.Term cupicef.prop (P x) => + Q0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (exist A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0) + v) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + p1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (exist A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0) + v) => + f0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Q0 + (eq__sig A P + (exist A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0) + v + (proj1__sig__eq A P + (exist A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0) + v p1) + (proj2__sig__eq A P + (exist A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0) + v p1))) => + Coq__Init__Logic.match____eq + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + (exist A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) x + p0) + (y : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (exist A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0) + y) => + cupicef.prod + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.prod cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (exist A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0) + y) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (exist A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + x p0) + y) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + (Q1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (exist A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + x p0) + y) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Q1 + (eq__sig A P + (exist A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0) + y + (proj1__sig__eq A P + (exist A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0) + y e) + (proj2__sig__eq A P + (exist A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0) + y e))) + (f1 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Q1 + (eq__sig A P + (exist A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0) + y + (proj1__sig__eq A P + (exist A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0) + y e) + (proj2__sig__eq A P + (exist A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0) + y e))) => + Q1 e))) + (Q1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a))) + (exist A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0) + (exist A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0)) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f1 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Q1 + (eq__sig A P + (exist A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0) + (exist A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0) + (proj1__sig__eq A P + (exist A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + x p0) + (exist A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + x p0) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (exist A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0))) + (proj2__sig__eq A P + (exist A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + x p0) + (exist A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + x p0) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (exist A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + x p0))))) => + f1) + v p1 Q0 f0) + u Q p (let__f0 A P u v Q f p). + +def eq__sig__rec : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + Q : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Univ cupicef.set) -> + f : + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) -> + cupicef.Term cupicef.set (Q (eq__sig A P u v p q))) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Term cupicef.set (Q p) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + Q : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Univ cupicef.set) => + eq__sig__rect A P u v + (cupicef.cast + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a))) + u v) => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a))) + u v) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I Q). + +def eq__sig__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + Q : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Univ cupicef.prop) -> + f : + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) -> + cupicef.Term cupicef.prop (Q (eq__sig A P u v p q))) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Term cupicef.prop (Q p) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + Q : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) -> + cupicef.Univ cupicef.prop) => + eq__sig__rec A P u v + (cupicef.cast + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + (cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a))) + u v) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a))) + u v) => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + cupicef.I Q). + +def eq__sig__hprop : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + P__hprop : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (P x) -> + q : cupicef.Term cupicef.prop (P x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + p q)) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + v)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + u v) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + P__hprop : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (P x) -> + q : cupicef.Term cupicef.prop (P x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + p q)) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + v)) => + eq__sig A P u v p + (P__hprop + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) v) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + v) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) v)). + +def eq__sig__uncurried__iff : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) + (sig cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) => + sig cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (sig cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type 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(cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v))) + u H) + (eq__sig__uncurried A P u v). + +def eq__sig__hprop__iff : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + P__hprop : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (P x) -> + q : cupicef.Term cupicef.prop (P x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + p q)) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + v))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + P__hprop : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (P x) -> + q : cupicef.Term cupicef.prop (P x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + p q)) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) + (x : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + v))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + v)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + v)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + u v) => + Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) + u v p) + (eq__sig__hprop A P P__hprop u v). + +def rew__sig : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) -> + cupicef.Univ cupicef.prop) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p)) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) (P y) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y) => + Q y p))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + sig (cupicef.type (cupicef.s cupicef.z)) (P a) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + Q a p)) + u y H) + (exist (P y) (Q y) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (proj1__sig (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + u) + y H) + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x a) => + Q a + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (proj1__sig (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p) + u) + a x0)) + (proj2__sig (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + u) + y H))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) -> + cupicef.Univ cupicef.prop) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p)) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) (P y0) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y0) => + Q y0 p))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + sig (cupicef.type (cupicef.s cupicef.z)) (P a) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + Q a p)) + u y0 e) + (exist (P y0) (Q y0) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (proj1__sig (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + u) + y0 e) + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x a) => + Q a + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (proj1__sig (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p) + u) + a x0)) + (proj2__sig (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + u) + y0 e))) + (match____sig cupicef.prop (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => Q x p) + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig (cupicef.type (cupicef.s cupicef.z)) (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + sig (cupicef.type (cupicef.s cupicef.z)) (P a) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + Q a p)) + s x + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + (exist (P x) (Q x) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (proj1__sig (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p) + s) + x + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + (proj2__sig (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + s))) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + q : cupicef.Term cupicef.prop (Q x x0) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p))) + (exist (P x) (Q x) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (proj1__sig (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p) + (exist (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p) + x0 q)) + x + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + (proj2__sig (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (exist (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p) + x0 q)))) + u) + y H. + +def sigT__of__sigT2__eq : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) u) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) v)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) => + Coq__Init__Logic.f__equal + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) + u v p. + +def projT1__of__sigT2__eq : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + v))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) => + projT1__eq A P + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) u) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) v) + (sigT__of__sigT2__eq A P Q u v p). + +def projT2__of__sigT2__eq : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + (projT1__of__sigT2__eq A P Q u v p)) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + v))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a))) + u + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + x : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u s) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s)) + (projT1__of__sigT2__eq A P Q u s x)) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + s))) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1__of__sigT2__eq A P Q u u + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u)))) + v p. + +def projT3__eq : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + (projT1__of__sigT2__eq A P Q u v p)) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) v)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a))) + u + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + x : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u s) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s)) + (projT1__of__sigT2__eq A P Q u s x)) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) s)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1__of__sigT2__eq A P Q u u + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u)))) + v p. + +def eq__existT2__uncurried : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + u1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + v1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + u2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P u1) -> + v2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P v1) -> + u3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q u1) -> + v3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q v1) -> + pqr : + cupicef.Term cupicef.prop + (sigT2 cupicef.prop cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 P u2 v1 p) + v2) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 Q u3 v1 p) + v3)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q)) + (existT2 A P Q u1 u2 u3) (existT2 A P Q v1 v2 v3)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + u1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + v1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + u2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P u1) => + v2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P v1) => + u3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q u1) => + v3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q v1) => + pqr : + cupicef.Term cupicef.prop + (sigT2 cupicef.prop cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 P u2 v1 p) + v2) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 Q u3 v1 p) + v3)) => + match____sigT2 cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1)) + (cupicef.cast + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 P u2 v1 p) + v2)) + (cupicef.cast + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 Q u3 v1 p) + v3)) + (s : + cupicef.Term cupicef.prop + (sigT2 cupicef.prop cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 P u2 v1 p) + v2) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 Q u3 v1 p) + v3)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q)) + (existT2 A P Q u1 u2 u3) (existT2 A P Q v1 v2 v3)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 P u2 v1 p) + v2) => + r : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 Q u3 v1 p) + v3) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 Q u3 v1 p) + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q v1) => + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 Q u3 v1 p) + y) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q)) + (existT2 A P Q u1 u2 u3) (existT2 A P Q v1 v2 y)) + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 P u2 v1 p) + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P v1) => + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 P u2 v1 p) + y) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q)) + (existT2 A P Q u1 u2 u3) + (existT2 A P Q v1 y + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 Q u3 v1 p))) + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 y) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q)) + (existT2 A P Q u1 u2 u3) + (existT2 A P Q y + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 P u2 y e) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 Q u3 y e))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q)) + (existT2 A P Q u1 u2 u3) (existT2 A P Q u1 u2 u3)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q)) + (existT2 A P Q u1 u2 u3) + (existT2 A P Q u1 + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 P u2 u1 + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 Q u3 u1 + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1)))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P Q)) + (existT2 A P Q u1 u2 u3))) + v1 p) + v2 q) + v3 r) + pqr. + +def eq__sigT2__uncurried : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + pqr : + cupicef.Term cupicef.prop + (sigT2 cupicef.prop cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a))) + u v) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + pqr : + cupicef.Term cupicef.prop + (sigT2 cupicef.prop cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) => + match____sigT2 cupicef.prop A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (sigT2 cupicef.prop cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + s)) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + s)) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (pqr0 : + cupicef.Term cupicef.prop + (sigT2 cupicef.prop cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type 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cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + u1 u2 u3))) + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A 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+ (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3))) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (projT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3))) + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s)) + p) + (projT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + 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+ P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + u1 u2 u3))) + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + 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(cupicef.s cupicef.z)) + A => + Q a) + (projT3 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3)) + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s)) + p) + (projT3 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + s))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + (existT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3) + s)) + (v1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + v2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P v1) => + v3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q v1) => + pqr1 : + cupicef.Term cupicef.prop + (sigT2 cupicef.prop cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3))) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v1 v2 v3)))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3))) + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v1 v2 v3)))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v1 v2 v3))))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3))) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3))) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v1 v2 v3))) + p) + (projT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v1 v2 v3)))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3))) + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v1 v2 v3)))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v1 v2 v3))))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3))) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (projT3 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3)) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v1 v2 v3))) + p) + (projT3 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (existT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v1 v2 v3)))) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + (existT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3) + (existT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v1 v2 v3)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + (existT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3) + (existT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v1 v2 v3)) + cupicef.I + (eq__existT2__uncurried A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + u1 v1 u2 v2 u3 v3 pqr1)) + v pqr0) + u pqr. + +def eq__sigT2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + r : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + v)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a))) + u v) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) => + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) => + r : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + v)) => + eq__sigT2__uncurried A P Q u v + (existT2 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (cupicef.cast + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (p0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + u)) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p0) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (cupicef.cast + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (p0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + u)) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p0) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + p q r). + +def eq__sigT2__hprop : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q__hprop : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q x) -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q x)) + p q)) -> + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + u) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + v)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a))) + u v) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q__hprop : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q x) -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q x)) + p q)) => + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + u) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + v)) => + eq__sigT2 A P Q u v + (projT1__eq A P + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) u) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) v) + p) + (projT2__eq A P + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) u) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) v) + p) + (Q__hprop + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + v)) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + (projT1__eq A P + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v) + p)) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) v)). + +def eq__sigT2__uncurried__iff : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) + (sigT2 cupicef.prop cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) => + sigT2 cupicef.prop cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + u)) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + u)) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (sigT2 cupicef.prop cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + u)) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s 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(cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (cupicef.cast + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (projT1 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (projT3 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v) + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + u H) + (eq__sigT2__uncurried A P Q u v). + +def let__f1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + R : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + r : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (R (eq__sigT2 A P Q u v p q r))) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (R + (eq__sigT2 A P Q u v (projT1__of__sigT2__eq A P Q u v p) + (projT2__of__sigT2__eq A P Q u v p) (projT3__eq A P Q u v p))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + R : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + r : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (R (eq__sigT2 A P Q u v p q r))) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) => + f (projT1__of__sigT2__eq A P Q u v p) (projT2__of__sigT2__eq A P Q u v p) + (projT3__eq A P Q u v p). + +def eq__sigT2__rect : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + R : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + r : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (R (eq__sigT2 A P Q u v p q r))) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (R p) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + R : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + r : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (projT1 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (R (eq__sigT2 A P Q u v p q r))) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) => + match____sigT2 (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + cupicef.prod + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I + (cupicef.prod cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + s v) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a))) + s v) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + (R0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a))) + s v) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + cupicef.prod cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + s v) + (p0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a))) + s v) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (R0 + (eq__sigT2 A P Q s v (projT1__of__sigT2__eq A P Q s v p0) + (projT2__of__sigT2__eq A P Q s v p0) + (projT3__eq A P Q s v p0))) + (f0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (R0 + (eq__sigT2 A P Q s v + (projT1__of__sigT2__eq A P Q s v p0) + (projT2__of__sigT2__eq A P Q s v p0) + (projT3__eq A P Q s v p0))) => + R0 p0)))) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q x) => + R0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + (existT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + v) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + p1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + (existT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + v) => + f0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (R0 + (eq__sigT2 A P Q + (existT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + v + (projT1__of__sigT2__eq A P Q + (existT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + v p1) + (projT2__of__sigT2__eq A P Q + (existT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + v p1) + (projT3__eq A P Q + (existT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + v p1))) => + Coq__Init__Logic.match____eq + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a))) + (existT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) x + p0 q) + (y : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a)) => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + (existT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + y) => + cupicef.prod + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.prod cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + (existT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + y) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + (existT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + x p0 q) + y) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + (R1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + (existT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + x p0 q) + y) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (R1 + (eq__sigT2 A P Q + (existT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + y + (projT1__of__sigT2__eq A P Q + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + y e) + (projT2__of__sigT2__eq A P Q + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + y e) + (projT3__eq A P Q + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + y e))) + (f1 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (R1 + (eq__sigT2 A P Q + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + y + (projT1__of__sigT2__eq A P Q + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + y e) + (projT2__of__sigT2__eq A P Q + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + y e) + (projT3__eq A P Q + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + y e))) => + R1 e))) + (R1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a))) + (existT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + (existT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q)) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f1 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (R1 + (eq__sigT2 A P Q + (existT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + (existT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + (projT1__of__sigT2__eq A P Q + (existT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + x p0 q) + (existT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + x p0 q) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q))) + (projT2__of__sigT2__eq A P Q + (existT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + x p0 q) + (existT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + x p0 q) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q))) + (projT3__eq A P Q + (existT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + x p0 q) + (existT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + x p0 q) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + (existT2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q))))) => + f1) + v p1 R0 f0) + u R p (let__f1 A P Q u v R f p). + +def eq__sigT2__rec : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + R : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Univ cupicef.set) -> + f : + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + r : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) -> + cupicef.Term cupicef.set (R (eq__sigT2 A P Q u v p q r))) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Term cupicef.set (R p) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + R : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Univ cupicef.set) => + eq__sigT2__rect A P Q u v + (cupicef.cast + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a))) + u v) => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a))) + u v) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I R). + +def eq__sigT2__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + R : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Univ cupicef.prop) -> + f : + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + r : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) -> + cupicef.Term cupicef.prop (R (eq__sigT2 A P Q u v p q r))) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Term cupicef.prop (R p) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + R : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Univ cupicef.prop) => + eq__sigT2__rec A P Q u v + (cupicef.cast + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + (cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a))) + u v) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a))) + u v) => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + cupicef.I R). + +def eq__sigT2__hprop__iff : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q__hprop : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q x) -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q x)) + p q)) -> + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + u) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + v))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q__hprop : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q x) -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q x)) + p q)) => + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) + (x : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + u) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + v))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + u) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + v)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) => + Coq__Init__Logic.f__equal + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a))) + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) + u v p) + (eq__sigT2__hprop A P Q Q__hprop u v). + +def eq__sigT2__nondep : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + C : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C)) -> + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) + v))) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) + u)) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) + v))) -> + r : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) u) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) v)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C))) + u v) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + C : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + u : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C)) => + v : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C)) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) + v))) => + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) + u)) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) + v))) => + r : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) u) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) v)) => + eq__sigT2 A (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) u v p + (Coq__Init__Logic.eq__trans + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B) a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C) a) + u)) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B) a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C) a) + v)) + p) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) + u)) + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) + v)) + (Coq__Init__Logic.rew__const A B + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B) a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C) a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B) a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C) a) + v)) + p + (projT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C) + u))) + q) + (Coq__Init__Logic.eq__trans + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B) a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C) a) + u)) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) + u) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B) a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C) a) + v)) + p) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) u) + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) v) + (Coq__Init__Logic.rew__const A C + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B) a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C) a) + u)) + (projT1 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + a) + (sigT__of__sigT2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B) a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C) a) + v)) + p + (projT3 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) + u)) + r). + +def rew__sigT2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) -> + cupicef.Univ cupicef.prop) -> + R : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) -> + cupicef.Univ cupicef.prop) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) cupicef.prop cupicef.prop + (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p)) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + cupicef.prop (P y) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y) => + Q y p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y) => + R y p))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + sigT2 (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + cupicef.prop (P a) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + Q a p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + R a p)) + u y H) + (existT2 (P y) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P y) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P y) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I (Q y)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P y) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P y) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I (R y)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (projT1 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p)) + (sigT__of__sigT2 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + R x p)) + u)) + y H) + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x a) => + Q a + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (projT1 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) + (sigT__of__sigT2 (P x) + (cupicef.cast + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + Q x p)) + (cupicef.cast + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + R x p)) + u)) + a x0)) + (projT2 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p)) + (sigT__of__sigT2 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + R x p)) + u)) + y H) + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x a) => + R a + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (projT1 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) + (sigT__of__sigT2 (P x) + (cupicef.cast + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + Q x p)) + (cupicef.cast + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + R x p)) + u)) + a x0)) + (projT3 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p)) + u) + y H))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) -> + cupicef.Univ cupicef.prop) => + R : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) -> + cupicef.Univ cupicef.prop) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) cupicef.prop cupicef.prop + (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p)) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + cupicef.prop (P y0) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y0) => + Q y0 p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y0) => + R y0 p))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + sigT2 (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + cupicef.prop (P a) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + Q a p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + R a p)) + u y0 e) + (existT2 (P y0) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P y0) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y0) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P y0) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y0) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I (Q y0)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P y0) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y0) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P y0) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y0) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I (R y0)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (projT1 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) + (sigT__of__sigT2 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + R x p)) + u)) + y0 e) + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x a) => + Q a + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (projT1 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + Q x p)) + (sigT__of__sigT2 (P x) + (cupicef.cast + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + Q x p)) + (cupicef.cast + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + R x p)) + u)) + a x0)) + (projT2 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) + (sigT__of__sigT2 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + R x p)) + u)) + y0 e) + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x a) => + R a + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (projT1 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + Q x p)) + (sigT__of__sigT2 (P x) + (cupicef.cast + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + Q x p)) + (cupicef.cast + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + R x p)) + u)) + a x0)) + (projT3 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + R x p)) + u) + y0 e))) + (match____sigT2 cupicef.prop (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p)) + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sigT2 (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + cupicef.prop (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + cupicef.prop (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + sigT2 (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + cupicef.prop (P a) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + Q a p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + R a p)) + s x + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + (existT2 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I (Q x)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I (R x)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (projT1 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) + (sigT__of__sigT2 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + Q x p)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + R x p)) + s)) + x + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + (projT2 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) + (sigT__of__sigT2 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + R x p)) + s)) + (projT3 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + R x p)) + s))) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + q : cupicef.Term cupicef.prop (Q x x0) => + r : cupicef.Term cupicef.prop (R x x0) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sigT2 (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + cupicef.prop (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p))) + (existT2 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I (Q x)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I (R x)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (projT1 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) + (sigT__of__sigT2 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + Q x p)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + R x p)) + (existT2 (P x) + (cupicef.cast + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + Q x p)) + (cupicef.cast + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + R x p)) + x0 q r))) + x + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + (projT2 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) + (sigT__of__sigT2 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + R x p)) + (existT2 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + Q x p)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + R x p)) + x0 q r))) + (projT3 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + R x p)) + (existT2 (P x) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P x) + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + R x p)) + x0 q r)))) + u) + y H. + +def sig__of__sig2__eq : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) u) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) v)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) => + Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a))) + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) + u v p. + +def proj1__sig__of__sig2__eq : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + v))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) => + proj1__sig__eq A P + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) u) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) v) + (sig__of__sig2__eq A P Q u v p). + +def proj2__sig__of__sig2__eq : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + (proj1__sig__of__sig2__eq A P Q u v p)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + v))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a))) + u + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + x : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u s) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s)) + (proj1__sig__of__sig2__eq A P Q u s x)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + s))) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig__of__sig2__eq A P Q u u + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u)))) + v p. + +def proj3__sig__eq : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + (proj1__sig__of__sig2__eq A P Q u v p)) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) v)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a))) + u + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + x : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u s) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s)) + (proj1__sig__of__sig2__eq A P Q u s x)) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) s)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig__of__sig2__eq A P Q u u + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u)))) + v p. + +def eq__exist2__uncurried : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + u1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + v1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + u2 : cupicef.Term cupicef.prop (P u1) -> + v2 : cupicef.Term cupicef.prop (P v1) -> + u3 : cupicef.Term cupicef.prop (Q u1) -> + v3 : cupicef.Term cupicef.prop (Q v1) -> + pqr : + cupicef.Term cupicef.prop + (sig2 cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + v3)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q)) + (exist2 A P Q u1 u2 u3) (exist2 A P Q v1 v2 v3)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + u1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + v1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + u2 : cupicef.Term cupicef.prop (P u1) => + v2 : cupicef.Term cupicef.prop (P v1) => + u3 : cupicef.Term cupicef.prop (Q u1) => + v3 : cupicef.Term cupicef.prop (Q v1) => + pqr : + cupicef.Term cupicef.prop + (sig2 cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + v3)) => + match____sig2 cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + v3) + (s : + cupicef.Term cupicef.prop + (sig2 cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + v3)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q)) + (exist2 A P Q u1 u2 u3) (exist2 A P Q v1 v2 v3)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) => + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) => + r : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + v3) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + (y : cupicef.Term cupicef.prop (Q v1) => + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + y) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q)) + (exist2 A P Q u1 u2 u3) (exist2 A P Q v1 v2 y)) + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + (y : cupicef.Term cupicef.prop (P v1) => + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + y) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q)) + (exist2 A P Q u1 u2 u3) + (exist2 A P Q v1 y + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__0 : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p))) + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 y) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q)) + (exist2 A P Q u1 u2 u3) + (exist2 A P Q y + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 y e) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 y e))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q)) + (exist2 A P Q u1 u2 u3) (exist2 A P Q u1 u2 u3)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q)) + (exist2 A P Q u1 u2 u3) + (exist2 A P Q u1 + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 u1 + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 u1 + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1)))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A P Q)) + (exist2 A P Q u1 u2 u3))) + v1 p) + v2 q) + v3 r) + pqr. + +def eq__sig2__uncurried : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + pqr : + cupicef.Term cupicef.prop + (sig2 cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a)) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a))) + u v) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + pqr : + cupicef.Term cupicef.prop + (sig2 cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a)) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) => + match____sig2 cupicef.prop A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (sig2 cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s 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cupicef.I A) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3))) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a)) + (proj3__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (exist2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + u1 u2 u3)) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)) + p) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) => + match____sig2 cupicef.prop A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (sig2 cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3))) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + 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cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3))) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3))) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s)) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3))) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (proj1__sig A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3))) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a)) + (proj3__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3)) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s)) + p) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s))) + (pqr1 : + cupicef.Term cupicef.prop + (sig2 cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3))) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + u1 u2 u3))) + (proj1__sig A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + s))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + 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(cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + (exist2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3) + s)) + (v1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + v2 : cupicef.Term cupicef.prop (P v1) => + v3 : cupicef.Term cupicef.prop (Q v1) => + pqr1 : + cupicef.Term cupicef.prop + (sig2 cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3))) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + (exist2 A + (a : + cupicef.Term + 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cupicef.z)) A => + Q a) + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3))) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a)) + (proj2__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3))) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v1 v2 v3))) + p) + (proj2__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v1 v2 v3)))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3))) + (proj1__sig A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v1 v2 v3)))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (proj1__sig A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v1 v2 v3))))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3))) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a)) + (proj3__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3)) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v1 v2 v3))) + p) + (proj3__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + (exist2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v1 v2 v3)))) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + (exist2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3) + (exist2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v1 v2 v3)) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + (exist2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u1 u2 u3) + (exist2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v1 v2 v3)) + cupicef.I + (eq__exist2__uncurried A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + u1 v1 u2 v2 u3 v3 pqr1)) + v pqr0) + u pqr. + +def eq__sig2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + r : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a)) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + v)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a))) + u v) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) => + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type 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=> + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) => + r : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a)) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + v)) => + eq__sig2__uncurried A P Q u v + (exist2 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (p0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p0) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (p0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a)) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p0) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + v)) + p q r). + +def eq__sig2__hprop : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q__hprop : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (Q x) -> + q : cupicef.Term cupicef.prop (Q x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q x)) + p q)) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + u) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + v)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a))) + u v) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q__hprop : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (Q x) -> + q : cupicef.Term cupicef.prop (Q x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q x)) + p q)) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + u) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + v)) => + eq__sig2 A P Q u v + (proj1__sig__eq A P + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) u) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) v) + p) + (proj2__sig__eq A P + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) u) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) v) + p) + (Q__hprop + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + v)) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + (proj1__sig__eq A P + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v) + p)) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) v)). + +def eq__sig2__uncurried__iff : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) + (sig2 cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a)) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) => + sig2 cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + u)) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) 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=> + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u0)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a)) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u0) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (exist2 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)) + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a)) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)))) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) + u H) + (eq__sig2__uncurried A P Q u v). + +def let__f2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + R : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + r : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a)) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (R (eq__sig2 A P Q u v p q r))) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (R + (eq__sig2 A P Q u v (proj1__sig__of__sig2__eq A P Q u v p) + (proj2__sig__of__sig2__eq A P Q u v p) (proj3__sig__eq A P Q u v p))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + R : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + r : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a)) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (R (eq__sig2 A P Q u v p q r))) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) => + f (proj1__sig__of__sig2__eq A P Q u v p) + (proj2__sig__of__sig2__eq A P Q u v p) (proj3__sig__eq A P Q u v p). + +def eq__sig2__rect : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + R : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + r : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a)) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (R (eq__sig2 A P Q u v p q r))) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (R p) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + R : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + r : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (proj1__sig A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a)) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (R (eq__sig2 A P Q u v p q r))) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) => + match____sig2 (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + cupicef.prod + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I + (cupicef.prod cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + s v) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a))) + s v) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + (R0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a))) + s v) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + cupicef.prod cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + s v) + (p0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a))) + s v) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (R0 + (eq__sig2 A P Q s v (proj1__sig__of__sig2__eq A P Q s v p0) + (proj2__sig__of__sig2__eq A P Q s v p0) + (proj3__sig__eq A P Q s v p0))) + (f0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (R0 + (eq__sig2 A P Q s v + (proj1__sig__of__sig2__eq A P Q s v p0) + (proj2__sig__of__sig2__eq A P Q s v p0) + (proj3__sig__eq A P Q s v p0))) => + R0 p0)))) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p0 : cupicef.Term cupicef.prop (P x) => + q : cupicef.Term cupicef.prop (Q x) => + R0 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + (exist2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + v) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + p1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + (exist2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + v) => + f0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (R0 + (eq__sig2 A P Q + (exist2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + v + (proj1__sig__of__sig2__eq A P Q + (exist2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + v p1) + (proj2__sig__of__sig2__eq A P Q + (exist2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + v p1) + (proj3__sig__eq A P Q + (exist2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + v p1))) => + Coq__Init__Logic.match____eq + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a))) + (exist2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) x + p0 q) + (y : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a)) => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + (exist2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + y) => + cupicef.prod + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.prod cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + (exist2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + y) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + (exist2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + x p0 q) + y) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + (R1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + (exist2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + x p0 q) + y) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (R1 + (eq__sig2 A P Q + (exist2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + y + (proj1__sig__of__sig2__eq A P Q + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + y e) + (proj2__sig__of__sig2__eq A P Q + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + y e) + (proj3__sig__eq A P Q + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + y e))) + (f1 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (R1 + (eq__sig2 A P Q + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + y + (proj1__sig__of__sig2__eq A P Q + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + y e) + (proj2__sig__of__sig2__eq A P Q + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + y e) + (proj3__sig__eq A P Q + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + y e))) => + R1 e))) + (R1 : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a))) + (exist2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + (exist2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q)) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f1 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (R1 + (eq__sig2 A P Q + (exist2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + (exist2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q) + (proj1__sig__of__sig2__eq A P Q + (exist2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + x p0 q) + (exist2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + x p0 q) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q))) + (proj2__sig__of__sig2__eq A P Q + (exist2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + x p0 q) + (exist2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + x p0 q) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q))) + (proj3__sig__eq A P Q + (exist2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + x p0 q) + (exist2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + x p0 q) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + (exist2 A + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + x p0 q))))) => + f1) + v p1 R0 f0) + u R p (let__f2 A P Q u v R f p). + +def eq__sig2__rec : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + R : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Univ cupicef.set) -> + f : + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + r : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a)) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) -> + cupicef.Term cupicef.set (R (eq__sig2 A P Q u v p q r))) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Term cupicef.set (R p) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + R : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Univ cupicef.set) => + eq__sig2__rect A P Q u v + (cupicef.cast + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a))) + u v) => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a))) + u v) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I R). + +def eq__sig2__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + R : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Univ cupicef.prop) -> + f : + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a)) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (proj2__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v))) -> + r : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Q + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a) + v)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a)) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) + p) + (proj3__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) -> + cupicef.Term cupicef.prop (R (eq__sig2 A P Q u v p q r))) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Term cupicef.prop (R p) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + R : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) -> + cupicef.Univ cupicef.prop) => + eq__sig2__rec A P Q u v + (cupicef.cast + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + (cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a))) + u v) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + A => + Q a))) + u v) => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + cupicef.I R). + +def eq__sig2__hprop__iff : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Q__hprop : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (Q x) -> + q : cupicef.Term cupicef.prop (Q x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q x)) + p q)) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + u) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + v))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Q__hprop : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (Q x) -> + q : cupicef.Term cupicef.prop (Q x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q x)) + p q)) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) + (x : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + u) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + v))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + u) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a) + v)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a))) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + u) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a) + v)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v)) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q a))) + u v) => + Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a))) + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig (cupicef.type (cupicef.s cupicef.z)) A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a))) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Q a)) + u v p) + (eq__sig2__hprop A P Q Q__hprop u v). + +def eq__sig2__nondep : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.prop -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C)) -> + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (sig__of__sig2 A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) + u)) + (proj1__sig A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (sig__of__sig2 A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) + v))) -> + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (proj2__sig A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (sig__of__sig2 A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) + u)) + (proj2__sig A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (sig__of__sig2 A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) + v))) -> + r : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + (proj3__sig A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) u) + (proj3__sig A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) v)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C))) + u v) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.prop => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C)) => + v : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C)) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (sig__of__sig2 A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C) + u)) + (proj1__sig A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (sig__of__sig2 A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C) + v))) => + q : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (proj2__sig A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (sig__of__sig2 A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C) + u)) + (proj2__sig A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (sig__of__sig2 A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C) + v))) => + r : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + (proj3__sig A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) + u) + (proj3__sig A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) + v)) => + eq__sig2 A (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) u v p + (Coq__Init__Logic.eq__trans + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B) a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C) a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B)) + (proj2__sig A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (sig__of__sig2 A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B) a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C) a) + v)) + p) + (proj2__sig A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (sig__of__sig2 A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) + u)) + (proj2__sig A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (sig__of__sig2 A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) + v)) + (Coq__Init__Logic.rew__const A + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I B) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B) a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C) a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B) a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C) a) + v)) + p + (proj2__sig A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (sig__of__sig2 A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C) + u))) + q) + (Coq__Init__Logic.eq__trans + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B) a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C) a) + u)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C)) + (proj3__sig A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) + u) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B) a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C) a) + v)) + p) + (proj3__sig A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) u) + (proj3__sig A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) v) + (Coq__Init__Logic.rew__const A + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I C) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B) a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C) a) + u)) + (proj1__sig A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + a) + (sig__of__sig2 A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + B) a) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + C) a) + v)) + p + (proj3__sig A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => C) + u)) + r). + +def rew__sig2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Q : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) -> + cupicef.Univ cupicef.prop) -> + R : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) -> + cupicef.Univ cupicef.prop) -> + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p)) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) (P y) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y) => + Q y p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y) => + R y p))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + sig2 (cupicef.type (cupicef.s cupicef.z)) (P a) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + Q a p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + R a p)) + u y H) + (exist2 (P y) (Q y) (R y) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (proj1__sig (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (sig__of__sig2 (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p) + u)) + y H) + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x a) => + Q a + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (proj1__sig (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p) + (sig__of__sig2 (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + R x p) + u)) + a x0)) + (proj2__sig (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (sig__of__sig2 (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p) + u)) + y H) + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x a) => + R a + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (proj1__sig (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p) + (sig__of__sig2 (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + R x p) + u)) + a x0)) + (proj3__sig (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p) + u) + y H))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Q : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) -> + cupicef.Univ cupicef.prop) => + R : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) -> + cupicef.Univ cupicef.prop) => + u : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p)) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) (P y0) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y0) => + Q y0 p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y0) => + R y0 p))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + sig2 (cupicef.type (cupicef.s cupicef.z)) (P a) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + Q a p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + R a p)) + u y0 e) + (exist2 (P y0) (Q y0) (R y0) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (proj1__sig (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (sig__of__sig2 (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + R x p) + u)) + y0 e) + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x a) => + Q a + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (proj1__sig (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p) + (sig__of__sig2 (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + Q x p) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + R x p) + u)) + a x0)) + (proj2__sig (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (sig__of__sig2 (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + R x p) + u)) + y0 e) + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x a) => + R a + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (proj1__sig (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p) + (sig__of__sig2 (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + Q x p) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + R x p) + u)) + a x0)) + (proj3__sig (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p) + u) + y0 e))) + (match____sig2 cupicef.prop (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => Q x p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => R x p) + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (sig2 (cupicef.type (cupicef.s cupicef.z)) (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + sig2 (cupicef.type (cupicef.s cupicef.z)) (P a) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + Q a p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) => + R a p)) + s x + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + (exist2 (P x) (Q x) (R x) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (proj1__sig (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p) + (sig__of__sig2 (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + R x p) + s)) + x + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + (proj2__sig (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (sig__of__sig2 (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + R x p) + s)) + (proj3__sig (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p) + s))) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + q : cupicef.Term cupicef.prop (Q x x0) => + r : cupicef.Term cupicef.prop (R x x0) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (sig2 (cupicef.type (cupicef.s cupicef.z)) (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p))) + (exist2 (P x) (Q x) (R x) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (proj1__sig (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p) + (sig__of__sig2 (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + R x p) + (exist2 (P x) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + Q x p) + (p : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) ( + P x) => + R x p) + x0 q r))) + x + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + (proj2__sig (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (sig__of__sig2 (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + R x p) + (exist2 (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + R x p) + x0 q r))) + (proj3__sig (P x) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Q x p) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + R x p) + (exist2 (P x) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + Q x p) + (p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P x) => + R x p) + x0 q r)))) + u) + y H. + +def sumbool : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + cupicef.Univ cupicef.set. + +def left : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.set (sumbool A B). + +def right : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.set (sumbool A B). + +[ ] cupicef.code_guarded? _ (left _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (right _ _ _) --> cupicef.guarded. + +def match____sumbool : + s : cupicef.Sort -> + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + P : (cupicef.Term cupicef.set (sumbool A B) -> cupicef.Univ s) -> + case__left : + (__ : cupicef.Term cupicef.prop A -> cupicef.Term s (P (left A B __))) -> + case__right : + (__ : cupicef.Term cupicef.prop B -> cupicef.Term s (P (right A B __))) -> + x : cupicef.Term cupicef.set (sumbool A B) -> + cupicef.Term s (P x). + +[ s, A, B, P, case__left, case__right, __] + match____sumbool s A B P case__left case__right (left A B __) + --> + case__left __. + +[ s, A, B, P, case__left, case__right, __] + match____sumbool s A B P case__left case__right (right A B __) + --> + case__right __. + +[ s, A, B, P, s'] + match____sumbool s A B + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____sumbool s' A B (x => P x). + +def sumbool__rect : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + P : + (s : cupicef.Term cupicef.set (sumbool A B) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : + (a : cupicef.Term cupicef.prop A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (left A B a))) -> + f0 : + (b : cupicef.Term cupicef.prop B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (right A B b))) -> + s : cupicef.Term cupicef.set (sumbool A B) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P s) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + P : + (s : cupicef.Term cupicef.set (sumbool A B) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : + (a : cupicef.Term cupicef.prop A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (left A B a))) => + f0 : + (b : cupicef.Term cupicef.prop B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (right A B b))) => + s : cupicef.Term cupicef.set (sumbool A B) => + match____sumbool (cupicef.type (cupicef.s cupicef.z)) A B + (s0 : cupicef.Term cupicef.set (sumbool A B) => P s0) f f0 s. + +def sumbool__ind : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + P : + (s : cupicef.Term cupicef.set (sumbool A B) -> cupicef.Univ cupicef.prop) -> + f : + (a : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop (P (left A B a))) -> + f0 : + (b : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop (P (right A B b))) -> + s : cupicef.Term cupicef.set (sumbool A B) -> + cupicef.Term cupicef.prop (P s) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + P : + (s : cupicef.Term cupicef.set (sumbool A B) -> cupicef.Univ cupicef.prop) => + f : + (a : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop (P (left A B a))) => + f0 : + (b : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop (P (right A B b))) => + s : cupicef.Term cupicef.set (sumbool A B) => + match____sumbool cupicef.prop A B + (s0 : cupicef.Term cupicef.set (sumbool A B) => P s0) f f0 s. + +def sumbool__rec : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + P : + (s : cupicef.Term cupicef.set (sumbool A B) -> cupicef.Univ cupicef.set) -> + f : + (a : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.set (P (left A B a))) -> + f0 : + (b : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.set (P (right A B b))) -> + s : cupicef.Term cupicef.set (sumbool A B) -> + cupicef.Term cupicef.set (P s) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + P : + (s : cupicef.Term cupicef.set (sumbool A B) -> cupicef.Univ cupicef.set) => + sumbool__rect A B + (cupicef.cast + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.set (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (sumbool A B) + (s : cupicef.Term cupicef.set (sumbool A B) => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (sumbool A B) + (s : cupicef.Term cupicef.set (sumbool A B) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P). + +def sumor : + Coq__Init__Specif__415 : cupicef.Sort -> + A : cupicef.Univ Coq__Init__Specif__415 -> + B : cupicef.Univ cupicef.prop -> + cupicef.Univ (cupicef.sup cupicef.set Coq__Init__Specif__415). + +sumor' : + A : cupicef.Univ cupicef.sinf -> + B : cupicef.Univ cupicef.prop -> + cupicef.Univ cupicef.sinf. + +[ Coq__Init__Specif__415, A, B] + cupicef.code (cupicef.cU _) (sumor Coq__Init__Specif__415 A B) + --> + sumor' (cupicef.code (cupicef.cU Coq__Init__Specif__415) A) B. + +def Coq__Init__Specif__415 : + cupicef.Nat + +:= cupicef.s cupicef.z. + +def inleft : + A : cupicef.Univ (cupicef.type Coq__Init__Specif__415) -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term (cupicef.type Coq__Init__Specif__415) A -> + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type Coq__Init__Specif__415)) + (sumor (cupicef.type Coq__Init__Specif__415) A B). + +def inright : + A : cupicef.Univ (cupicef.type Coq__Init__Specif__415) -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop B -> + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type Coq__Init__Specif__415)) + (sumor (cupicef.type Coq__Init__Specif__415) A B). + +[ ] cupicef.code_guarded? _ (inleft _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (inright _ _ _) --> cupicef.guarded. + +def match____sumor : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type Coq__Init__Specif__415) -> + B : cupicef.Univ cupicef.prop -> + P : + (cupicef.Term + (cupicef.sup cupicef.set (cupicef.type Coq__Init__Specif__415)) + (sumor (cupicef.type Coq__Init__Specif__415) A B) -> + cupicef.Univ s) -> + case__inleft : + (__ : cupicef.Term (cupicef.type Coq__Init__Specif__415) A -> + cupicef.Term s (P (inleft A B __))) -> + case__inright : + (__ : cupicef.Term cupicef.prop B -> cupicef.Term s (P (inright A B __))) -> + x : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type Coq__Init__Specif__415)) + (sumor (cupicef.type Coq__Init__Specif__415) A B) -> + cupicef.Term s (P x). + +[ s, A, B, P, case__inleft, case__inright, __] + match____sumor s A B P case__inleft case__inright (inleft A B __) + --> + case__inleft __. + +[ s, A, B, P, case__inleft, case__inright, __] + match____sumor s A B P case__inleft case__inright (inright A B __) + --> + case__inright __. + +[ s, A, B, P, s'] + match____sumor s A B + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____sumor s' A B (x => P x). + +def sumor__rect : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ cupicef.prop -> + P : + (s : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (sumor (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (inleft A B a))) -> + f0 : + (b : cupicef.Term cupicef.prop B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (inright A B b))) -> + s : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (sumor (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P s) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ cupicef.prop => + P : + (s : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (sumor (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (inleft A B a))) => + f0 : + (b : cupicef.Term cupicef.prop B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P (inright A B b))) => + s : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (sumor (cupicef.type (cupicef.s cupicef.z)) A B) => + match____sumor (cupicef.type (cupicef.s cupicef.z)) A B + (s0 : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (sumor (cupicef.type (cupicef.s cupicef.z)) A B) => + P s0) + f f0 s. + +def sumor__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ cupicef.prop -> + P : + (s : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (sumor (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ cupicef.prop) -> + f : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (P (inleft A B a))) -> + f0 : + (b : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop (P (inright A B b))) -> + s : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (sumor (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Term cupicef.prop (P s) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ cupicef.prop => + P : + (s : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (sumor (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ cupicef.prop) => + f : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (P (inleft A B a))) => + f0 : + (b : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop (P (inright A B b))) => + s : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (sumor (cupicef.type (cupicef.s cupicef.z)) A B) => + match____sumor cupicef.prop A B + (s0 : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (sumor (cupicef.type (cupicef.s cupicef.z)) A B) => + P s0) + f f0 s. + +def sumor__rec : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ cupicef.prop -> + P : + (s : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (sumor (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ cupicef.set) -> + f : + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set (P (inleft A B a))) -> + f0 : + (b : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.set (P (inright A B b))) -> + s : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (sumor (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Term cupicef.set (P s) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ cupicef.prop => + P : + (s : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (sumor (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ cupicef.set) => + sumor__rect A B + (cupicef.cast + (cupicef.rule + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (sumor (cupicef.type (cupicef.s cupicef.z)) A B) + (s : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (sumor (cupicef.type (cupicef.s cupicef.z)) A B) => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (sumor (cupicef.type (cupicef.s cupicef.z)) A B) + (s : + cupicef.Term + (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (sumor (cupicef.type (cupicef.s cupicef.z)) A B) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P). + +def let__s : + S : cupicef.Univ cupicef.set -> + S_27_ : cupicef.Univ cupicef.set -> + R : + (__ : cupicef.Term cupicef.set S -> + __0 : cupicef.Term cupicef.set S_27_ -> + cupicef.Univ cupicef.prop) -> + H : + (x : cupicef.Term cupicef.set S -> + cupicef.Term cupicef.set + (sig cupicef.set S_27_ (y : cupicef.Term cupicef.set S_27_ => R x y))) -> + z : cupicef.Term cupicef.set S -> + cupicef.Term cupicef.set + (sig cupicef.set S_27_ (y : cupicef.Term cupicef.set S_27_ => R z y)) + +:= S : cupicef.Univ cupicef.set => + S_27_ : cupicef.Univ cupicef.set => + R : + (__ : cupicef.Term cupicef.set S -> + __0 : cupicef.Term cupicef.set S_27_ -> + cupicef.Univ cupicef.prop) => + H : + (x : cupicef.Term cupicef.set S -> + cupicef.Term cupicef.set + (sig cupicef.set S_27_ (y : cupicef.Term cupicef.set S_27_ => R x y))) => + z : cupicef.Term cupicef.set S => + H z. + +def Choice : + S : cupicef.Univ cupicef.set -> + S_27_ : cupicef.Univ cupicef.set -> + R : + (__ : cupicef.Term cupicef.set S -> + __0 : cupicef.Term cupicef.set S_27_ -> + cupicef.Univ cupicef.prop) -> + __ : + (x : cupicef.Term cupicef.set S -> + cupicef.Term cupicef.set + (sig cupicef.set S_27_ (y : cupicef.Term cupicef.set S_27_ => R x y))) -> + cupicef.Term cupicef.set + (sig cupicef.set + (cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I S + (__0 : cupicef.Term cupicef.set S => S_27_)) + (f : + (__0 : cupicef.Term cupicef.set S -> cupicef.Term cupicef.set S_27_) => + cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I S + (z : cupicef.Term cupicef.set S => R z (f z)))) + +:= S : cupicef.Univ cupicef.set => + S_27_ : cupicef.Univ cupicef.set => + R : + (__ : cupicef.Term cupicef.set S -> + __0 : cupicef.Term cupicef.set S_27_ -> + cupicef.Univ cupicef.prop) => + H : + (x : cupicef.Term cupicef.set S -> + cupicef.Term cupicef.set + (sig cupicef.set S_27_ (y : cupicef.Term cupicef.set S_27_ => R x y))) => + exist + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I S + (__ : cupicef.Term cupicef.set S => S_27_))) + (f : (__ : cupicef.Term cupicef.set S -> cupicef.Term cupicef.set S_27_) => + cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I S + (z : cupicef.Term cupicef.set S => R z (f z))) + (z : cupicef.Term cupicef.set S => + proj1__sig + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I S_27_) + (y : cupicef.Term cupicef.set S_27_ => R z y) (H z)) + (z : cupicef.Term cupicef.set S => + match____sig cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I S_27_) + (y : cupicef.Term cupicef.set S_27_ => R z y) + (s0 : + cupicef.Term cupicef.set + (sig cupicef.set S_27_ + (y : cupicef.Term cupicef.set S_27_ => R z y)) => + R z + (proj1__sig + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I S_27_) + (y : cupicef.Term cupicef.set S_27_ => R z y) s0)) + (x : cupicef.Term cupicef.set S_27_ => + r : cupicef.Term cupicef.prop (R z x) => + r) + (let__s S S_27_ R H z)). + +def let__s0 : + S : cupicef.Univ cupicef.set -> + S_27_ : cupicef.Univ cupicef.set -> + R_27_ : + (__ : cupicef.Term cupicef.set S -> + __0 : cupicef.Term cupicef.set S_27_ -> + cupicef.Univ cupicef.set) -> + H : + (x : cupicef.Term cupicef.set S -> + cupicef.Term cupicef.set + (sigT cupicef.set cupicef.set S_27_ + (y : cupicef.Term cupicef.set S_27_ => R_27_ x y))) -> + z : cupicef.Term cupicef.set S -> + cupicef.Term cupicef.set + (sigT cupicef.set cupicef.set S_27_ + (y : cupicef.Term cupicef.set S_27_ => R_27_ z y)) + +:= S : cupicef.Univ cupicef.set => + S_27_ : cupicef.Univ cupicef.set => + R_27_ : + (__ : cupicef.Term cupicef.set S -> + __0 : cupicef.Term cupicef.set S_27_ -> + cupicef.Univ cupicef.set) => + H : + (x : cupicef.Term cupicef.set S -> + cupicef.Term cupicef.set + (sigT cupicef.set cupicef.set S_27_ + (y : cupicef.Term cupicef.set S_27_ => R_27_ x y))) => + z : cupicef.Term cupicef.set S => + H z. + +def Choice2 : + S : cupicef.Univ cupicef.set -> + S_27_ : cupicef.Univ cupicef.set -> + R_27_ : + (__ : cupicef.Term cupicef.set S -> + __0 : cupicef.Term cupicef.set S_27_ -> + cupicef.Univ cupicef.set) -> + __ : + (x : cupicef.Term cupicef.set S -> + cupicef.Term cupicef.set + (sigT cupicef.set cupicef.set S_27_ + (y : cupicef.Term cupicef.set S_27_ => R_27_ x y))) -> + cupicef.Term cupicef.set + (sigT cupicef.set cupicef.set + (cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I S + (__0 : cupicef.Term cupicef.set S => S_27_)) + (f : + (__0 : cupicef.Term cupicef.set S -> cupicef.Term cupicef.set S_27_) => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I S + (z : cupicef.Term cupicef.set S => R_27_ z (f z)))) + +:= S : cupicef.Univ cupicef.set => + S_27_ : cupicef.Univ cupicef.set => + R_27_ : + (__ : cupicef.Term cupicef.set S -> + __0 : cupicef.Term cupicef.set S_27_ -> + cupicef.Univ cupicef.set) => + H : + (x : cupicef.Term cupicef.set S -> + cupicef.Term cupicef.set + (sigT cupicef.set cupicef.set S_27_ + (y : cupicef.Term cupicef.set S_27_ => R_27_ x y))) => + existT + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I S + (__ : cupicef.Term cupicef.set S => S_27_))) + (cupicef.cast + (cupicef.rule (cupicef.rule cupicef.set cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.rule cupicef.set cupicef.set) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.rule cupicef.set cupicef.set) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.rule cupicef.set cupicef.set) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I S + (__ : cupicef.Term cupicef.set S => S_27_)) + (f : + (__ : cupicef.Term cupicef.set S -> + cupicef.Term cupicef.set S_27_) => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.rule cupicef.set cupicef.set) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.rule cupicef.set cupicef.set) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I S + (__ : cupicef.Term cupicef.set S => S_27_)) + (__ : + (__ : cupicef.Term cupicef.set S -> + cupicef.Term cupicef.set S_27_) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I + (f : + (__ : cupicef.Term cupicef.set S -> cupicef.Term cupicef.set S_27_) => + cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I S + (z : cupicef.Term cupicef.set S => R_27_ z (f z)))) + (z : cupicef.Term cupicef.set S => + projT1 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I S_27_) + (cupicef.cast + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.set (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + cupicef.I S_27_ + (y : cupicef.Term cupicef.set S_27_ => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I S_27_ + (__ : cupicef.Term cupicef.set S_27_ => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I (y : cupicef.Term cupicef.set S_27_ => R_27_ z y)) + (H z)) + (z : cupicef.Term cupicef.set S => + match____sigT cupicef.set + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I S_27_) + (cupicef.cast + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.set (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + cupicef.I S_27_ + (y : cupicef.Term cupicef.set S_27_ => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I S_27_ + (__ : cupicef.Term cupicef.set S_27_ => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I (y : cupicef.Term cupicef.set S_27_ => R_27_ z y)) + (s0 : + cupicef.Term cupicef.set + (sigT cupicef.set cupicef.set S_27_ + (y : cupicef.Term cupicef.set S_27_ => R_27_ z y)) => + R_27_ z + (projT1 + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I S_27_) + (cupicef.cast + (cupicef.rule cupicef.set + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I S_27_ + (y : cupicef.Term cupicef.set S_27_ => + cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I S_27_ + (__ : cupicef.Term cupicef.set S_27_ => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I (y : cupicef.Term cupicef.set S_27_ => R_27_ z y)) + s0)) + (x : cupicef.Term cupicef.set S_27_ => + r : cupicef.Term cupicef.set (R_27_ z x) => + r) + (let__s0 S S_27_ R_27_ H z)). + +def let__s1 : + S : cupicef.Univ cupicef.set -> + R1 : (__ : cupicef.Term cupicef.set S -> cupicef.Univ cupicef.prop) -> + R2 : (__ : cupicef.Term cupicef.set S -> cupicef.Univ cupicef.prop) -> + H : + (x : cupicef.Term cupicef.set S -> + cupicef.Term cupicef.set (sumbool (R1 x) (R2 x))) -> + z : cupicef.Term cupicef.set S -> + cupicef.Term cupicef.set (sumbool (R1 z) (R2 z)) + +:= S : cupicef.Univ cupicef.set => + R1 : (__ : cupicef.Term cupicef.set S -> cupicef.Univ cupicef.prop) => + R2 : (__ : cupicef.Term cupicef.set S -> cupicef.Univ cupicef.prop) => + H : + (x : cupicef.Term cupicef.set S -> + cupicef.Term cupicef.set (sumbool (R1 x) (R2 x))) => + z : cupicef.Term cupicef.set S => + H z. + +def bool__choice : + S : cupicef.Univ cupicef.set -> + R1 : (__ : cupicef.Term cupicef.set S -> cupicef.Univ cupicef.prop) -> + R2 : (__ : cupicef.Term cupicef.set S -> cupicef.Univ cupicef.prop) -> + __ : + (x : cupicef.Term cupicef.set S -> + cupicef.Term cupicef.set (sumbool (R1 x) (R2 x))) -> + cupicef.Term cupicef.set + (sig cupicef.set + (cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I S + (__0 : cupicef.Term cupicef.set S => Coq__Init__Datatypes.bool)) + (f : + (__0 : cupicef.Term cupicef.set S -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool) => + cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I S + (x : cupicef.Term cupicef.set S => + Coq__Init__Logic.or + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (f x) Coq__Init__Datatypes.true) + (R1 x)) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (f x) Coq__Init__Datatypes.false) + (R2 x))))) + +:= S : cupicef.Univ cupicef.set => + R1 : (__ : cupicef.Term cupicef.set S -> cupicef.Univ cupicef.prop) => + R2 : (__ : cupicef.Term cupicef.set S -> cupicef.Univ cupicef.prop) => + H : + (x : cupicef.Term cupicef.set S -> + cupicef.Term cupicef.set (sumbool (R1 x) (R2 x))) => + exist + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I S + (__ : cupicef.Term cupicef.set S => Coq__Init__Datatypes.bool))) + (f : + (__ : cupicef.Term cupicef.set S -> + cupicef.Term cupicef.set Coq__Init__Datatypes.bool) => + cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I S + (x : cupicef.Term cupicef.set S => + Coq__Init__Logic.or + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (f x) Coq__Init__Datatypes.true) + (R1 x)) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (f x) Coq__Init__Datatypes.false) + (R2 x)))) + (z : cupicef.Term cupicef.set S => + match____sumbool cupicef.set (R1 z) (R2 z) + (s : cupicef.Term cupicef.set (sumbool (R1 z) (R2 z)) => + Coq__Init__Datatypes.bool) + (__ : cupicef.Term cupicef.prop (R1 z) => Coq__Init__Datatypes.true) + (__ : cupicef.Term cupicef.prop (R2 z) => Coq__Init__Datatypes.false) + (H z)) + (z : cupicef.Term cupicef.set S => + match____sumbool cupicef.prop (R1 z) (R2 z) + (s0 : cupicef.Term cupicef.set (sumbool (R1 z) (R2 z)) => + Coq__Init__Logic.or + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (match____sumbool cupicef.set (R1 z) (R2 z) + (s1 : cupicef.Term cupicef.set (sumbool (R1 z) (R2 z)) => + Coq__Init__Datatypes.bool) + (__ : cupicef.Term cupicef.prop (R1 z) => + Coq__Init__Datatypes.true) + (__ : cupicef.Term cupicef.prop (R2 z) => + Coq__Init__Datatypes.false) + s0) + Coq__Init__Datatypes.true) + (R1 z)) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + (match____sumbool cupicef.set (R1 z) (R2 z) + (s1 : cupicef.Term cupicef.set (sumbool (R1 z) (R2 z)) => + Coq__Init__Datatypes.bool) + (__ : cupicef.Term cupicef.prop (R1 z) => + Coq__Init__Datatypes.true) + (__ : cupicef.Term cupicef.prop (R2 z) => + Coq__Init__Datatypes.false) + s0) + Coq__Init__Datatypes.false) + (R2 z))) + (r : cupicef.Term cupicef.prop (R1 z) => + Coq__Init__Logic.or__introl + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (R1 z)) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) + (R2 z)) + (Coq__Init__Logic.conj + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) + (R1 z) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true) + r)) + (r : cupicef.Term cupicef.prop (R2 z) => + Coq__Init__Logic.or__intror + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) + (R1 z)) + (Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (R2 z)) + (Coq__Init__Logic.conj + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) + (R2 z) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false) + r)) + (let__s1 S R1 R2 H z)). + +def let__f3 : + X : cupicef.Univ cupicef.set -> + R : + (__ : cupicef.Term cupicef.set X -> + __0 : cupicef.Term cupicef.set X -> + cupicef.Univ cupicef.prop) -> + H : + (x : cupicef.Term cupicef.set X -> + cupicef.Term cupicef.set + (sig cupicef.set X (y : cupicef.Term cupicef.set X => R x y))) -> + x0 : cupicef.Term cupicef.set X -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set X + +:= X : cupicef.Univ cupicef.set => + R : + (__ : cupicef.Term cupicef.set X -> + __0 : cupicef.Term cupicef.set X -> + cupicef.Univ cupicef.prop) => + H : + (x : cupicef.Term cupicef.set X -> + cupicef.Term cupicef.set + (sig cupicef.set X (y : cupicef.Term cupicef.set X => R x y))) => + x0 : cupicef.Term cupicef.set X => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.set + (cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => X)))) + (c => + c + (f => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.match____nat cupicef.set + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => X) x0 + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + proj1__sig + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I X) + (y : cupicef.Term cupicef.set X => R (f n_27_) y) (H (f n_27_))) + n)) + cupicef.0. + +def dependent__choice : + X : cupicef.Univ cupicef.set -> + R : + (__ : cupicef.Term cupicef.set X -> + __0 : cupicef.Term cupicef.set X -> + cupicef.Univ cupicef.prop) -> + __ : + (x : cupicef.Term cupicef.set X -> + cupicef.Term cupicef.set + (sig cupicef.set X (y : cupicef.Term cupicef.set X => R x y))) -> + x0 : cupicef.Term cupicef.set X -> + cupicef.Term cupicef.set + (sig cupicef.set + (cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (__0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => X)) + (f : + (__0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set X) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + (f Coq__Init__Datatypes.O) x0) + (cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I + Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + R (f n) (f (Coq__Init__Datatypes.S n)))))) + +:= X : cupicef.Univ cupicef.set => + R : + (__ : cupicef.Term cupicef.set X -> + __0 : cupicef.Term cupicef.set X -> + cupicef.Univ cupicef.prop) => + H : + (x : cupicef.Term cupicef.set X -> + cupicef.Term cupicef.set + (sig cupicef.set X (y : cupicef.Term cupicef.set X => R x y))) => + x0 : cupicef.Term cupicef.set X => + exist + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (cupicef.prod cupicef.set cupicef.set + (cupicef.rule cupicef.set cupicef.set) cupicef.I + Coq__Init__Datatypes.nat + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => X))) + (f0 : + (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + cupicef.Term cupicef.set X) => + Coq__Init__Logic.and + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + (f0 Coq__Init__Datatypes.O) x0) + (cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I + Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + R (f0 n) (f0 (Coq__Init__Datatypes.S n))))) + (let__f3 X R H x0) + (Coq__Init__Logic.conj + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + (let__f3 X R H x0 Coq__Init__Datatypes.O) x0) + (cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I + Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + R (let__f3 X R H x0 n) + (let__f3 X R H x0 (Coq__Init__Datatypes.S n)))) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x0) + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + Coq__Init__Datatypes.nat__ind + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + R (let__f3 X R H x0 n0) + (let__f3 X R H x0 (Coq__Init__Datatypes.S n0))) + (cupicef.cast cupicef.prop cupicef.prop + ((y : cupicef.Term cupicef.set X => R x0 y) + (proj1__sig + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I X) + (y : cupicef.Term cupicef.set X => R x0 y) (H x0))) + (R (let__f3 X R H x0 Coq__Init__Datatypes.O) + (let__f3 X R H x0 + (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) + cupicef.I + (proj2__sig + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I X) + (y : cupicef.Term cupicef.set X => R x0 y) (H x0))) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + cupicef.Term cupicef.prop + (R (let__f3 X R H x0 n0) + (let__f3 X R H x0 (Coq__Init__Datatypes.S n0))) => + cupicef.cast cupicef.prop cupicef.prop + ((y : cupicef.Term cupicef.set X => + R + (proj1__sig + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I X) + (y0 : cupicef.Term cupicef.set X => + R (let__f3 X R H x0 n0) y0) + (H (let__f3 X R H x0 n0))) + y) + (proj1__sig + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I X) + (y : cupicef.Term cupicef.set X => + R + (proj1__sig + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I X) + (y0 : cupicef.Term cupicef.set X => + R (let__f3 X R H x0 n0) y0) + (H (let__f3 X R H x0 n0))) + y) + (H + (proj1__sig + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I X) + (y : cupicef.Term cupicef.set X => + R (let__f3 X R H x0 n0) y) + (H (let__f3 X R H x0 n0)))))) + (R (let__f3 X R H x0 (Coq__Init__Datatypes.S n0)) + (let__f3 X R H x0 + (Coq__Init__Datatypes.S (Coq__Init__Datatypes.S n0)))) + cupicef.I + (proj2__sig + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I X) + (y : cupicef.Term cupicef.set X => + R + (proj1__sig + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I X) + (y0 : cupicef.Term cupicef.set X => + R (let__f3 X R H x0 n0) y0) + (H (let__f3 X R H x0 n0))) + y) + (H + (proj1__sig + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I X) + (y : cupicef.Term cupicef.set X => + R (let__f3 X R H x0 n0) y) + (H (let__f3 X R H x0 n0)))))) + n)). + +def Exc : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Univ (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + Coq__Init__Datatypes.option (cupicef.type (cupicef.s cupicef.z)) A. + +def value : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Datatypes.option (cupicef.type (cupicef.s cupicef.z)) A) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + Coq__Init__Datatypes.Some A. + +def error : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Term (cupicef.sup cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Datatypes.option (cupicef.type (cupicef.s cupicef.z)) A) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + Coq__Init__Datatypes.None A. + +def except : + P : cupicef.Univ cupicef.set -> + f : cupicef.Term cupicef.prop Coq__Init__Logic.False -> + cupicef.Term cupicef.set P + +:= Coq__Init__Logic.False__rec. + +def absurd__set : + A : cupicef.Univ cupicef.prop -> + C : cupicef.Univ cupicef.set -> + __ : cupicef.Term cupicef.prop A -> + __0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + cupicef.Term cupicef.set C + +:= A : cupicef.Univ cupicef.prop => + C : cupicef.Univ cupicef.set => + h1 : cupicef.Term cupicef.prop A => + h2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + Coq__Init__Logic.False__rec C (h2 h1). + +(; End of translation. ;) diff --git a/out/Coq__Init__Tactics.dk b/out/Coq__Init__Tactics.dk new file mode 100644 index 000000000..53b774e53 --- /dev/null +++ b/out/Coq__Init__Tactics.dk @@ -0,0 +1,87 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +thm decide__left : + C : cupicef.Univ cupicef.prop -> + decide : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool C (Coq__Init__Logic.not C)) -> + __ : cupicef.Term cupicef.prop C -> + P : + (__0 : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool C (Coq__Init__Logic.not C)) -> + cupicef.Univ cupicef.prop) -> + __0 : + (H : cupicef.Term cupicef.prop C -> + cupicef.Term cupicef.prop + (P (Coq__Init__Specif.left C (Coq__Init__Logic.not C) H))) -> + cupicef.Term cupicef.prop (P decide) + +:= C : cupicef.Univ cupicef.prop => + decide : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool C (Coq__Init__Logic.not C)) => + H : cupicef.Term cupicef.prop C => + P : + (__ : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool C (Coq__Init__Logic.not C)) -> + cupicef.Univ cupicef.prop) => + H0 : + (H0 : cupicef.Term cupicef.prop C -> + cupicef.Term cupicef.prop + (P (Coq__Init__Specif.left C (Coq__Init__Logic.not C) H0))) => + Coq__Init__Specif.match____sumbool cupicef.prop C (Coq__Init__Logic.not C) + (s : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool C (Coq__Init__Logic.not C)) => + P s) + (c : cupicef.Term cupicef.prop C => H0 c) + (n : cupicef.Term cupicef.prop (Coq__Init__Logic.not C) => + Coq__Init__Logic.False__ind + (P (Coq__Init__Specif.right C (Coq__Init__Logic.not C) n)) (n H)) + decide. + +thm decide__right : + C : cupicef.Univ cupicef.prop -> + decide : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool C (Coq__Init__Logic.not C)) -> + __ : cupicef.Term cupicef.prop (Coq__Init__Logic.not C) -> + P : + (__0 : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool C (Coq__Init__Logic.not C)) -> + cupicef.Univ cupicef.prop) -> + __0 : + (H : cupicef.Term cupicef.prop (Coq__Init__Logic.not C) -> + cupicef.Term cupicef.prop + (P (Coq__Init__Specif.right C (Coq__Init__Logic.not C) H))) -> + cupicef.Term cupicef.prop (P decide) + +:= C : cupicef.Univ cupicef.prop => + decide : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool C (Coq__Init__Logic.not C)) => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.not C) => + P : + (__ : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool C (Coq__Init__Logic.not C)) -> + cupicef.Univ cupicef.prop) => + H0 : + (H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not C) -> + cupicef.Term cupicef.prop + (P (Coq__Init__Specif.right C (Coq__Init__Logic.not C) H0))) => + Coq__Init__Specif.match____sumbool cupicef.prop C (Coq__Init__Logic.not C) + (s : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool C (Coq__Init__Logic.not C)) => + P s) + (c : cupicef.Term cupicef.prop C => + Coq__Init__Logic.False__ind + (P (Coq__Init__Specif.left C (Coq__Init__Logic.not C) c)) (H c)) + (n : cupicef.Term cupicef.prop (Coq__Init__Logic.not C) => H0 n) decide. + +(; End of translation. ;) diff --git a/out/Coq__Init__Tauto.dk b/out/Coq__Init__Tauto.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Init__Tauto.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Init__Wf.dk b/out/Coq__Init__Wf.dk new file mode 100644 index 000000000..ed349ffad --- /dev/null +++ b/out/Coq__Init__Wf.dk @@ -0,0 +1,1621 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def Acc : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop. + +def Acc__intro : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (Acc A R y)) -> + cupicef.Term cupicef.prop (Acc A R x). + +[ ] cupicef.code_guarded? _ (Acc__intro _ _ _ _) --> cupicef.guarded. + +def match____Acc : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : (cupicef.Term cupicef.prop (Acc A R x) -> cupicef.Univ s) -> + case__Acc__intro : + (__ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (Acc A R y)) -> + cupicef.Term s (P (Acc__intro A R x __))) -> + x0 : cupicef.Term cupicef.prop (Acc A R x) -> + cupicef.Term s (P x0). + +[ s, A, R, x, P, case__Acc__intro, __] + match____Acc s A R x P case__Acc__intro (Acc__intro A R x __) + --> + case__Acc__intro __. + +[ s, A, R, x, P, s'] + match____Acc s A R x + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____Acc s' A R x (x => P x). + +def Acc__rect : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (Acc A R y)) -> + __0 : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + a : cupicef.Term cupicef.prop (Acc A R x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (Acc A R y)) -> + __0 : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x)) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (Acc A R x) + (a : cupicef.Term cupicef.prop (Acc A R x) => P x))))) + (c => + c + (F => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + a : cupicef.Term cupicef.prop (Acc A R x) => + match____Acc (cupicef.type (cupicef.s cupicef.z)) A R x + (__ : cupicef.Term cupicef.prop (Acc A R x) => P x) + (a0 : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (Acc A R y)) => + f x a0 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + r : cupicef.Term cupicef.prop (R y x) => + F y (a0 y r))) + a)) + cupicef.0. + +def Acc__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (Acc A R y)) -> + __0 : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (P y)) -> + cupicef.Term cupicef.prop (P x)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + a : cupicef.Term cupicef.prop (Acc A R x) -> + cupicef.Term cupicef.prop (P x) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (Acc A R y)) -> + __0 : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (P y)) -> + cupicef.Term cupicef.prop (P x)) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Acc A R x) + (a : cupicef.Term cupicef.prop (Acc A R x) => P x))))) + (c => + c + (F => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + a : cupicef.Term cupicef.prop (Acc A R x) => + match____Acc cupicef.prop A R x + (__ : cupicef.Term cupicef.prop (Acc A R x) => P x) + (a0 : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (Acc A R y)) => + f x a0 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + r : cupicef.Term cupicef.prop (R y x) => + F y (a0 y r))) + a)) + cupicef.0. + +def Acc__rec : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.set) -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (Acc A R y)) -> + __0 : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.set (P y)) -> + cupicef.Term cupicef.set (P x)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + a : cupicef.Term cupicef.prop (Acc A R x) -> + cupicef.Term cupicef.set (P x) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.set) => + Acc__rect A R + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P). + +def Acc__inv : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (Acc A R x) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (Acc A R y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : cupicef.Term cupicef.prop (Acc A R x) => + match____Acc cupicef.prop A R x + (__ : cupicef.Term cupicef.prop (Acc A R x) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R y x) + (__0 : cupicef.Term cupicef.prop (R y x) => Acc A R y))) + (H0 : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (Acc A R y)) => + H0) + H. + +def well__founded : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => Acc A R a). + +def well__founded__induction__type : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Rwf : cupicef.Term cupicef.prop (well__founded A R) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + __ : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Rwf : cupicef.Term cupicef.prop (well__founded A R) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + X : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x)) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Acc__rect A R P + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (Acc A R y)) => + X0 : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) => + X x X0) + a (Rwf a). + +def well__founded__induction : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Rwf : cupicef.Term cupicef.prop (well__founded A R) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.set) -> + __ : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.set (P y)) -> + cupicef.Term cupicef.set (P x)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set (P a) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Rwf : cupicef.Term cupicef.prop (well__founded A R) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.set) => + well__founded__induction__type A R Rwf + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P). + +def well__founded__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Rwf : cupicef.Term cupicef.prop (well__founded A R) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + __ : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (P y)) -> + cupicef.Term cupicef.prop (P x)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (P a) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Rwf : cupicef.Term cupicef.prop (well__founded A R) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + well__founded__induction__type A R Rwf + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P). + +def Fix__F : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + F : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + a : cupicef.Term cupicef.prop (Acc A R x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + F : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x)) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (Acc A R x) + (a : cupicef.Term cupicef.prop (Acc A R x) => P x))))) + (c => + c + (Fix__F0 => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + a : cupicef.Term cupicef.prop (Acc A R x) => + F x + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + h : cupicef.Term cupicef.prop (R y x) => + Fix__F0 y (Acc__inv A R x a y h)))) + cupicef.0. + +def Acc__inv__dep : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + a : cupicef.Term cupicef.prop (Acc A R x) -> + cupicef.Univ cupicef.prop) -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + a : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (Acc A R y)) -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + r : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (P y (a y r))) -> + cupicef.Term cupicef.prop (P x (Acc__intro A R x a))) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + a : cupicef.Term cupicef.prop (Acc A R x) -> + cupicef.Term cupicef.prop (P x a) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + a : cupicef.Term cupicef.prop (Acc A R x) -> + cupicef.Univ cupicef.prop) => + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + a : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (Acc A R y)) -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + r : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (P y (a y r))) -> + cupicef.Term cupicef.prop (P x (Acc__intro A R x a))) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Acc A R x) + (a : cupicef.Term cupicef.prop (Acc A R x) => P x a))))) + (c => + c + (F => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + a : cupicef.Term cupicef.prop (Acc A R x) => + match____Acc cupicef.prop A R x + (a0 : cupicef.Term cupicef.prop (Acc A R x) => P x a0) + (a0 : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop (Acc A R y)) => + f x a0 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + r : cupicef.Term cupicef.prop (R y x) => + F y (a0 y r))) + a)) + cupicef.0. + +thm Fix__F__eq : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + F : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + r : cupicef.Term cupicef.prop (Acc A R x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (F x + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p : cupicef.Term cupicef.prop (R y x) => + Fix__F A R P F y (Acc__inv A R x r y p))) + (Fix__F A R P F x r)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + F : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + r : cupicef.Term cupicef.prop (Acc A R x) => + Acc__inv__dep A R + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + r0 : cupicef.Term cupicef.prop (Acc A R x0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x0)) + (F x0 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p : cupicef.Term cupicef.prop (R y x0) => + Fix__F A R P F y (Acc__inv A R x0 r0 y p))) + (Fix__F A R P F x0 r0)) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + a : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x0) -> + cupicef.Term cupicef.prop (Acc A R y)) => + H : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + r0 : cupicef.Term cupicef.prop (R y x0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P y)) + (F y + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p : cupicef.Term cupicef.prop (R y0 y) => + Fix__F A R P F y0 (Acc__inv A R y (a y r0) y0 p))) + (Fix__F A R P F y (a y r0)))) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x0)) + (Fix__F A R P F x0 (Acc__intro A R x0 a))) + x r. + +def Fix : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Rwf : cupicef.Term cupicef.prop (well__founded A R) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + F : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Rwf : cupicef.Term cupicef.prop (well__founded A R) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + F : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Fix__F A R P F x (Rwf x). + +def let__a : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Rwf : cupicef.Term cupicef.prop (well__founded A R) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + F : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x)) -> + F__ext : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + f : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + g : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P y)) + (f y p) (g y p))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (F x f) (F x g))) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (Acc A R x) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Rwf : cupicef.Term cupicef.prop (well__founded A R) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + F : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x)) => + F__ext : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + f : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + g : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P y)) + (f y p) (g y p))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (F x f) (F x g))) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Rwf x. + +thm Fix__F__inv : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Rwf : cupicef.Term cupicef.prop (well__founded A R) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + F : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x)) -> + F__ext : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + f : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + g : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P y)) + (f y p) (g y p))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (F x f) (F x g))) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + r : cupicef.Term cupicef.prop (Acc A R x) -> + s : cupicef.Term cupicef.prop (Acc A R x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (Fix__F A R P F x r) (Fix__F A R P F x s)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Rwf : cupicef.Term cupicef.prop (well__founded A R) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + F : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x)) => + F__ext : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + f : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + g : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P y)) + (f y p) (g y p))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (F x f) (F x g))) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Acc__ind A R + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (Acc A R x0) + (r : cupicef.Term cupicef.prop (Acc A R x0) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (Acc A R x0) + (s : cupicef.Term cupicef.prop (Acc A R x0) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x0)) + (Fix__F A R P F x0 r) (Fix__F A R P F x0 s)))) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x0) -> + cupicef.Term cupicef.prop (Acc A R y)) => + H0 : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x0) -> + r : cupicef.Term cupicef.prop (Acc A R y) -> + s : cupicef.Term cupicef.prop (Acc A R y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P y)) + (Fix__F A R P F y r) (Fix__F A R P F y s))) => + r : cupicef.Term cupicef.prop (Acc A R x0) => + s : cupicef.Term cupicef.prop (Acc A R x0) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x0)) + (F x0 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p : cupicef.Term cupicef.prop (R y x0) => + Fix__F A R P F y (Acc__inv A R x0 r y p))) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x0)) + p (Fix__F A R P F x0 s)) + (Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x0)) + (F x0 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p : cupicef.Term cupicef.prop (R y x0) => + Fix__F A R P F y (Acc__inv A R x0 s y p))) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x0) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x0)) + (F x0 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p0 : cupicef.Term cupicef.prop (R y x0) => + Fix__F A R P F y (Acc__inv A R x0 r y p0))) + p) + (F__ext x0 + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p : cupicef.Term cupicef.prop (R y x0) => + Fix__F A R P F y (Acc__inv A R x0 r y p)) + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p : cupicef.Term cupicef.prop (R y x0) => + Fix__F A R P F y (Acc__inv A R x0 s y p)) + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p : cupicef.Term cupicef.prop (R y x0) => + H0 y p (Acc__inv A R x0 r y p) (Acc__inv A R x0 s y p))) + (Fix__F A R P F x0 s) (Fix__F__eq A R P F x0 s)) + (Fix__F A R P F x0 r) (Fix__F__eq A R P F x0 r)) + x (let__a A R Rwf P F F__ext x). + +thm Fix__eq : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + Rwf : cupicef.Term cupicef.prop (well__founded A R) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + F : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x)) -> + F__ext : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + f : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + g : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P y)) + (f y p) (g y p))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (F x f) (F x g))) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (Fix A R Rwf P F x) + (F x + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p : cupicef.Term cupicef.prop (R y x) => + Fix A R Rwf P F y))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + Rwf : cupicef.Term cupicef.prop (well__founded A R) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + F : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x)) => + F__ext : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + f : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + g : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y)) -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : cupicef.Term cupicef.prop (R y x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P y)) + (f y p) (g y p))) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (F x f) (F x g))) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.cast cupicef.prop cupicef.prop + ((p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + p + (F x + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p0 : cupicef.Term cupicef.prop (R y x) => + Fix__F A R P F y (Rwf y)))) + (Fix__F A R P F x (Rwf x))) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (Fix A R Rwf P F x) + (F x + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p : cupicef.Term cupicef.prop (R y x) => + Fix A R Rwf P F y))) + cupicef.I + (Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (F x + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p : cupicef.Term cupicef.prop (R y x) => + Fix__F A R P F y (Acc__inv A R x (Rwf x) y p))) + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + p + (F x + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p0 : cupicef.Term cupicef.prop (R y x) => + Fix__F A R P F y (Rwf y)))) + (F__ext x + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p : cupicef.Term cupicef.prop (R y x) => + Fix__F A R P F y (Acc__inv A R x (Rwf x) y p)) + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p : cupicef.Term cupicef.prop (R y x) => + Fix__F A R P F y (Rwf y)) + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p : cupicef.Term cupicef.prop (R y x) => + Fix__F__inv A R Rwf P F F__ext y (Acc__inv A R x (Rwf x) y p) + (Rwf y))) + (Fix__F A R P F x (Rwf x)) (Fix__F__eq A R P F x (Rwf x))). + +def Fix__F__2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + __0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ cupicef.prop) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + F : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __ : + cupicef.Term cupicef.prop + (R (Coq__Init__Datatypes.pair A B y y_27_) + (Coq__Init__Datatypes.pair A B x x_27_)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y y_27_)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x x_27_)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + a : + cupicef.Term cupicef.prop + (Acc + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) + R (Coq__Init__Datatypes.pair A B x x_27_)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x x_27_) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + __0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ cupicef.prop) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + F : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __ : + cupicef.Term cupicef.prop + (R (Coq__Init__Datatypes.pair A B y y_27_) + (Coq__Init__Datatypes.pair A B x x_27_)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y y_27_)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x x_27_)) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.2 + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I B + (x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + cupicef.prod cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Acc + (Coq__Init__Datatypes.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) + R (Coq__Init__Datatypes.pair A B x x_27_)) + (a : + cupicef.Term cupicef.prop + (Acc + (Coq__Init__Datatypes.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) + R (Coq__Init__Datatypes.pair A B x x_27_)) => + P x x_27_)))))) + (c => + c + (Fix__F__3 => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + a : + cupicef.Term cupicef.prop + (Acc + (Coq__Init__Datatypes.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) + R (Coq__Init__Datatypes.pair A B x x_27_)) => + F x x_27_ + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + h : + cupicef.Term cupicef.prop + (R (Coq__Init__Datatypes.pair A B y y_27_) + (Coq__Init__Datatypes.pair A B x x_27_)) => + Fix__F__3 y y_27_ + (Acc__inv + (Coq__Init__Datatypes.prod + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) + R (Coq__Init__Datatypes.pair A B x x_27_) a + (Coq__Init__Datatypes.pair A B y y_27_) h)))) + cupicef.0. + +def well__founded__induction__type__2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + __0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ cupicef.prop) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + Rwf : + cupicef.Term cupicef.prop + (well__founded + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) + R) -> + __ : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __ : + cupicef.Term cupicef.prop + (R (Coq__Init__Datatypes.pair A B y y_27_) + (Coq__Init__Datatypes.pair A B x x_27_)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y y_27_)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x x_27_)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P a b) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + __0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ cupicef.prop) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + Rwf : + cupicef.Term cupicef.prop + (well__founded + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) + R) => + X : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __ : + cupicef.Term cupicef.prop + (R (Coq__Init__Datatypes.pair A B y y_27_) + (Coq__Init__Datatypes.pair A B x x_27_)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P y y_27_)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x x_27_)) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + Fix__F__2 A B R P X a b (Rwf (Coq__Init__Datatypes.pair A B a b)). + +def Acc__intro__generator : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + wf : cupicef.Term cupicef.prop (well__founded A R) -> + cupicef.Term cupicef.prop (well__founded A R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.0 cupicef.prop + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Datatypes.nat + (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (well__founded A R) + (wf : cupicef.Term cupicef.prop (well__founded A R) => + well__founded A R))))) + (c => + c + (Acc__intro__generator0 => + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + wf : cupicef.Term cupicef.prop (well__founded A R) => + Coq__Init__Datatypes.match____nat cupicef.prop + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + well__founded A R) + wf + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Acc__intro A R x + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + __ : cupicef.Term cupicef.prop (R y x) => + Acc__intro__generator0 n0 (Acc__intro__generator0 n0 wf) y)) + n)) + cupicef.0. + +(; End of translation. ;) diff --git a/out/Coq__Logic__Decidable.dk b/out/Coq__Logic__Decidable.dk new file mode 100644 index 000000000..822838e0e --- /dev/null +++ b/out/Coq__Logic__Decidable.dk @@ -0,0 +1,9614 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def decidable : + P : cupicef.Univ cupicef.prop -> cupicef.Univ cupicef.prop + +:= P : cupicef.Univ cupicef.prop => + Coq__Init__Logic.or P (Coq__Init__Logic.not P). + +def let__H : + P : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or P (Coq__Init__Logic.not P)) -> + H0 : + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= P : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or P (Coq__Init__Logic.not P)) => + H0 : + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) => + H0 H1. + +thm dec__not__not : + P : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (decidable P) -> + __0 : + (__0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop P + +:= cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (P : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop)) + cupicef.I (Coq__Init__Logic.or P (Coq__Init__Logic.not P)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or P (Coq__Init__Logic.not P)) => + cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not P) + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) => + Coq__Init__Logic.False)) + (H0 : + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + P)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (P : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop)) + cupicef.I (decidable P) + (__ : cupicef.Term cupicef.prop (decidable P) => + cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not P) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) => + Coq__Init__Logic.False)) + (__0 : + (__0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + P)))) + cupicef.I + (P : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or P (Coq__Init__Logic.not P)) => + H0 : + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.or__ind P (Coq__Init__Logic.not P) P + (H1 : cupicef.Term cupicef.prop P => H1) + (H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) => + Coq__Init__Logic.False__ind P (let__H P H H0 H1)) + H). + +thm dec__True : + cupicef.Term cupicef.prop (decidable Coq__Init__Logic.True) + +:= cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.or Coq__Init__Logic.True + (Coq__Init__Logic.not Coq__Init__Logic.True)) + (decidable Coq__Init__Logic.True) cupicef.I + (Coq__Init__Logic.or__introl Coq__Init__Logic.True + (Coq__Init__Logic.not Coq__Init__Logic.True) Coq__Init__Logic.I). + +thm dec__False : + cupicef.Term cupicef.prop (decidable Coq__Init__Logic.False) + +:= cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.or Coq__Init__Logic.False + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.False + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False))) + (decidable Coq__Init__Logic.False) cupicef.I + (Coq__Init__Logic.or__intror Coq__Init__Logic.False + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.False + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False)) + (H : cupicef.Term cupicef.prop Coq__Init__Logic.False => H)). + +def let__H0 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) -> + H4 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) => + H4 : cupicef.Term cupicef.prop A => + H1 H4. + +def let__H1 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) -> + H4 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) => + H4 : cupicef.Term cupicef.prop B => + H2 H4. + +thm dec__or : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (decidable A) -> + __0 : cupicef.Term cupicef.prop (decidable B) -> + cupicef.Term cupicef.prop (decidable (Coq__Init__Logic.or A B)) + +:= cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + Coq__Init__Logic.or (Coq__Init__Logic.or A B) + (Coq__Init__Logic.not (Coq__Init__Logic.or A B))))))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (decidable A) + (__ : cupicef.Term cupicef.prop (decidable A) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (decidable B) + (__0 : cupicef.Term cupicef.prop (decidable B) => + decidable (Coq__Init__Logic.or A B)))))) + cupicef.I + (A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + Coq__Init__Logic.or__ind A (Coq__Init__Logic.not A) + (Coq__Init__Logic.or (Coq__Init__Logic.or A B) + (Coq__Init__Logic.not (Coq__Init__Logic.or A B))) + (H1 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.or__ind B (Coq__Init__Logic.not B) + (Coq__Init__Logic.or (Coq__Init__Logic.or A B) + (Coq__Init__Logic.not (Coq__Init__Logic.or A B))) + (H2 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.or__introl (Coq__Init__Logic.or A B) + (Coq__Init__Logic.not (Coq__Init__Logic.or A B)) + (Coq__Init__Logic.or__introl A B H1)) + (H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + Coq__Init__Logic.or__introl (Coq__Init__Logic.or A B) + (Coq__Init__Logic.not (Coq__Init__Logic.or A B)) + (Coq__Init__Logic.or__introl A B H1)) + H0) + (H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + Coq__Init__Logic.or__ind B (Coq__Init__Logic.not B) + (Coq__Init__Logic.or (Coq__Init__Logic.or A B) + (Coq__Init__Logic.not (Coq__Init__Logic.or A B))) + (H2 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.or__introl (Coq__Init__Logic.or A B) + (Coq__Init__Logic.not (Coq__Init__Logic.or A B)) + (Coq__Init__Logic.or__intror A B H2)) + (H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + Coq__Init__Logic.or__intror (Coq__Init__Logic.or A B) + (Coq__Init__Logic.not (Coq__Init__Logic.or A B)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or A B) + (H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not (Coq__Init__Logic.or A B)) cupicef.I + (H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) => + Coq__Init__Logic.or__ind A B Coq__Init__Logic.False + (H4 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H0 A B H H0 H1 H2 H3 H4)) + (H4 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H1 A B H H0 H1 H2 H3 H4)) + H3))) + H0) + H). + +def let__H2 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : cupicef.Term cupicef.prop A -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) -> + H4 : cupicef.Term cupicef.prop A -> + H5 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : cupicef.Term cupicef.prop A => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) => + H4 : cupicef.Term cupicef.prop A => + H5 : cupicef.Term cupicef.prop B => + H2 H5. + +def let__H3 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H2 : cupicef.Term cupicef.prop B -> + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) -> + H4 : cupicef.Term cupicef.prop A -> + H5 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H2 : cupicef.Term cupicef.prop B => + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) => + H4 : cupicef.Term cupicef.prop A => + H5 : cupicef.Term cupicef.prop B => + H1 H4. + +def let__H4 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) -> + H4 : cupicef.Term cupicef.prop A -> + H5 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) => + H4 : cupicef.Term cupicef.prop A => + H5 : cupicef.Term cupicef.prop B => + H1 H4. + +def let__H5 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) -> + H4 : cupicef.Term cupicef.prop A -> + H5 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) => + H4 : cupicef.Term cupicef.prop A => + H5 : cupicef.Term cupicef.prop B => + H2 H5. + +thm dec__and : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (decidable A) -> + __0 : cupicef.Term cupicef.prop (decidable B) -> + cupicef.Term cupicef.prop (decidable (Coq__Init__Logic.and A B)) + +:= cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + Coq__Init__Logic.or (Coq__Init__Logic.and A B) + (Coq__Init__Logic.not (Coq__Init__Logic.and A B))))))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (decidable A) + (__ : cupicef.Term cupicef.prop (decidable A) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (decidable B) + (__0 : cupicef.Term cupicef.prop (decidable B) => + decidable (Coq__Init__Logic.and A B)))))) + cupicef.I + (A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + Coq__Init__Logic.or__ind A (Coq__Init__Logic.not A) + (Coq__Init__Logic.or (Coq__Init__Logic.and A B) + (Coq__Init__Logic.not (Coq__Init__Logic.and A B))) + (H1 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.or__ind B (Coq__Init__Logic.not B) + (Coq__Init__Logic.or (Coq__Init__Logic.and A B) + (Coq__Init__Logic.not (Coq__Init__Logic.and A B))) + (H2 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.or__introl (Coq__Init__Logic.and A B) + (Coq__Init__Logic.not (Coq__Init__Logic.and A B)) + (Coq__Init__Logic.conj A B H1 H2)) + (H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + Coq__Init__Logic.or__intror (Coq__Init__Logic.and A B) + (Coq__Init__Logic.not (Coq__Init__Logic.and A B)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and A B) + (H3 : + cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not (Coq__Init__Logic.and A B)) cupicef.I + (H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) => + Coq__Init__Logic.and__ind A B Coq__Init__Logic.False + (H4 : cupicef.Term cupicef.prop A => + H5 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H2 A B H H0 H1 H2 H3 H4 H5)) + H3))) + H0) + (H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + Coq__Init__Logic.or__ind B (Coq__Init__Logic.not B) + (Coq__Init__Logic.or (Coq__Init__Logic.and A B) + (Coq__Init__Logic.not (Coq__Init__Logic.and A B))) + (H2 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.or__intror (Coq__Init__Logic.and A B) + (Coq__Init__Logic.not (Coq__Init__Logic.and A B)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and A B) + (H3 : + cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not (Coq__Init__Logic.and A B)) cupicef.I + (H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) => + Coq__Init__Logic.and__ind A B Coq__Init__Logic.False + (H4 : cupicef.Term cupicef.prop A => + H5 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H3 A B H H0 H1 H2 H3 H4 H5)) + H3))) + (H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + Coq__Init__Logic.or__intror (Coq__Init__Logic.and A B) + (Coq__Init__Logic.not (Coq__Init__Logic.and A B)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and A B) + (H3 : + cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not (Coq__Init__Logic.and A B)) cupicef.I + (H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) => + Coq__Init__Logic.and__ind A B Coq__Init__Logic.False + (H4 : cupicef.Term cupicef.prop A => + H5 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H5 A B H H0 H1 H2 H3 H4 H5)) + H3))) + H0) + H). + +def let__H6 : + A : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop A -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop A => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 H0. + +def let__H7 : + A : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : cupicef.Term cupicef.prop A => + H0 H1. + +thm dec__not : + A : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (decidable A) -> + cupicef.Term cupicef.prop (decidable (Coq__Init__Logic.not A)) + +:= cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + Coq__Init__Logic.or (Coq__Init__Logic.not A) + (Coq__Init__Logic.not (Coq__Init__Logic.not A))))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (decidable A) + (__ : cupicef.Term cupicef.prop (decidable A) => + decidable (Coq__Init__Logic.not A)))) + cupicef.I + (A : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + Coq__Init__Logic.or__ind A (Coq__Init__Logic.not A) + (Coq__Init__Logic.or (Coq__Init__Logic.not A) + (Coq__Init__Logic.not (Coq__Init__Logic.not A))) + (H0 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.or__intror (Coq__Init__Logic.not A) + (Coq__Init__Logic.not (Coq__Init__Logic.not A)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not A) + (H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not (Coq__Init__Logic.not A)) cupicef.I + (H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H6 A H H0 H1)))) + (H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + Coq__Init__Logic.or__introl (Coq__Init__Logic.not A) + (Coq__Init__Logic.not (Coq__Init__Logic.not A)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (H1 : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + (Coq__Init__Logic.not A) cupicef.I + (H1 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H7 A H H0 H1)))) + H). + +def let__H8 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : cupicef.Term cupicef.prop A -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H3 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : cupicef.Term cupicef.prop A => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H3 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H3 H1. + +def let__H9 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : cupicef.Term cupicef.prop A -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H3 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : cupicef.Term cupicef.prop A => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H3 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H2 (let__H8 A B H H0 H1 H2 H3). + +def let__H10 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H2 : cupicef.Term cupicef.prop B -> + H3 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H2 : cupicef.Term cupicef.prop B => + H3 : cupicef.Term cupicef.prop A => + H1 H3. + +def let__H11 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H3 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H3 : cupicef.Term cupicef.prop A => + H1 H3. + +thm dec__imp : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (decidable A) -> + __0 : cupicef.Term cupicef.prop (decidable B) -> + cupicef.Term cupicef.prop + (decidable + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__1 : cupicef.Term cupicef.prop A => B))) + +:= cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + A (__ : cupicef.Term cupicef.prop A => B)))))))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (decidable A) + (__ : cupicef.Term cupicef.prop (decidable A) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (decidable B) + (__0 : cupicef.Term cupicef.prop (decidable B) => + decidable + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__1 : cupicef.Term cupicef.prop A => B))))))) + cupicef.I + (A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + Coq__Init__Logic.or__ind A (Coq__Init__Logic.not A) + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)))) + (H1 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.or__ind B (Coq__Init__Logic.not B) + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)))) + (H2 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.or__introl + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) + (H3 : cupicef.Term cupicef.prop A => H2)) + (H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + Coq__Init__Logic.or__intror + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) + (cupicef.cast + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.prop + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (H3 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) + cupicef.I + (H3 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H9 A B H H0 H1 H2 H3)))) + H0) + (H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + Coq__Init__Logic.or__ind B (Coq__Init__Logic.not B) + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)))) + (H2 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.or__introl + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) + (H3 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind B (let__H10 A B H H0 H1 H2 H3))) + (H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + Coq__Init__Logic.or__introl + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) + (H3 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind B (let__H11 A B H H0 H1 H2 H3))) + H0) + H). + +def let__H12 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : cupicef.Term cupicef.prop A -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff A B) -> + H4 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + H5 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : cupicef.Term cupicef.prop A => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff A B) => + H4 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H5 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + H4 H1. + +def let__H13 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : cupicef.Term cupicef.prop A -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff A B) -> + H4 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + H5 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : cupicef.Term cupicef.prop A => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff A B) => + H4 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H5 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + H2 (let__H12 A B H H0 H1 H2 H3 H4 H5). + +def let__H14 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : cupicef.Term cupicef.prop A -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff A B) -> + H4 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + H5 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop A + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : cupicef.Term cupicef.prop A => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff A B) => + H4 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H5 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + H5 (let__H12 A B H H0 H1 H2 H3 H4 H5). + +def let__H15 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H2 : cupicef.Term cupicef.prop B -> + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff A B) -> + H4 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + H5 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop A + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H2 : cupicef.Term cupicef.prop B => + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff A B) => + H4 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H5 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + H5 H2. + +def let__H16 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H2 : cupicef.Term cupicef.prop B -> + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff A B) -> + H4 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + H5 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H2 : cupicef.Term cupicef.prop B => + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff A B) => + H4 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H5 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + H1 (let__H15 A B H H0 H1 H2 H3 H4 H5). + +def let__H17 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H2 : cupicef.Term cupicef.prop B -> + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff A B) -> + H4 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + H5 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H2 : cupicef.Term cupicef.prop B => + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff A B) => + H4 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H5 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + H4 (let__H15 A B H H0 H1 H2 H3 H4 H5). + +def let__H18 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H3 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H3 : cupicef.Term cupicef.prop A => + H1 H3. + +def let__H19 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H3 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H3 : cupicef.Term cupicef.prop B => + H2 H3. + +thm dec__iff : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (decidable A) -> + __0 : cupicef.Term cupicef.prop (decidable B) -> + cupicef.Term cupicef.prop (decidable (Coq__Init__Logic.iff A B)) + +:= cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + Coq__Init__Logic.or (Coq__Init__Logic.iff A B) + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B))))))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (decidable A) + (__ : cupicef.Term cupicef.prop (decidable A) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (decidable B) + (__0 : cupicef.Term cupicef.prop (decidable B) => + decidable (Coq__Init__Logic.iff A B)))))) + cupicef.I + (A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + Coq__Init__Logic.or__ind A (Coq__Init__Logic.not A) + (Coq__Init__Logic.or (Coq__Init__Logic.iff A B) + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B))) + (H1 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.or__ind B (Coq__Init__Logic.not B) + (Coq__Init__Logic.or (Coq__Init__Logic.iff A B) + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B))) + (H2 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.or__introl (Coq__Init__Logic.iff A B) + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)) + (H3 : cupicef.Term cupicef.prop A => H2) + (H3 : cupicef.Term cupicef.prop B => H1))) + (H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + Coq__Init__Logic.or__intror (Coq__Init__Logic.iff A B) + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.iff A B) + (H3 : + cupicef.Term cupicef.prop (Coq__Init__Logic.iff A B) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) cupicef.I + (H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff A B) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)) + Coq__Init__Logic.False + (H4 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + H5 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop A) => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H13 A B H H0 H1 H2 H3 H4 H5)) + H3))) + H0) + (H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + Coq__Init__Logic.or__ind B (Coq__Init__Logic.not B) + (Coq__Init__Logic.or (Coq__Init__Logic.iff A B) + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B))) + (H2 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.or__intror (Coq__Init__Logic.iff A B) + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.iff A B) + (H3 : + cupicef.Term cupicef.prop (Coq__Init__Logic.iff A B) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) cupicef.I + (H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.iff A B) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)) + Coq__Init__Logic.False + (H4 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + H5 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop A) => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H16 A B H H0 H1 H2 H3 H4 H5)) + H3))) + (H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + Coq__Init__Logic.or__introl (Coq__Init__Logic.iff A B) + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)) + (H3 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind B (let__H18 A B H H0 H1 H2 H3)) + (H3 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False__ind A (let__H19 A B H H0 H1 H2 H3)))) + H0) + H). + +def let__H20 : + P : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or P (Coq__Init__Logic.not P)) -> + H0 : + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Coq__Init__Logic.not P)) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= P : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or P (Coq__Init__Logic.not P)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.not P)) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) => + H0 H1. + +thm not__not : + P : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (decidable P) -> + __0 : + cupicef.Term cupicef.prop (Coq__Init__Logic.not (Coq__Init__Logic.not P)) -> + cupicef.Term cupicef.prop P + +:= cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (P : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (Coq__Init__Logic.or P (Coq__Init__Logic.not P)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or P (Coq__Init__Logic.not P)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Logic.not P)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.not P)) => + P)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (P : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (decidable P) + (__ : cupicef.Term cupicef.prop (decidable P) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Logic.not P)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.not P)) => + P)))) + cupicef.I + (P : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or P (Coq__Init__Logic.not P)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.not P)) => + Coq__Init__Logic.or__ind P (Coq__Init__Logic.not P) P + (H1 : cupicef.Term cupicef.prop P => H1) + (H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not P) => + Coq__Init__Logic.False__ind P (let__H20 P H H0 H1)) + H). + +def let__H21 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.or A B)) -> + __ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.or A B)) => + H0 : cupicef.Term cupicef.prop A => + H (Coq__Init__Logic.or__introl A B H0). + +def let__H22 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.or A B)) -> + __ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.or A B)) => + H1 : cupicef.Term cupicef.prop B => + H (Coq__Init__Logic.or__intror A B H1). + +def let__H23 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.or A B)) -> + H2 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.or A B)) => + H2 : cupicef.Term cupicef.prop A => + let__H21 A B H H2. + +def let__H24 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.or A B)) -> + H2 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.or A B)) => + H2 : cupicef.Term cupicef.prop B => + let__H22 A B H H2. + +thm not__or : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.or A B)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (Coq__Init__Logic.not A) (Coq__Init__Logic.not B)) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.or A B)) => + Coq__Init__Logic.conj (Coq__Init__Logic.not A) (Coq__Init__Logic.not B) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (H2 : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + (Coq__Init__Logic.not A) cupicef.I + (H2 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H23 A B H H2))) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (H2 : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + (Coq__Init__Logic.not B) cupicef.I + (H2 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H24 A B H H2))). + +def let__H25 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.and A B)) -> + H1 : cupicef.Term cupicef.prop A -> + __ : cupicef.Term cupicef.prop A -> + __0 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.and A B)) => + H1 : cupicef.Term cupicef.prop A => + H2 : cupicef.Term cupicef.prop A => + H3 : cupicef.Term cupicef.prop B => + H0 (Coq__Init__Logic.conj A B H2 H3). + +def let__H26 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.and A B)) -> + H1 : cupicef.Term cupicef.prop A -> + __ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.and A B)) => + H1 : cupicef.Term cupicef.prop A => + let__H25 A B H H0 H1 H1. + +def let__H27 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.and A B)) -> + H1 : cupicef.Term cupicef.prop A -> + H2 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.and A B)) => + H1 : cupicef.Term cupicef.prop A => + H2 : cupicef.Term cupicef.prop B => + let__H26 A B H H0 H1 H2. + +def let__H28 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.and A B)) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + __ : cupicef.Term cupicef.prop A -> + __0 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.and A B)) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H2 : cupicef.Term cupicef.prop A => + H3 : cupicef.Term cupicef.prop B => + H0 (Coq__Init__Logic.conj A B H2 H3). + +def let__H29 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.and A B)) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H2 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.and A B)) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H2 : cupicef.Term cupicef.prop A => + H1 H2. + +def let__H30 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.and A B)) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H2 : cupicef.Term cupicef.prop A -> + __ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.and A B)) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H2 : cupicef.Term cupicef.prop A => + let__H28 A B H H0 H1 H2. + +thm not__and : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (decidable A) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.and A B)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Logic.not A) (Coq__Init__Logic.not B)) + +:= cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Logic.and A B)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.and A B)) => + Coq__Init__Logic.or (Coq__Init__Logic.not A) + (Coq__Init__Logic.not B)))))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (decidable A) + (__ : cupicef.Term cupicef.prop (decidable A) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Logic.and A B)) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.and A B)) => + Coq__Init__Logic.or (Coq__Init__Logic.not A) + (Coq__Init__Logic.not B)))))) + cupicef.I + (A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.and A B)) => + Coq__Init__Logic.or__ind A (Coq__Init__Logic.not A) + (Coq__Init__Logic.or (Coq__Init__Logic.not A) + (Coq__Init__Logic.not B)) + (H1 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.or__intror (Coq__Init__Logic.not A) + (Coq__Init__Logic.not B) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (H2 : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + (Coq__Init__Logic.not B) cupicef.I + (H2 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H27 A B H H0 H1 H2)))) + (H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + Coq__Init__Logic.or__introl (Coq__Init__Logic.not A) + (Coq__Init__Logic.not B) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (H2 : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + (Coq__Init__Logic.not A) cupicef.I + (H2 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H29 A B H H0 H1 H2)))) + H). + +def let__H33 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) -> + H1 : cupicef.Term cupicef.prop A -> + H2 : cupicef.Term cupicef.prop B -> + H3 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H4 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) => + H1 : cupicef.Term cupicef.prop A => + H2 : cupicef.Term cupicef.prop B => + H3 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop A => + H3 H2. + +def let__H32 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) -> + H1 : cupicef.Term cupicef.prop A -> + H2 : cupicef.Term cupicef.prop B -> + __ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) => + H1 : cupicef.Term cupicef.prop A => + H2 : cupicef.Term cupicef.prop B => + (H3 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind B (let__H33 A B H H0 H1 H2 H3 H4)) + (y : cupicef.Term cupicef.prop B => + H0 (x : cupicef.Term cupicef.prop A => y)). + +def let__H31 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) -> + H1 : cupicef.Term cupicef.prop A -> + H2 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) => + H1 : cupicef.Term cupicef.prop A => + H2 : cupicef.Term cupicef.prop B => + H0 (let__H32 A B H H0 H1 H2). + +def let__H36 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H2 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H3 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H2 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 : cupicef.Term cupicef.prop A => + H1 H3. + +def let__H35 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + __ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + (H2 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind B (let__H36 A B H H0 H1 H2 H3)) + (y : cupicef.Term cupicef.prop B => + H0 (x : cupicef.Term cupicef.prop A => y)). + +def let__H34 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H0 (let__H35 A B H H0 H1). + +def let__H39 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H2 : cupicef.Term cupicef.prop B -> + H3 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H4 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H2 : cupicef.Term cupicef.prop B => + H3 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop A => + H3 H2. + +def let__H40 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H2 : cupicef.Term cupicef.prop B -> + H3 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H4 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H2 : cupicef.Term cupicef.prop B => + H3 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop A => + H1 H4. + +def let__H38 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H2 : cupicef.Term cupicef.prop B -> + __ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H2 : cupicef.Term cupicef.prop B => + (H3 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind B (let__H40 A B H H0 H1 H2 H3 H4)) + (y : cupicef.Term cupicef.prop B => + H0 (x : cupicef.Term cupicef.prop A => y)). + +def let__H37 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H2 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H2 : cupicef.Term cupicef.prop B => + H0 (let__H38 A B H H0 H1 H2). + +thm not__imp : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (decidable A) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B))) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.and A (Coq__Init__Logic.not B)) + +:= cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) => + Coq__Init__Logic.and A (Coq__Init__Logic.not B)))))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (decidable A) + (__ : cupicef.Term cupicef.prop (decidable A) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B))) + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B))) => + Coq__Init__Logic.and A (Coq__Init__Logic.not B)))))) + cupicef.I + (A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) => + Coq__Init__Logic.or__ind A (Coq__Init__Logic.not A) + (Coq__Init__Logic.and A (Coq__Init__Logic.not B)) + (H1 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.conj A (Coq__Init__Logic.not B) H1 + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (H2 : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + (Coq__Init__Logic.not B) cupicef.I + (H2 : cupicef.Term cupicef.prop B => + (H3 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind Coq__Init__Logic.False H3) + (let__H31 A B H H0 H1 H2)))) + (H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + Coq__Init__Logic.conj A (Coq__Init__Logic.not B) + ((H2 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind A H2) (let__H34 A B H H0 H1)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (H2 : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + (Coq__Init__Logic.not B) cupicef.I + (H2 : cupicef.Term cupicef.prop B => + (H3 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind Coq__Init__Logic.False H3) + (let__H37 A B H H0 H1 H2)))) + H). + +def let__H41 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + H1 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H1 : cupicef.Term cupicef.prop A => + H0 H1. + +def let__H42 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H2 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H2 : cupicef.Term cupicef.prop A => + H0 H2. + +def let__H43 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H2 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H2 : cupicef.Term cupicef.prop A => + H1 H2. + +thm imp__simp : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (decidable A) -> + __0 : (__0 : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.or (Coq__Init__Logic.not A) B) + +:= cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop)) + cupicef.I (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (H0 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.or (Coq__Init__Logic.not A) B))))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop)) + cupicef.I (decidable A) + (__ : cupicef.Term cupicef.prop (decidable A) => + cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B)) + (__0 : + (__0 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.or (Coq__Init__Logic.not A) B))))) + cupicef.I + (A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + Coq__Init__Logic.or__ind A (Coq__Init__Logic.not A) + (Coq__Init__Logic.or (Coq__Init__Logic.not A) B) + (H1 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.or__intror (Coq__Init__Logic.not A) B + (let__H41 A B H H0 H1)) + (H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + Coq__Init__Logic.or__introl (Coq__Init__Logic.not A) B + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (H2 : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + (Coq__Init__Logic.not A) cupicef.I + (H2 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H43 A B H H0 H1 H2)))) + H). + +def let__H44 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) -> + H2 : cupicef.Term cupicef.prop A -> + H3 : cupicef.Term cupicef.prop B -> + __ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + __0 : (__0 : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) => + H2 : cupicef.Term cupicef.prop A => + H3 : cupicef.Term cupicef.prop B => + H4 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H5 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + H1 + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)) + H4 H5). + +def let__H47 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) -> + H2 : cupicef.Term cupicef.prop A -> + H3 : cupicef.Term cupicef.prop B -> + H4 : + (y : cupicef.Term cupicef.prop B -> + __ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H5 : cupicef.Term cupicef.prop A -> + __ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) => + H2 : cupicef.Term cupicef.prop A => + H3 : cupicef.Term cupicef.prop B => + H4 : + (y : cupicef.Term cupicef.prop B -> + __ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H5 : cupicef.Term cupicef.prop A => + H4 H3. + +def let__H46 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) -> + H2 : cupicef.Term cupicef.prop A -> + H3 : cupicef.Term cupicef.prop B -> + __ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) => + H2 : cupicef.Term cupicef.prop A => + H3 : cupicef.Term cupicef.prop B => + (H4 : + (y : cupicef.Term cupicef.prop B -> + __ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H5 : cupicef.Term cupicef.prop A => + H3) + (y : cupicef.Term cupicef.prop B => + let__H44 A B H H0 H1 H2 H3 (x : cupicef.Term cupicef.prop A => y)). + +def let__H45 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) -> + H2 : cupicef.Term cupicef.prop A -> + H3 : cupicef.Term cupicef.prop B -> + __ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) => + H2 : cupicef.Term cupicef.prop A => + H3 : cupicef.Term cupicef.prop B => + let__H44 A B H H0 H1 H2 H3 (let__H46 A B H H0 H1 H2 H3). + +def let__H50 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) -> + H2 : cupicef.Term cupicef.prop A -> + H3 : cupicef.Term cupicef.prop B -> + H4 : + (__ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H5 : + (y : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H6 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) => + H2 : cupicef.Term cupicef.prop A => + H3 : cupicef.Term cupicef.prop B => + H4 : + (__ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H5 : + (y : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H6 : cupicef.Term cupicef.prop B => + H5 H2. + +def let__H49 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) -> + H2 : cupicef.Term cupicef.prop A -> + H3 : cupicef.Term cupicef.prop B -> + H4 : + (__ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + __ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop A + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) => + H2 : cupicef.Term cupicef.prop A => + H3 : cupicef.Term cupicef.prop B => + H4 : + (__ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + (H5 : + (y : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H6 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False__ind A (let__H50 A B H H0 H1 H2 H3 H4 H5 H6)) + (y : cupicef.Term cupicef.prop A => + H4 (x : cupicef.Term cupicef.prop B => y)). + +def let__H48 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) -> + H2 : cupicef.Term cupicef.prop A -> + H3 : cupicef.Term cupicef.prop B -> + H4 : + (__ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) => + H2 : cupicef.Term cupicef.prop A => + H3 : cupicef.Term cupicef.prop B => + H4 : + (__ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 (let__H49 A B H H0 H1 H2 H3 H4). + +def let__H51 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) -> + H2 : cupicef.Term cupicef.prop A -> + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + __ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + __0 : (__0 : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) => + H2 : cupicef.Term cupicef.prop A => + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H4 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H5 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + H1 + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)) + H4 H5). + +def let__H52 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) -> + H2 : cupicef.Term cupicef.prop A -> + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H4 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) => + H2 : cupicef.Term cupicef.prop A => + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H4 : cupicef.Term cupicef.prop B => + H3 H4. + +def let__H53 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H3 : cupicef.Term cupicef.prop B -> + __ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + __0 : (__0 : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H3 : cupicef.Term cupicef.prop B => + H4 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H5 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + H1 + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)) + H4 H5). + +def let__H56 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H3 : cupicef.Term cupicef.prop B -> + H4 : + (y : cupicef.Term cupicef.prop B -> + __ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H5 : cupicef.Term cupicef.prop A -> + __ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H3 : cupicef.Term cupicef.prop B => + H4 : + (y : cupicef.Term cupicef.prop B -> + __ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H5 : cupicef.Term cupicef.prop A => + H4 H3. + +def let__H57 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H3 : cupicef.Term cupicef.prop B -> + H4 : + (y : cupicef.Term cupicef.prop B -> + __ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H5 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H3 : cupicef.Term cupicef.prop B => + H4 : + (y : cupicef.Term cupicef.prop B -> + __ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H5 : cupicef.Term cupicef.prop A => + H2 H5. + +def let__H55 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H3 : cupicef.Term cupicef.prop B -> + __ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H3 : cupicef.Term cupicef.prop B => + (H4 : + (y : cupicef.Term cupicef.prop B -> + __ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H5 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind B (let__H57 A B H H0 H1 H2 H3 H4 H5)) + (y : cupicef.Term cupicef.prop B => + let__H53 A B H H0 H1 H2 H3 (x : cupicef.Term cupicef.prop A => y)). + +def let__H54 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H3 : cupicef.Term cupicef.prop B -> + __ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H3 : cupicef.Term cupicef.prop B => + let__H53 A B H H0 H1 H2 H3 (let__H55 A B H H0 H1 H2 H3). + +def let__H58 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H3 : cupicef.Term cupicef.prop B -> + H4 : + (__ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H5 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H3 : cupicef.Term cupicef.prop B => + H4 : + (__ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H5 : cupicef.Term cupicef.prop A => + H2 H5. + +def let__H59 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + __ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + __0 : (__0 : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H4 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H5 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + H1 + (Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)) + H4 H5). + +def let__H62 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H4 : + (y : cupicef.Term cupicef.prop B -> + __ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H5 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H4 : + (y : cupicef.Term cupicef.prop B -> + __ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H5 : cupicef.Term cupicef.prop A => + H2 H5. + +def let__H61 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + __ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + (H4 : + (y : cupicef.Term cupicef.prop B -> + __ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H5 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind B (let__H62 A B H H0 H1 H2 H3 H4 H5)) + (y : cupicef.Term cupicef.prop B => + let__H59 A B H H0 H1 H2 H3 (x : cupicef.Term cupicef.prop A => y)). + +def let__H60 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + __ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + let__H59 A B H H0 H1 H2 H3 (let__H61 A B H H0 H1 H2 H3). + +def let__H65 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H4 : + (__ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H5 : + (y : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H6 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H4 : + (__ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H5 : + (y : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H6 : cupicef.Term cupicef.prop B => + H3 H6. + +def let__H64 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H4 : + (__ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + __ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop A + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H4 : + (__ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + (H5 : + (y : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H6 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False__ind A (let__H65 A B H H0 H1 H2 H3 H4 H5 H6)) + (y : cupicef.Term cupicef.prop A => + H4 (x : cupicef.Term cupicef.prop B => y)). + +def let__H63 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) -> + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H4 : + (__ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) => + H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H4 : + (__ : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 (let__H64 A B H H0 H1 H2 H3 H4). + +thm not__iff : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (decidable A) -> + __0 : cupicef.Term cupicef.prop (decidable B) -> + __1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (Coq__Init__Logic.and A (Coq__Init__Logic.not B)) + (Coq__Init__Logic.and (Coq__Init__Logic.not A) B)) + +:= cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) => + Coq__Init__Logic.or + (Coq__Init__Logic.and A (Coq__Init__Logic.not B)) + (Coq__Init__Logic.and (Coq__Init__Logic.not A) B))))))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I (decidable A) + (__ : cupicef.Term cupicef.prop (decidable A) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (decidable B) + (__0 : cupicef.Term cupicef.prop (decidable B) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) + (__1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) => + Coq__Init__Logic.or + (Coq__Init__Logic.and A (Coq__Init__Logic.not B)) + (Coq__Init__Logic.and (Coq__Init__Logic.not A) B))))))) + cupicef.I + (A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not (Coq__Init__Logic.iff A B)) => + Coq__Init__Logic.or__ind A (Coq__Init__Logic.not A) + (Coq__Init__Logic.or + (Coq__Init__Logic.and A (Coq__Init__Logic.not B)) + (Coq__Init__Logic.and (Coq__Init__Logic.not A) B)) + (H2 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.or__ind B (Coq__Init__Logic.not B) + (Coq__Init__Logic.or + (Coq__Init__Logic.and A (Coq__Init__Logic.not B)) + (Coq__Init__Logic.and (Coq__Init__Logic.not A) B)) + (H3 : cupicef.Term cupicef.prop B => + (H4 : + (__ : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + (H5 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.or + (Coq__Init__Logic.and A (Coq__Init__Logic.not B)) + (Coq__Init__Logic.and (Coq__Init__Logic.not A) B)) + H5) + (let__H48 A B H H0 H1 H2 H3 H4)) + (let__H45 A B H H0 H1 H2 H3)) + (H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + Coq__Init__Logic.or__introl + (Coq__Init__Logic.and A (Coq__Init__Logic.not B)) + (Coq__Init__Logic.and (Coq__Init__Logic.not A) B) + (Coq__Init__Logic.conj A (Coq__Init__Logic.not B) H2 + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (H4 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not B) cupicef.I + (H4 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H52 A B H H0 H1 H2 H3 H4))))) + H0) + (H2 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + Coq__Init__Logic.or__ind B (Coq__Init__Logic.not B) + (Coq__Init__Logic.or + (Coq__Init__Logic.and A (Coq__Init__Logic.not B)) + (Coq__Init__Logic.and (Coq__Init__Logic.not A) B)) + (H3 : cupicef.Term cupicef.prop B => + (H4 : + (__ : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.or__intror + (Coq__Init__Logic.and A (Coq__Init__Logic.not B)) + (Coq__Init__Logic.and (Coq__Init__Logic.not A) B) + (Coq__Init__Logic.conj (Coq__Init__Logic.not A) B + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (H5 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not A) cupicef.I + (H5 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H58 A B H H0 H1 H2 H3 H4 H5))) + H3)) + (let__H54 A B H H0 H1 H2 H3)) + (H3 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + (H4 : + (__ : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + (H5 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.or + (Coq__Init__Logic.and A (Coq__Init__Logic.not B)) + (Coq__Init__Logic.and (Coq__Init__Logic.not A) B)) + H5) + (let__H63 A B H H0 H1 H2 H3 H4)) + (let__H60 A B H H0 H1 H2 H3)) + H0) + H). + +def let__H67 : + H : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.True + +:= H : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.I. + +def let__H66 : + H : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= H : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H (let__H67 H). + +thm not__true__iff : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.True + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True => + Coq__Init__Logic.False)) + Coq__Init__Logic.False) + +:= Coq__Init__Logic.conj + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.True + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True => + Coq__Init__Logic.False)) + (__ : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.False)) + (cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Logic.False + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.True + (__0 : cupicef.Term cupicef.prop Coq__Init__Logic.True => + Coq__Init__Logic.False))) + (H : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + (H1 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind Coq__Init__Logic.False H1) (let__H66 H)) + (H : cupicef.Term cupicef.prop Coq__Init__Logic.False => + H0 : cupicef.Term cupicef.prop Coq__Init__Logic.True => + Coq__Init__Logic.False__ind Coq__Init__Logic.False H). + +thm not__false__iff : + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.False + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False)) + Coq__Init__Logic.True) + +:= Coq__Init__Logic.conj + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.False + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False)) + (__ : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.True)) + (cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Coq__Init__Logic.True + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.True => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + Coq__Init__Logic.False + (__0 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False))) + (H : + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.I) + (H : cupicef.Term cupicef.prop Coq__Init__Logic.True => + H0 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind Coq__Init__Logic.False H0). + +def let__H68 : + A : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop A -> + H1 : cupicef.Term cupicef.prop A -> + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop A => + H1 : cupicef.Term cupicef.prop A => + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 H0. + +def let__H69 : + A : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H1 H0. + +def let__H70 : + A : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : cupicef.Term cupicef.prop A -> + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : cupicef.Term cupicef.prop A => + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H0 H1. + +def let__H71 : + A : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : cupicef.Term cupicef.prop A -> + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : cupicef.Term cupicef.prop A => + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 H1. + +thm not__not__iff : + A : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (decidable A) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + (__0 : + (__0 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.False)) + A) + +:= cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + Coq__Init__Logic.iff + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.False)) + A))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (decidable A) + (__ : cupicef.Term cupicef.prop (decidable A) => + Coq__Init__Logic.iff + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + (__0 : + (__0 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.False)) + A))) + cupicef.I + (A : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + Coq__Init__Logic.or__ind A (Coq__Init__Logic.not A) + (Coq__Init__Logic.iff + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.False)) + A) + (H0 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.conj + (cupicef.prod + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.False)) + (__ : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + A)) + (cupicef.prod cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop)) + cupicef.I A + (__ : cupicef.Term cupicef.prop A => + cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + (__0 : + (__0 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.False))) + (H1 : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H0) + (H1 : cupicef.Term cupicef.prop A => + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H68 A H H0 H1 H2))) + (H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + Coq__Init__Logic.conj + (cupicef.prod + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.False)) + (__ : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + A)) + (cupicef.prod cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop)) + cupicef.I A + (__ : cupicef.Term cupicef.prop A => + cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + (__0 : + (__0 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.False))) + (H1 : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.False__ind A (let__H69 A H H0 H1)) + (H1 : cupicef.Term cupicef.prop A => + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H71 A H H0 H1 H2))) + H). + +def let__H72 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop A -> + H1 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H3 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop A => + H1 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 : cupicef.Term cupicef.prop B => + H2 H0. + +def let__H73 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop A -> + H1 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H3 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop A + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop A => + H1 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 : cupicef.Term cupicef.prop B => + H1 H3. + +def let__H74 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + __0 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H2 : cupicef.Term cupicef.prop B -> + __ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + __0 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 : cupicef.Term cupicef.prop B => + H1 H0. + +def let__H75 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + __0 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H2 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + __0 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 : cupicef.Term cupicef.prop B => + let__H74 A B H H0 H1 H2 H2. + +def let__H76 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H3 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop A + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 : cupicef.Term cupicef.prop B => + H1 H3. + +def let__H77 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H3 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 : cupicef.Term cupicef.prop B => + H0 (let__H76 A B H H0 H1 H2 H3). + +def let__H78 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H3 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 : cupicef.Term cupicef.prop B => + H2 (let__H76 A B H H0 H1 H2 H3). + +thm contrapositive : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (decidable A) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + (__0 : + (__0 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__1 : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => A))) + +:= cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + Coq__Init__Logic.iff + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)))))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (decidable A) + (__ : cupicef.Term cupicef.prop (decidable A) => + Coq__Init__Logic.iff + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + (__0 : + (__0 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__1 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => A)))))) + cupicef.I + (A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + Coq__Init__Logic.or__ind A (Coq__Init__Logic.not A) + (Coq__Init__Logic.iff + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A))) + (H0 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.conj + (cupicef.prod + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) + (__ : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + __0 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => A))) + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)) + (__ : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop A) => + cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + (__0 : + (__0 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__1 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False)))) + (H1 : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + __0 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 : cupicef.Term cupicef.prop B => + H0) + (H1 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop A) => + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H72 A B H H0 H1 H2 H3))) + (H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + Coq__Init__Logic.conj + (cupicef.prod + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) + (__ : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + __0 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => A))) + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)) + (__ : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop A) => + cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + (__0 : + (__0 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__1 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False)))) + (H1 : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + __0 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False__ind A (let__H75 A B H H0 H1 H2)) + (H1 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop A) => + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H78 A B H H0 H1 H2 H3))) + H). + +def let__H79 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop A -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + B) -> + H2 : cupicef.Term cupicef.prop A -> + H3 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop A => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + B) => + H2 : cupicef.Term cupicef.prop A => + H3 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 H0. + +def let__H80 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop A -> + H1 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop A => + H1 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H1 H0. + +def let__H81 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + B) -> + H2 : cupicef.Term cupicef.prop A -> + H3 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + B) => + H2 : cupicef.Term cupicef.prop A => + H3 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H0 H2. + +def let__H82 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + B) -> + H2 : cupicef.Term cupicef.prop A -> + H3 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + B) => + H2 : cupicef.Term cupicef.prop A => + H3 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 H2. + +def let__H83 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + B) -> + H2 : cupicef.Term cupicef.prop A -> + H3 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + B) => + H2 : cupicef.Term cupicef.prop A => + H3 : cupicef.Term cupicef.prop B => + H0 H2. + +def let__H84 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + H2 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H2 : cupicef.Term cupicef.prop A => + H0 H2. + +def let__H85 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + H2 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H2 : cupicef.Term cupicef.prop A => + H1 H2. + +thm or__not__l__iff__1 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (decidable A) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + B) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B))) + +:= cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + Coq__Init__Logic.iff + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + B) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)))))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (decidable A) + (__ : cupicef.Term cupicef.prop (decidable A) => + Coq__Init__Logic.iff + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + B) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B)))))) + cupicef.I + (A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + Coq__Init__Logic.or__ind A (Coq__Init__Logic.not A) + (Coq__Init__Logic.iff + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + B) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) + (H0 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + B) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + B) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B))) + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + B)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + B) => + H2 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.or__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + B B + (H3 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.False__ind B (let__H79 A B H H0 H1 H2 H3)) + (H3 : cupicef.Term cupicef.prop B => H3) H1) + (H1 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.or__intror + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + B (let__H80 A B H H0 H1))) + (H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + B) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + B) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B))) + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + B)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + B) => + H2 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.or__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + B B + (H3 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.False__ind B (let__H82 A B H H0 H1 H2 H3)) + (H3 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False__ind B (let__H83 A B H H0 H1 H2 H3)) + H1) + (H1 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.or__introl + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + B + (H2 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H84 A B H H0 H1 H2)))) + H). + +def let__H86 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H0 : cupicef.Term cupicef.prop B -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + B) -> + H2 : cupicef.Term cupicef.prop A -> + H3 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H0 : cupicef.Term cupicef.prop B => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + B) => + H2 : cupicef.Term cupicef.prop A => + H3 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 H2. + +def let__H87 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + B) -> + H2 : cupicef.Term cupicef.prop A -> + H3 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + B) => + H2 : cupicef.Term cupicef.prop A => + H3 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 H2. + +def let__H88 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + B) -> + H2 : cupicef.Term cupicef.prop A -> + H3 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + B) => + H2 : cupicef.Term cupicef.prop A => + H3 : cupicef.Term cupicef.prop B => + H0 H3. + +def let__H89 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H1 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + H2 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H1 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H2 : cupicef.Term cupicef.prop A => + H1 H2. + +def let__H90 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H1 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + H2 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H1 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H2 : cupicef.Term cupicef.prop A => + H0 (let__H89 A B H H0 H1 H2). + +thm or__not__l__iff__2 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (decidable B) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + B) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B))) + +:= cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + Coq__Init__Logic.iff + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + B) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)))))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (decidable B) + (__ : cupicef.Term cupicef.prop (decidable B) => + Coq__Init__Logic.iff + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + B) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B)))))) + cupicef.I + (A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + Coq__Init__Logic.or__ind B (Coq__Init__Logic.not B) + (Coq__Init__Logic.iff + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + B) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B))) + (H0 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + B) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + B) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B))) + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + B)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + B) => + H2 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.or__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + B B + (H3 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.False__ind B (let__H86 A B H H0 H1 H2 H3)) + (H3 : cupicef.Term cupicef.prop B => H3) H1) + (H1 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.or__intror + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + B H0)) + (H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + B) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + B) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B))) + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + B)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + B) => + H2 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.or__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + B B + (H3 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.False__ind B (let__H87 A B H H0 H1 H2 H3)) + (H3 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False__ind B (let__H88 A B H H0 H1 H2 H3)) + H1) + (H1 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.or__introl + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + B + (H2 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H90 A B H H0 H1 H2)))) + H). + +def let__H91 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop A -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) -> + H2 : cupicef.Term cupicef.prop B -> + H3 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop A => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) => + H2 : cupicef.Term cupicef.prop B => + H3 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 H2. + +def let__H92 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) -> + H2 : cupicef.Term cupicef.prop B -> + H3 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) => + H2 : cupicef.Term cupicef.prop B => + H3 : cupicef.Term cupicef.prop A => + H0 H3. + +def let__H93 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) -> + H2 : cupicef.Term cupicef.prop B -> + H3 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) => + H2 : cupicef.Term cupicef.prop B => + H3 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 H2. + +def let__H94 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + H2 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop A + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + H2 : cupicef.Term cupicef.prop B => + H1 H2. + +def let__H95 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + H2 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + H2 : cupicef.Term cupicef.prop B => + H0 (let__H94 A B H H0 H1 H2). + +thm or__not__r__iff__1 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (decidable A) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => A))) + +:= cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + Coq__Init__Logic.iff + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)))))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (decidable A) + (__ : cupicef.Term cupicef.prop (decidable A) => + Coq__Init__Logic.iff + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => A)))))) + cupicef.I + (A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + Coq__Init__Logic.or__ind A (Coq__Init__Logic.not A) + (Coq__Init__Logic.iff + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A))) + (H0 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => A))) + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)) + (__ : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop A) => + Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False)))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) => + H2 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.or__ind A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + A (H3 : cupicef.Term cupicef.prop A => H3) + (H3 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.False__ind A (let__H91 A B H H0 H1 H2 H3)) + H1) + (H1 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop A) => + Coq__Init__Logic.or__introl A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + H0)) + (H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => A))) + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)) + (__ : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop A) => + Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False)))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) => + H2 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.or__ind A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + A + (H3 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind A (let__H92 A B H H0 H1 H2 H3)) + (H3 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.False__ind A (let__H93 A B H H0 H1 H2 H3)) + H1) + (H1 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop A) => + Coq__Init__Logic.or__intror A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + (H2 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H95 A B H H0 H1 H2)))) + H). + +def let__H96 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H0 : cupicef.Term cupicef.prop B -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) -> + H2 : cupicef.Term cupicef.prop B -> + H3 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H0 : cupicef.Term cupicef.prop B => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) => + H2 : cupicef.Term cupicef.prop B => + H3 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 H0. + +def let__H97 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H0 : cupicef.Term cupicef.prop B -> + H1 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + cupicef.Term cupicef.prop A + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H0 : cupicef.Term cupicef.prop B => + H1 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + H1 H0. + +def let__H98 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) -> + H2 : cupicef.Term cupicef.prop B -> + H3 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) => + H2 : cupicef.Term cupicef.prop B => + H3 : cupicef.Term cupicef.prop A => + H0 H2. + +def let__H99 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) -> + H2 : cupicef.Term cupicef.prop B -> + H3 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) => + H2 : cupicef.Term cupicef.prop B => + H3 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H0 H2. + +def let__H100 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) -> + H2 : cupicef.Term cupicef.prop B -> + H3 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) => + H2 : cupicef.Term cupicef.prop B => + H3 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 H2. + +def let__H101 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H1 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + H2 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H1 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + H2 : cupicef.Term cupicef.prop B => + H0 H2. + +def let__H102 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) -> + H1 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) -> + H2 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop A + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + H1 : (__ : cupicef.Term cupicef.prop B -> cupicef.Term cupicef.prop A) => + H2 : cupicef.Term cupicef.prop B => + H1 H2. + +thm or__not__r__iff__2 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (decidable B) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => A))) + +:= cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + Coq__Init__Logic.iff + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)))))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (decidable B) + (__ : cupicef.Term cupicef.prop (decidable B) => + Coq__Init__Logic.iff + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => A)))))) + cupicef.I + (A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or B (Coq__Init__Logic.not B)) => + Coq__Init__Logic.or__ind B (Coq__Init__Logic.not B) + (Coq__Init__Logic.iff + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A))) + (H0 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => A))) + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)) + (__ : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop A) => + Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False)))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) => + H2 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.or__ind A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + A (H3 : cupicef.Term cupicef.prop A => H3) + (H3 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.False__ind A (let__H96 A B H H0 H1 H2 H3)) + H1) + (H1 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop A) => + Coq__Init__Logic.or__introl A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + (let__H97 A B H H0 H1))) + (H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not B) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => A))) + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => A)) + (__ : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop A) => + Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False)))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) => + H2 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.or__ind A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + A + (H3 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind A (let__H98 A B H H0 H1 H2 H3)) + (H3 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.False__ind A (let__H100 A B H H0 H1 H2 H3)) + H1) + (H1 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop A) => + Coq__Init__Logic.or__intror A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + (H2 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H101 A B H H0 H1 H2)))) + H). + +def let__H103 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop A -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) -> + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H3 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop A => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) => + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 : cupicef.Term cupicef.prop A => + H2 H0. + +def let__H104 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop A -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) -> + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H3 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop A => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) => + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 : cupicef.Term cupicef.prop B => + H2 H0. + +def let__H105 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop B) => + H1 H0. + +def let__H106 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) -> + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H3 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) => + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 : cupicef.Term cupicef.prop A => + H0 H3. + +def let__H107 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) -> + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H3 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) => + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 : cupicef.Term cupicef.prop A => + H2 H3. + +thm imp__not__l : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (decidable A) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + (__0 : + (__0 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + B)) + (Coq__Init__Logic.or A B)) + +:= cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + Coq__Init__Logic.iff + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + B)) + (Coq__Init__Logic.or A B))))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (decidable A) + (__ : cupicef.Term cupicef.prop (decidable A) => + Coq__Init__Logic.iff + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + (__0 : + (__0 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + B)) + (Coq__Init__Logic.or A B))))) + cupicef.I + (A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + Coq__Init__Logic.or__ind A (Coq__Init__Logic.not A) + (Coq__Init__Logic.iff + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + B)) + (Coq__Init__Logic.or A B)) + (H0 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.conj + (cupicef.prod + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + B)) + (__ : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.or A B)) + (cupicef.prod cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop)) + cupicef.I (Coq__Init__Logic.or A B) + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) => + cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + (__0 : + (__0 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + B))) + (H1 : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.or__introl A B H0) + (H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) => + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.or__ind A B B + (H3 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind B (let__H103 A B H H0 H1 H2 H3)) + (H3 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False__ind B (let__H104 A B H H0 H1 H2 H3)) + H1)) + (H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + Coq__Init__Logic.conj + (cupicef.prod + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + B)) + (__ : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.or A B)) + (cupicef.prod cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop)) + cupicef.I (Coq__Init__Logic.or A B) + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) => + cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + (__0 : + (__0 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + B))) + (H1 : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.or__intror A B (let__H105 A B H H0 H1)) + (H1 : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) => + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.or__ind A B B + (H3 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind B (let__H107 A B H H0 H1 H2 H3)) + (H3 : cupicef.Term cupicef.prop B => H3) H1)) + H). + +def let__H108 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + __ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H0 : cupicef.Term cupicef.prop A => + H (Coq__Init__Logic.or__introl A B H0). + +def let__H109 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + __ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H1 : cupicef.Term cupicef.prop B => + H (Coq__Init__Logic.or__intror A B H1). + +def let__H110 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H2 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 : cupicef.Term cupicef.prop A => + let__H108 A B H H2. + +def let__H111 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H2 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 : cupicef.Term cupicef.prop B => + let__H109 A B H H2. + +def let__H112 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) -> + H1 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H2 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H3 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) => + H1 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 : cupicef.Term cupicef.prop A => + H1 H3. + +def let__H113 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) -> + H1 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H2 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H3 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) => + H1 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 : cupicef.Term cupicef.prop B => + H2 H3. + +thm not__or__iff : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or A B) + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)))) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + Coq__Init__Logic.conj + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or A B) + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) => + Coq__Init__Logic.False)) + (__ : + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)))) + (cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or A B) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) => + Coq__Init__Logic.False))) + (H : + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + (H2 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H110 A B H H2)) + (H2 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H111 A B H H2))) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.or A B) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => Coq__Init__Logic.False)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + Coq__Init__Logic.False + (H1 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.or__ind A B Coq__Init__Logic.False + (H3 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H112 A B H H0 H1 H2 H3)) + (H3 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H113 A B H H0 H1 H2 H3)) + H0) + H). + +def let__H114 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H0 : cupicef.Term cupicef.prop A -> + H1 : cupicef.Term cupicef.prop B -> + __ : cupicef.Term cupicef.prop A -> + __0 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H0 : cupicef.Term cupicef.prop A => + H1 : cupicef.Term cupicef.prop B => + H2 : cupicef.Term cupicef.prop A => + H3 : cupicef.Term cupicef.prop B => + H (Coq__Init__Logic.conj A B H2 H3). + +def let__H115 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H0 : cupicef.Term cupicef.prop A -> + H1 : cupicef.Term cupicef.prop B -> + __ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H0 : cupicef.Term cupicef.prop A => + H1 : cupicef.Term cupicef.prop B => + let__H114 A B H H0 H1 H0. + +def let__H116 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H0 : cupicef.Term cupicef.prop A -> + H1 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H0 : cupicef.Term cupicef.prop A => + H1 : cupicef.Term cupicef.prop B => + let__H115 A B H H0 H1 H1. + +def let__H117 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + (__ : cupicef.Term cupicef.prop A -> + __0 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) -> + H1 : cupicef.Term cupicef.prop A -> + H2 : cupicef.Term cupicef.prop B -> + __ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + (__ : cupicef.Term cupicef.prop A -> + __0 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) => + H1 : cupicef.Term cupicef.prop A => + H2 : cupicef.Term cupicef.prop B => + H H1. + +def let__H118 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + (__ : cupicef.Term cupicef.prop A -> + __0 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) -> + H1 : cupicef.Term cupicef.prop A -> + H2 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + (__ : cupicef.Term cupicef.prop A -> + __0 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) => + H1 : cupicef.Term cupicef.prop A => + H2 : cupicef.Term cupicef.prop B => + let__H117 A B H H0 H1 H2 H2. + +thm not__and__iff : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and A B) + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) => + Coq__Init__Logic.False)) + (cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (__ : cupicef.Term cupicef.prop A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)))) + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + Coq__Init__Logic.conj + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and A B) + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) => + Coq__Init__Logic.False)) + (__ : + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (__0 : cupicef.Term cupicef.prop A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__1 : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)))) + (cupicef.prod + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (__ : cupicef.Term cupicef.prop A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) + (__ : + (__ : cupicef.Term cupicef.prop A -> + __0 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.and A B) + (__0 : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) => + Coq__Init__Logic.False))) + (H : + (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H0 : cupicef.Term cupicef.prop A => + H1 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H116 A B H H0 H1)) + (H : + (__ : cupicef.Term cupicef.prop A -> + __0 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.and A B) => + Coq__Init__Logic.and__ind A B Coq__Init__Logic.False + (H1 : cupicef.Term cupicef.prop A => + H2 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H118 A B H H0 H1 H2)) + H0). + +def let__H121 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop A -> + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H2 : cupicef.Term cupicef.prop B -> + H3 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H4 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop A => + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 : cupicef.Term cupicef.prop B => + H3 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop A => + H3 H2. + +def let__H120 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop A -> + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H2 : cupicef.Term cupicef.prop B -> + __ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop A => + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 : cupicef.Term cupicef.prop B => + (H3 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind B (let__H121 A B H H0 H1 H2 H3 H4)) + (y : cupicef.Term cupicef.prop B => + H1 (x : cupicef.Term cupicef.prop A => y)). + +def let__H119 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop A -> + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H2 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop A => + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 : cupicef.Term cupicef.prop B => + H1 (let__H120 A B H H0 H1 H2). + +def let__H122 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop A -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) -> + H2 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + H3 : cupicef.Term cupicef.prop A -> + H4 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop A => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) => + H2 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H3 : cupicef.Term cupicef.prop A => + H4 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 H0. + +def let__H123 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop A -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) -> + H2 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + H3 : cupicef.Term cupicef.prop A -> + H4 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop A => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) => + H2 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H3 : cupicef.Term cupicef.prop A => + H4 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 (let__H122 A B H H0 H1 H2 H3 H4). + +def let__H126 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H2 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H3 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 : cupicef.Term cupicef.prop A => + H0 H3. + +def let__H125 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + __ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + (H2 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind B (let__H126 A B H H0 H1 H2 H3)) + (y : cupicef.Term cupicef.prop B => + H1 (x : cupicef.Term cupicef.prop A => y)). + +def let__H124 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H1 (let__H125 A B H H0 H1). + +def let__H129 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H2 : cupicef.Term cupicef.prop B -> + H3 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H4 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 : cupicef.Term cupicef.prop B => + H3 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop A => + H3 H2. + +def let__H130 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H2 : cupicef.Term cupicef.prop B -> + H3 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H4 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 : cupicef.Term cupicef.prop B => + H3 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop A => + H0 H4. + +def let__H128 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H2 : cupicef.Term cupicef.prop B -> + __ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 : cupicef.Term cupicef.prop B => + (H3 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind B (let__H130 A B H H0 H1 H2 H3 H4)) + (y : cupicef.Term cupicef.prop B => + H1 (x : cupicef.Term cupicef.prop A => y)). + +def let__H127 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H2 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 : cupicef.Term cupicef.prop B => + H1 (let__H128 A B H H0 H1 H2). + +def let__H131 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) -> + H2 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + H3 : cupicef.Term cupicef.prop A -> + H4 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) => + H2 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H3 : cupicef.Term cupicef.prop A => + H4 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H0 H3. + +def let__H132 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) -> + H2 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + H3 : cupicef.Term cupicef.prop A -> + H4 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) => + H2 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H3 : cupicef.Term cupicef.prop A => + H4 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 H3. + +def let__H133 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) -> + H2 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + H3 : cupicef.Term cupicef.prop A -> + H4 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False))) => + H2 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H3 : cupicef.Term cupicef.prop A => + H4 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 (let__H132 A B H H0 H1 H2 H3 H4). + +thm not__imp__iff : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (decidable A) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B)) + (__0 : + (__0 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.and A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)))) + +:= cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + Coq__Init__Logic.iff + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.and A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))))))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (decidable A) + (__ : cupicef.Term cupicef.prop (decidable A) => + Coq__Init__Logic.iff + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B)) + (__0 : + (__0 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.and A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))))))) + cupicef.I + (A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + Coq__Init__Logic.or__ind A (Coq__Init__Logic.not A) + (Coq__Init__Logic.iff + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.and A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)))) + (H0 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.conj + (cupicef.prod + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.False)) + (__ : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.and A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False)))) + (cupicef.prod cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop)) + cupicef.I + (Coq__Init__Logic.and A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) => + cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B)) + (__0 : + (__0 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.False))) + (H1 : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.conj A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + H0 + (H2 : cupicef.Term cupicef.prop B => + (H3 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind Coq__Init__Logic.False H3) + (let__H119 A B H H0 H1 H2))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) => + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.and__ind A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + Coq__Init__Logic.False + (H3 : cupicef.Term cupicef.prop A => + H4 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H123 A B H H0 H1 H2 H3 H4)) + H1)) + (H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + Coq__Init__Logic.conj + (cupicef.prod + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.False)) + (__ : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.and A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False)))) + (cupicef.prod cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop)) + cupicef.I + (Coq__Init__Logic.and A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) => + cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B)) + (__0 : + (__0 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.False))) + (H1 : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.conj A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + ((H2 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind A H2) (let__H124 A B H H0 H1)) + (H2 : cupicef.Term cupicef.prop B => + (H3 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind Coq__Init__Logic.False H3) + (let__H127 A B H H0 H1 H2))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False))) => + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.and__ind A + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + Coq__Init__Logic.False + (H3 : cupicef.Term cupicef.prop A => + H4 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H133 A B H H0 H1 H2 H3 H4)) + H1)) + H). + +def let__H136 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop A -> + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H2 : cupicef.Term cupicef.prop B -> + H3 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H4 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop A => + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 : cupicef.Term cupicef.prop B => + H3 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop A => + H3 H2. + +def let__H135 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop A -> + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H2 : cupicef.Term cupicef.prop B -> + __ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop A => + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 : cupicef.Term cupicef.prop B => + (H3 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind B (let__H136 A B H H0 H1 H2 H3 H4)) + (y : cupicef.Term cupicef.prop B => + H1 (x : cupicef.Term cupicef.prop A => y)). + +def let__H134 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop A -> + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H2 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop A => + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 : cupicef.Term cupicef.prop B => + H1 (let__H135 A B H H0 H1 H2). + +def let__H137 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop A -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + A) -> + H2 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + H3 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H4 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop A => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + A) => + H2 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H3 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop A => + H2 H0. + +def let__H138 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop A -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + A) -> + H2 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + H3 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H4 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop A => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + A) => + H2 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H3 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop A => + H3 (let__H137 A B H H0 H1 H2 H3 H4). + +def let__H141 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H2 : cupicef.Term cupicef.prop B -> + H3 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H4 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 : cupicef.Term cupicef.prop B => + H3 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop A => + H3 H2. + +def let__H142 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H2 : cupicef.Term cupicef.prop B -> + H3 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H4 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 : cupicef.Term cupicef.prop B => + H3 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop A => + H0 H4. + +def let__H140 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H2 : cupicef.Term cupicef.prop B -> + __ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 : cupicef.Term cupicef.prop B => + (H3 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind B (let__H142 A B H H0 H1 H2 H3 H4)) + (y : cupicef.Term cupicef.prop B => + H1 (x : cupicef.Term cupicef.prop A => y)). + +def let__H139 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H2 : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 : cupicef.Term cupicef.prop B => + H1 (let__H140 A B H H0 H1 H2). + +def let__H145 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H2 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H3 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H2 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 : cupicef.Term cupicef.prop A => + H0 H3. + +def let__H144 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + __ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + (H2 : + (y : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H3 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind B (let__H145 A B H H0 H1 H2 H3)) + (y : cupicef.Term cupicef.prop B => + H1 (x : cupicef.Term cupicef.prop A => y)). + +def let__H143 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + (__ : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H1 (let__H144 A B H H0 H1). + +def let__H146 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + A) -> + H2 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + H3 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H4 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + A) => + H2 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H3 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop A => + H0 H4. + +def let__H147 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + A) -> + H2 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + H3 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H4 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + A) => + H2 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H3 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop A => + H2 H4. + +def let__H148 : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) -> + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) -> + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + A) -> + H2 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) -> + H3 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) -> + H4 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop Coq__Init__Logic.False + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + A) => + H2 : (__ : cupicef.Term cupicef.prop A -> cupicef.Term cupicef.prop B) => + H3 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop A => + H3 (let__H147 A B H H0 H1 H2 H3 H4). + +thm not__imp__rev__iff : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + __ : cupicef.Term cupicef.prop (decidable A) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B)) + (__0 : + (__0 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + A)) + +:= cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + Coq__Init__Logic.iff + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False)) + A))))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (A : cupicef.Univ cupicef.prop => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (decidable A) + (__ : cupicef.Term cupicef.prop (decidable A) => + Coq__Init__Logic.iff + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B)) + (__0 : + (__0 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False)) + A))))) + cupicef.I + (A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or A (Coq__Init__Logic.not A)) => + Coq__Init__Logic.or__ind A (Coq__Init__Logic.not A) + (Coq__Init__Logic.iff + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + A)) + (H0 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.conj + (cupicef.prod + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.False)) + (__ : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False)) + A)) + (cupicef.prod cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop)) + cupicef.I + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False)) + A) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False)) + A) => + cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B)) + (__0 : + (__0 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.False))) + (H1 : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + A + (H2 : cupicef.Term cupicef.prop B => + (H3 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind Coq__Init__Logic.False H3) + (let__H134 A B H H0 H1 H2)) + H0) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False)) + A) => + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + A Coq__Init__Logic.False + (H3 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H138 A B H H0 H1 H2 H3 H4)) + H1)) + (H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.not A) => + Coq__Init__Logic.conj + (cupicef.prod + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B)) + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.False)) + (__ : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__0 : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False)) + A)) + (cupicef.prod cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop)) + cupicef.I + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False)) + A) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False)) + A) => + cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__0 : cupicef.Term cupicef.prop A => B)) + (__0 : + (__0 : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.False))) + (H1 : + (__ : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + A + (H2 : cupicef.Term cupicef.prop B => + (H3 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind Coq__Init__Logic.False H3) + (let__H139 A B H H0 H1 H2)) + ((H2 : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.False__ind A H2) (let__H143 A B H H0 H1))) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => + Coq__Init__Logic.False)) + A) => + H2 : + (__ : cupicef.Term cupicef.prop A -> + cupicef.Term cupicef.prop B) => + Coq__Init__Logic.and__ind + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I B + (__ : cupicef.Term cupicef.prop B => Coq__Init__Logic.False)) + A Coq__Init__Logic.False + (H3 : + (__ : cupicef.Term cupicef.prop B -> + cupicef.Term cupicef.prop Coq__Init__Logic.False) => + H4 : cupicef.Term cupicef.prop A => + Coq__Init__Logic.False__ind Coq__Init__Logic.False + (let__H148 A B H H0 H1 H2 H3 H4)) + H1)) + H). + +def let__e : + X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + Y : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + A : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y -> + cupicef.Univ cupicef.prop) -> + Hdec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y -> + cupicef.Term cupicef.prop + (decidable + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Y) + y y_27_))) -> + H : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex Y + (Coq__Init__Logic.unique Y + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y => + A x y)))) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex Y + (Coq__Init__Logic.unique Y + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y => A x y0))) + +:= X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + Y : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + A : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y -> + cupicef.Univ cupicef.prop) => + Hdec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y -> + cupicef.Term cupicef.prop + (decidable + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Y) + y y_27_))) => + H : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex Y + (Coq__Init__Logic.unique Y + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y => + A x y)))) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y => + H x. + +def let__d : + X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + Y : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + A : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y -> + cupicef.Univ cupicef.prop) -> + Hdec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y -> + cupicef.Term cupicef.prop + (decidable + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Y) + y y_27_))) -> + H : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex Y + (Coq__Init__Logic.unique Y + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y => + A x y)))) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y -> + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.unique Y + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y => A x y0) + y_27_) -> + Hex : cupicef.Term cupicef.prop (A x y_27_) -> + Huniq : + (x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y -> + __ : cupicef.Term cupicef.prop (A x x_27_) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Y) + y_27_ x_27_)) -> + cupicef.Term cupicef.prop + (decidable + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Y) + y y_27_)) + +:= X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + Y : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + A : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y -> + cupicef.Univ cupicef.prop) => + Hdec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y -> + cupicef.Term cupicef.prop + (decidable + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Y) + y y_27_))) => + H : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex Y + (Coq__Init__Logic.unique Y + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y => + A x y)))) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y => + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.unique Y + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y => A x y0) + y_27_) => + Hex : cupicef.Term cupicef.prop (A x y_27_) => + Huniq : + (x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y -> + __ : cupicef.Term cupicef.prop (A x x_27_) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Y) + y_27_ x_27_)) => + Hdec y y_27_. + +thm dec__functional__relation : + X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + Y : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + A : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y -> + cupicef.Univ cupicef.prop) -> + __ : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y -> + cupicef.Term cupicef.prop + (decidable + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Y) + y y_27_))) -> + __0 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex Y + (Coq__Init__Logic.unique Y + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y => + A x y)))) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y -> + cupicef.Term cupicef.prop (decidable (A x y)) + +:= X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + Y : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + A : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y -> + cupicef.Univ cupicef.prop) => + Hdec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y -> + cupicef.Term cupicef.prop + (decidable + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Y) + y y_27_))) => + H : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex Y + (Coq__Init__Logic.unique Y + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y => + A x y)))) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y => + Coq__Init__Logic.match____ex cupicef.prop Y + (Coq__Init__Logic.unique Y + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y => A x y0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.ex Y + (Coq__Init__Logic.unique Y + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y => + A x y0))) => + decidable (A x y)) + (y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.unique Y + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y => + A x y0) + y_27_) => + Coq__Init__Logic.match____and cupicef.prop + ((y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y => A x y0) + y_27_) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Y + (x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + ((y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y => + A x y0) x_27_) + (__ : + cupicef.Term cupicef.prop + ((y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y => + A x y0) x_27_) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Y) + y_27_ x_27_))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and (A x y_27_) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I Y + (x_27_ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (A x x_27_) + (__ : cupicef.Term cupicef.prop (A x x_27_) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Y) + y_27_ x_27_)))) => + decidable (A x y)) + (Hex : cupicef.Term cupicef.prop (A x y_27_) => + Huniq : + (x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y -> + __ : cupicef.Term cupicef.prop (A x x_27_) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Y) + y_27_ x_27_)) => + Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Y) + y y_27_) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Y) + y y_27_)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Y) + y y_27_) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Y) + y y_27_))) => + decidable (A x y)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Y) + y y_27_) => + Coq__Init__Logic.eq__ind__r Y y_27_ + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y => + decidable (A x y0)) + (Coq__Init__Logic.or__introl (A x y_27_) + (Coq__Init__Logic.not (A x y_27_)) Hex) + y H1) + (Hnot : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Y) + y y_27_)) => + Coq__Init__Logic.or__intror (A x y) + (Coq__Init__Logic.not (A x y)) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (A x y) + (H1 : cupicef.Term cupicef.prop (A x y) => + Coq__Init__Logic.False)) + (Coq__Init__Logic.not (A x y)) cupicef.I + (H1 : cupicef.Term cupicef.prop (A x y) => + (H2 : + (__ : cupicef.Term cupicef.prop (A x y_27_) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Y) + y_27_ y_27_)) => + (H3 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Y) + y_27_ y_27_) => + (H4 : + (y_27_0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) Y -> + cupicef.Term cupicef.prop + (decidable + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Y) + y_27_ y_27_0))) => + (H5 : + (__ : cupicef.Term cupicef.prop (A x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Y) + y_27_ y)) => + (H6 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Y) + y_27_ y) => + Hnot + (Coq__Init__Logic.eq__sym + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Y) + y_27_ y H6)) + (H5 H1)) + (Huniq y)) + (Hdec y_27_)) + (H2 Hex)) + (Huniq y_27_)))) + (let__d X Y A Hdec H x y y_27_ H0 Hex Huniq)) + H0) + (let__e X Y A Hdec H x y). + +(; End of translation. ;) diff --git a/out/Coq__Logic__EqdepFacts.dk b/out/Coq__Logic__EqdepFacts.dk new file mode 100644 index 000000000..ae6136ea7 --- /dev/null +++ b/out/Coq__Logic__EqdepFacts.dk @@ -0,0 +1,12316 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def eq__dep : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + cupicef.Univ cupicef.prop. + +def eq__dep__intro : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + cupicef.Term cupicef.prop (eq__dep U P p x p x). + +[ ] cupicef.code_guarded? _ (eq__dep__intro _ _ _ _) --> cupicef.guarded. + +def match____eq__dep : + s : cupicef.Sort -> + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + P0 : + (q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + cupicef.Term cupicef.prop (eq__dep U P p x q __) -> + cupicef.Univ s) -> + case__eq__dep__intro : cupicef.Term s (P0 p x (eq__dep__intro U P p x)) -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + x0 : cupicef.Term cupicef.prop (eq__dep U P p x q __) -> + cupicef.Term s (P0 q __ x0). + +[ s, U, P, p, x, P0, case__eq__dep__intro] + match____eq__dep s U P p x P0 case__eq__dep__intro {p} {x} + (eq__dep__intro U P p x) + --> + case__eq__dep__intro. + +[ s, U, P, p, x, P0, s'] + match____eq__dep s U P p x + (q => + __ => + x => + cupicef.uncode _ (cupicef.code (cupicef.cU s') (P0 q __ x))) + --> + match____eq__dep s' U P p x (q => __ => x => P0 q __ x). + +def eq__dep__rect : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + P0 : + (q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P0 p x) -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + p0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + e : cupicef.Term cupicef.prop (eq__dep U P p x q p0) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P0 q p0) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + P0 : + (q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P0 p x) => + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + p0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) => + e : cupicef.Term cupicef.prop (eq__dep U P p x q p0) => + match____eq__dep (cupicef.type (cupicef.s cupicef.z)) U P p x + (q0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + p1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q0) => + __ : cupicef.Term cupicef.prop (eq__dep U P p x q0 p1) => + P0 q0 p1) + f q p0 e. + +def eq__dep__ind : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + P0 : + (q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + cupicef.Univ cupicef.prop) -> + f : cupicef.Term cupicef.prop (P0 p x) -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + p0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + e : cupicef.Term cupicef.prop (eq__dep U P p x q p0) -> + cupicef.Term cupicef.prop (P0 q p0) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + P0 : + (q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + cupicef.Univ cupicef.prop) => + f : cupicef.Term cupicef.prop (P0 p x) => + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + p0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) => + e : cupicef.Term cupicef.prop (eq__dep U P p x q p0) => + match____eq__dep cupicef.prop U P p x + (q0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + p1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q0) => + __ : cupicef.Term cupicef.prop (eq__dep U P p x q0 p1) => + P0 q0 p1) + f q p0 e. + +def eq__dep__rec : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + P0 : + (q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + cupicef.Univ cupicef.set) -> + f : cupicef.Term cupicef.set (P0 p x) -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + p0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + e : cupicef.Term cupicef.prop (eq__dep U P p x q p0) -> + cupicef.Term cupicef.set (P0 q p0) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + P0 : + (q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + cupicef.Univ cupicef.set) => + eq__dep__rect U P p x + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I U + (q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I (P q) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I U + (q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I (P q) + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I))) + cupicef.I P0). + +thm eq__dep__refl : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + cupicef.Term cupicef.prop (eq__dep U P p x p x) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + eq__dep__intro U P. + +thm eq__dep__sym : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + __ : cupicef.Term cupicef.prop (eq__dep U P p x q y) -> + cupicef.Term cupicef.prop (eq__dep U P q y p x) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) => + H : cupicef.Term cupicef.prop (eq__dep U P p x q y) => + match____eq__dep cupicef.prop U P p x + (q0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + p0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q0) => + __ : cupicef.Term cupicef.prop (eq__dep U P p x q0 p0) => + eq__dep U P q0 p0 p x) + (eq__dep__intro U P p x) q y H. + +thm eq__dep__trans : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + r : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P r) -> + __ : cupicef.Term cupicef.prop (eq__dep U P p x q y) -> + __0 : cupicef.Term cupicef.prop (eq__dep U P q y r z) -> + cupicef.Term cupicef.prop (eq__dep U P p x r z) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + r : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P r) => + H : cupicef.Term cupicef.prop (eq__dep U P p x q y) => + match____eq__dep cupicef.prop U P p x + (q0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + p0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q0) => + __ : cupicef.Term cupicef.prop (eq__dep U P p x q0 p0) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eq__dep U P q0 p0 r z) + (__0 : cupicef.Term cupicef.prop (eq__dep U P q0 p0 r z) => + eq__dep U P p x r z)) + (H0 : cupicef.Term cupicef.prop (eq__dep U P p x r z) => H0) q y H. + +def eq__indd : + A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) -> + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + P : + (a : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + e : cupicef.Term cupicef.prop (Coq__Init__Logic.eq A x a) -> + cupicef.Univ cupicef.prop) -> + f : cupicef.Term cupicef.prop (P x (Coq__Init__Logic.eq__refl A x)) -> + y : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + e : cupicef.Term cupicef.prop (Coq__Init__Logic.eq A x y) -> + cupicef.Term cupicef.prop (P y e) + +:= A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) => + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + P : + (a : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + e : cupicef.Term cupicef.prop (Coq__Init__Logic.eq A x a) -> + cupicef.Univ cupicef.prop) => + f : cupicef.Term cupicef.prop (P x (Coq__Init__Logic.eq__refl A x)) => + y : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + e : cupicef.Term cupicef.prop (Coq__Init__Logic.eq A x y) => + Coq__Init__Logic.match____eq cupicef.prop A x + (y0 : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + e0 : cupicef.Term cupicef.prop (Coq__Init__Logic.eq A x y0) => + P y0 e0) + f y e. + +def eq__dep1 : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + cupicef.Univ cupicef.prop. + +def eq__dep1__intro : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + h : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + q p) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + q P y p h)) -> + cupicef.Term cupicef.prop (eq__dep1 U P p x q y). + +[ ] cupicef.code_guarded? _ (eq__dep1__intro _ _ _ _ _ _ _ _) --> cupicef.guarded. + +def match____eq__dep1 : + s : cupicef.Sort -> + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + P0 : (cupicef.Term cupicef.prop (eq__dep1 U P p x q y) -> cupicef.Univ s) -> + case__eq__dep1__intro : + (h : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + q p) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + q P y p h)) -> + cupicef.Term s (P0 (eq__dep1__intro U P p x q y h __))) -> + x0 : cupicef.Term cupicef.prop (eq__dep1 U P p x q y) -> + cupicef.Term s (P0 x0). + +[ s, U, P, p, x, q, y, P0, case__eq__dep1__intro, h, __] + match____eq__dep1 s U P p x q y P0 case__eq__dep1__intro + (eq__dep1__intro U P p x q y h __) + --> + case__eq__dep1__intro h __. + +[ s, U, P, p, x, q, y, P0, s'] + match____eq__dep1 s U P p x q y + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P0 x))) + --> + match____eq__dep1 s' U P p x q y (x => P0 x). + +def eq__dep1__rect : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + P0 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (h : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + q p) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + q P y p h)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) P0) -> + e : cupicef.Term cupicef.prop (eq__dep1 U P p x q y) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) P0 + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) => + P0 : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (h : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + q p) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + q P y p h)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) P0) => + e : cupicef.Term cupicef.prop (eq__dep1 U P p x q y) => + match____eq__dep1 (cupicef.type (cupicef.s cupicef.z)) U P p x q y + (__ : cupicef.Term cupicef.prop (eq__dep1 U P p x q y) => P0) f e. + +def eq__dep1__ind : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + P0 : cupicef.Univ cupicef.prop -> + f : + (h : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + q p) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + q P y p h)) -> + cupicef.Term cupicef.prop P0) -> + e : cupicef.Term cupicef.prop (eq__dep1 U P p x q y) -> + cupicef.Term cupicef.prop P0 + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) => + P0 : cupicef.Univ cupicef.prop => + f : + (h : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + q p) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + q P y p h)) -> + cupicef.Term cupicef.prop P0) => + e : cupicef.Term cupicef.prop (eq__dep1 U P p x q y) => + match____eq__dep1 cupicef.prop U P p x q y + (__ : cupicef.Term cupicef.prop (eq__dep1 U P p x q y) => P0) f e. + +def eq__dep1__rec : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + P0 : cupicef.Univ cupicef.set -> + f : + (h : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + q p) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + q P y p h)) -> + cupicef.Term cupicef.set P0) -> + e : cupicef.Term cupicef.prop (eq__dep1 U P p x q y) -> + cupicef.Term cupicef.set P0 + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) => + P0 : cupicef.Univ cupicef.set => + eq__dep1__rect U P p x q y + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I P0). + +thm eq__dep1__dep : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + __ : cupicef.Term cupicef.prop (eq__dep1 U P p x q y) -> + cupicef.Term cupicef.prop (eq__dep U P p x q y) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) => + H : cupicef.Term cupicef.prop (eq__dep1 U P p x q y) => + match____eq__dep1 cupicef.prop U P p x q y + (__ : cupicef.Term cupicef.prop (eq__dep1 U P p x q y) => + eq__dep U P p x q y) + (eq__qp : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + q p) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + q P y p eq__qp)) => + eq__indd + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + q + (p0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + eq__qp0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + q p0) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (P p0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p0) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p0)) + x0 + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + q P y p0 eq__qp0)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p0)) + x0 + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + q P y p0 eq__qp0)) => + eq__dep U P p0 x0 q y))) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P q)) + x0 + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + q P y q + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + q))) => + Coq__Init__Logic.eq__ind__r (P q) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + q P y q + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + q)) + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) => + eq__dep U P q x1 q y) + (eq__dep__intro U P q y) x0 H1) + p eq__qp x H0) + H. + +thm eq__dep__dep1 : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + __ : cupicef.Term cupicef.prop (eq__dep U P p x q y) -> + cupicef.Term cupicef.prop (eq__dep1 U P p x q y) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) => + H : cupicef.Term cupicef.prop (eq__dep U P p x q y) => + match____eq__dep cupicef.prop U P p x + (q0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + p0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q0) => + __ : cupicef.Term cupicef.prop (eq__dep U P p x q0 p0) => + eq__dep1 U P p x q0 p0) + (eq__dep1__intro U P p x p x + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + p) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x x) + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + p P x p + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + p))) + cupicef.I + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x))) + q y H. + +def let__H0 : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.existT U P p x) + (Coq__Init__Specif.existT U P q y)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.existT U P p x) (Coq__Init__Specif.existT U P q y)) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.existT U P p x) + (Coq__Init__Specif.existT U P q y)) => + H. + +thm eq__sigT__eq__dep : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.existT U P p x) + (Coq__Init__Specif.existT U P q y)) -> + cupicef.Term cupicef.prop (eq__dep U P p x q y) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.existT U P p x) + (Coq__Init__Specif.existT U P q y)) => + Coq__Init__Logic.eq__ind__r + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.existT U P q y) + (s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) U P) => + eq__dep U P (Coq__Init__Specif.projT1 U P s) + (Coq__Init__Specif.projT2 U P s) q y) + (eq__dep__intro U P q y) (Coq__Init__Specif.existT U P p x) + (let__H0 U P p q x y H). + +thm eq__dep__eq__sigT : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + __ : cupicef.Term cupicef.prop (eq__dep U P p x q y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.existT U P p x) (Coq__Init__Specif.existT U P q y)) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) => + H : cupicef.Term cupicef.prop (eq__dep U P p x q y) => + match____eq__dep cupicef.prop U P p x + (q0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q0) => + __ : cupicef.Term cupicef.prop (eq__dep U P p x q0 y0) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.existT U P p x) + (Coq__Init__Specif.existT U P q0 y0)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.existT U P p x)) + q y H. + +thm eq__sigT__iff__eq__dep : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.existT U P p x) + (Coq__Init__Specif.existT U P q y)) + (eq__dep U P p x q y)) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.existT U P p x) + (Coq__Init__Specif.existT U P q y)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.existT U P p x) + (Coq__Init__Specif.existT U P q y)) => + eq__dep U P p x q y)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eq__dep U P p x q y) + (__ : cupicef.Term cupicef.prop (eq__dep U P p x q y) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.existT U P p x) + (Coq__Init__Specif.existT U P q y))) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.existT U P p x) + (Coq__Init__Specif.existT U P q y)) => + eq__sigT__eq__dep U P p q x y H) + (H : cupicef.Term cupicef.prop (eq__dep U P p x q y) => + eq__dep__eq__sigT U P p q x y H). + +def let__H1 : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ cupicef.prop) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term cupicef.prop (P p) -> + y : cupicef.Term cupicef.prop (P q) -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.exist U P p x) (Coq__Init__Specif.exist U P q y)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.exist U P p x) (Coq__Init__Specif.exist U P q y)) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ cupicef.prop) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + x : cupicef.Term cupicef.prop (P p) => + y : cupicef.Term cupicef.prop (P q) => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.exist U P p x) (Coq__Init__Specif.exist U P q y)) => + H. + +thm eq__sig__eq__dep : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ cupicef.prop) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term cupicef.prop (P p) -> + y : cupicef.Term cupicef.prop (P q) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.exist U P p x) (Coq__Init__Specif.exist U P q y)) -> + cupicef.Term cupicef.prop + (eq__dep U + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I U + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I U + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + p x q y) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ cupicef.prop) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + x : cupicef.Term cupicef.prop (P p) => + y : cupicef.Term cupicef.prop (P q) => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.exist U P p x) (Coq__Init__Specif.exist U P q y)) => + Coq__Init__Logic.eq__ind__r + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) U P) + (Coq__Init__Specif.exist U P q y) + (s : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) U P) => + eq__dep U + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + (Coq__Init__Specif.proj1__sig U P s) + (Coq__Init__Specif.proj2__sig U P s) q y) + (eq__dep__intro U + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + q y) + (Coq__Init__Specif.exist U P p x) (let__H1 U P p q x y H). + +thm eq__dep__eq__sig : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ cupicef.prop) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term cupicef.prop (P p) -> + y : cupicef.Term cupicef.prop (P q) -> + __ : + cupicef.Term cupicef.prop + (eq__dep U + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + p x q y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.exist U P p x) (Coq__Init__Specif.exist U P q y)) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ cupicef.prop) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + x : cupicef.Term cupicef.prop (P p) => + y : cupicef.Term cupicef.prop (P q) => + H : + cupicef.Term cupicef.prop + (eq__dep U + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + p x q y) => + match____eq__dep cupicef.prop U + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P) + p x + (q0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + y0 : cupicef.Term cupicef.prop (P q0) => + __ : + cupicef.Term cupicef.prop + (eq__dep U + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + p x q0 y0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.exist U P p x) (Coq__Init__Specif.exist U P q0 y0)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.exist U P p x)) + q y H. + +thm eq__sig__iff__eq__dep : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ cupicef.prop) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term cupicef.prop (P p) -> + y : cupicef.Term cupicef.prop (P q) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.exist U P p x) (Coq__Init__Specif.exist U P q y)) + (eq__dep U + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + p x q y)) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ cupicef.prop) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + x : cupicef.Term cupicef.prop (P p) => + y : cupicef.Term cupicef.prop (P q) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.exist U P p x) + (Coq__Init__Specif.exist U P q y)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig + (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.exist U P p x) + (Coq__Init__Specif.exist U P q y)) => + eq__dep U + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I U + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I U + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + p x q y)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eq__dep U + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + p x q y) + (__ : + cupicef.Term cupicef.prop + (eq__dep U + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I U + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I U + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + p x q y) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.exist U P p x) + (Coq__Init__Specif.exist U P q y))) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) U + P)) + (Coq__Init__Specif.exist U P p x) + (Coq__Init__Specif.exist U P q y)) => + eq__sig__eq__dep U P p q x y H) + (H : + cupicef.Term cupicef.prop + (eq__dep U + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + p x q y) => + eq__dep__eq__sig U P p q x y H). + +def internal__eq__sym__internal : + A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) -> + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + y : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.eq A x y) -> + cupicef.Term cupicef.prop (Coq__Init__Logic.eq A y x) + +:= A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) => + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + y : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.eq A x y) => + Coq__Init__Logic.match____eq cupicef.prop A x + (y0 : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.eq A x y0) => + Coq__Init__Logic.eq A y0 x) + (Coq__Init__Logic.eq__refl A x) y H. + +def internal__eq__sym__involutive : + A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) -> + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + y : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.eq A x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Logic.eq A x y)) + (internal__eq__sym__internal A y x + (internal__eq__sym__internal A x y H)) + H) + +:= A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) => + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + y : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.eq A x y) => + Coq__Init__Logic.match____eq cupicef.prop A x + (y0 : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.eq A x y0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Logic.eq A x y0)) + (internal__eq__sym__internal A y0 x + (internal__eq__sym__internal A x y0 H0)) + H0) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Logic.eq A x x)) + (Coq__Init__Logic.eq__refl A x)) + y H. + +def internal__eq__rew__r__dep : + A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) -> + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + y : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + P : + (y0 : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.eq A y0 y) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + HC : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P y (Coq__Init__Logic.eq__refl A y)) -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.eq A x y) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x H) + +:= A : cupicef.Univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) => + x : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + y : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + P : + (y0 : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A -> + H : cupicef.Term cupicef.prop (Coq__Init__Logic.eq A y0 y) -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + HC : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (P y (Coq__Init__Logic.eq__refl A y)) => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.eq A x y) => + Coq__Init__Logic.match____eq (cupicef.type (cupicef.s cupicef.z)) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Logic.eq A x y)) + (internal__eq__sym__internal A y x (internal__eq__sym__internal A x y H)) + (H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.eq A x y) => + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Logic.eq A x y)) + (internal__eq__sym__internal A y x + (internal__eq__sym__internal A x y H)) + H0) => + P x H0) + (Coq__Init__Logic.match____eq (cupicef.type (cupicef.s cupicef.z)) A y + (y0 : cupicef.Term (cupicef.type (cupicef.s (cupicef.s cupicef.z))) A => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.eq A y y0) => + P y0 (internal__eq__sym__internal A y y0 H0)) + HC x (internal__eq__sym__internal A x y H)) + H (internal__eq__sym__involutive A x y H). + +def let__s : + X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + H1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x1) -> + H2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x2) -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x1 H1) + (Coq__Init__Specif.existT X P x2 H2)) -> + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P) + +:= X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + H1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x1) => + H2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x2) => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x1 H1) + (Coq__Init__Specif.existT X P x2 H2)) => + Coq__Init__Specif.existT X P x2 H2. + +def eq__sigT__sig__eq : + X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + H1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x1) -> + H2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x2) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x1 H1) + (Coq__Init__Specif.existT X P x2 H2)) + (Coq__Init__Specif.sig cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 x2) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 x2) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x2)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 P H1 x2 H) + H2))) + +:= X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + H1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x1) => + H2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x2) => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x1 H1) + (Coq__Init__Specif.existT X P x2 H2)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x1 H1) + (Coq__Init__Specif.existT X P x2 H2)) => + Coq__Init__Specif.sig cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 x2) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 x2) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x2)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 P H1 x2 H) + H2))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Specif.sig cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 x2) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 x2) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x2)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 P H1 x2 H) + H2)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Specif.sig cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 x2) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 x2) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x2)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 P H1 x2 H) + H2)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x1 H1) + (Coq__Init__Specif.existT X P x2 H2))) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x1 H1) + (Coq__Init__Specif.existT X P x2 H2)) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x1 H1) + (y : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P) => + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x1 H1) y) => + Coq__Init__Specif.sig cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 (Coq__Init__Specif.projT1 X P y)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 (Coq__Init__Specif.projT1 X P y)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P (Coq__Init__Specif.projT1 X P y))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 P H1 (Coq__Init__Specif.projT1 X P y) H0) + (Coq__Init__Specif.projT2 X P y))) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Specif.sig cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 x1) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 x1) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 P H1 x1 H0) + H1)) + (Coq__Init__Specif.sig cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 + (Coq__Init__Specif.projT1 X P + (Coq__Init__Specif.existT X P x1 H1))) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 + (Coq__Init__Specif.projT1 X P + (Coq__Init__Specif.existT X P x1 H1))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (Coq__Init__Specif.projT1 X P + (Coq__Init__Specif.existT X P x1 H1)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 P H1 + (Coq__Init__Specif.projT1 X P + (Coq__Init__Specif.existT X P x1 H1)) + H0) + (Coq__Init__Specif.projT2 X P + (Coq__Init__Specif.existT X P x1 H1)))) + cupicef.I + (Coq__Init__Specif.exist + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 x1)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 x1) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x1)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 P H1 x1 H0) + H1) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x1)) + H1))) + (Coq__Init__Specif.existT X P x2 H2) H) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Specif.sig cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 x2) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 x2) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x2)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 P H1 x2 H) + H2)) => + Coq__Init__Specif.match____sig cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 x2)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 x2) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x2)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 P H1 x2 H0) + H2) + (s : + cupicef.Term cupicef.prop + (Coq__Init__Specif.sig cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 x2) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 x2) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x2)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 P H1 x2 H0) + H2)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x1 H1) + (Coq__Init__Specif.existT X P x2 H2)) + (x : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 x2) => + internal__eq__rew__r__dep + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 x2 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I X + (x3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + cupicef.prod cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x3 x2) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x3 x2) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))))) + cupicef.I X + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + cupicef.prod cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.prop + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + y x2) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + y x2) => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I))) + cupicef.I + (x3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x3 x2) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (P x3) + (H3 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x3) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x2)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x3 P H3 x2 x0) + H2) + (e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x2)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x3 P H3 x2 x0) + H2) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x3 H3) + (Coq__Init__Specif.existT X P x2 H2))))) + (H3 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x2) => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x2)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x2 P H3 x2 + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x2)) + H2) => + Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x2)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x2 P H3 x2 + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x2)) + (H4 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x2) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x2 H3) + (Coq__Init__Specif.existT X P x2 H4)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x2 + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x2 P H3 x2 + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x2)))) + H2 e) + x H1) + H). + +def let__s0 : + X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + H1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x1) -> + H2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x2) -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x1 H1) + (Coq__Init__Specif.existT X P x2 H2)) -> + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P) + +:= X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + H1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x1) => + H2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x2) => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x1 H1) + (Coq__Init__Specif.existT X P x2 H2)) => + Coq__Init__Specif.existT X P x2 H2. + +def eq__sigT__fst : + X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + H1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x1) -> + H2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x2) -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x1 H1) + (Coq__Init__Specif.existT X P x2 H2)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 x2) + +:= X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + H1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x1) => + H2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x2) => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x1 H1) + (Coq__Init__Specif.existT X P x2 H2)) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x1 H1) + (y : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P) => + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x1 H1) y) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 (Coq__Init__Specif.projT1 X P y)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + (Coq__Init__Specif.projT1 X P (Coq__Init__Specif.existT X P x1 H1))) + (Coq__Init__Specif.existT X P x2 H2) H. + +def let__s1 : + X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + H1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x1) -> + H2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x2) -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x1 H1) + (Coq__Init__Specif.existT X P x2 H2)) -> + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P) + +:= X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + H1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x1) => + H2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x2) => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x1 H1) + (Coq__Init__Specif.existT X P x2 H2)) => + Coq__Init__Specif.existT X P x2 H2. + +def eq__sigT__snd : + X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + H1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x1) -> + H2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x2) -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x1 H1) + (Coq__Init__Specif.existT X P x2 H2)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x2)) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 P H1 x2 (eq__sigT__fst X P x1 x2 H1 H2 H)) + H2) + +:= X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + H1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x1) => + H2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P x2) => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x1 H1) + (Coq__Init__Specif.existT X P x2 H2)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (Coq__Init__Specif.projT1 X P + (Coq__Init__Specif.existT X P x2 H2)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 P H1 + (Coq__Init__Specif.projT1 X P (Coq__Init__Specif.existT X P x2 H2)) + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x1 H1) + (y : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sigT + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P) => + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x1 H1) y) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 (Coq__Init__Specif.projT1 X P y)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + (Coq__Init__Specif.projT1 X P + (Coq__Init__Specif.existT X P x1 H1))) + (Coq__Init__Specif.existT X P x2 H2) H)) + (Coq__Init__Specif.projT2 X P (Coq__Init__Specif.existT X P x2 H2))) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x2)) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 P H1 x2 (eq__sigT__fst X P x1 x2 H1 H2 H)) + H2) + cupicef.I + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x1 H1) + (y : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P) => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x1 H1) y) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P (Coq__Init__Specif.projT1 X P y))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 P H1 (Coq__Init__Specif.projT1 X P y) + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x1 H1) + (y0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sigT + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P) => + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.existT X P x1 H1) y0) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 (Coq__Init__Specif.projT1 X P y0)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + (Coq__Init__Specif.projT1 X P + (Coq__Init__Specif.existT X P x1 H1))) + y e)) + (Coq__Init__Specif.projT2 X P y)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (Coq__Init__Specif.projT1 X P + (Coq__Init__Specif.existT X P x1 H1)))) + (Coq__Init__Specif.projT2 X P (Coq__Init__Specif.existT X P x1 H1))) + (Coq__Init__Specif.existT X P x2 H2) H). + +def let__s2 : + X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + cupicef.Univ cupicef.prop) -> + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + H1 : cupicef.Term cupicef.prop (P x1) -> + H2 : cupicef.Term cupicef.prop (P x2) -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.exist X P x1 H1) + (Coq__Init__Specif.exist X P x2 H2)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) X P) + +:= X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + cupicef.Univ cupicef.prop) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + H1 : cupicef.Term cupicef.prop (P x1) => + H2 : cupicef.Term cupicef.prop (P x2) => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.exist X P x1 H1) + (Coq__Init__Specif.exist X P x2 H2)) => + Coq__Init__Specif.exist X P x2 H2. + +def eq__sig__fst : + X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + cupicef.Univ cupicef.prop) -> + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + H1 : cupicef.Term cupicef.prop (P x1) -> + H2 : cupicef.Term cupicef.prop (P x2) -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.exist X P x1 H1) + (Coq__Init__Specif.exist X P x2 H2)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 x2) + +:= X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + cupicef.Univ cupicef.prop) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + H1 : cupicef.Term cupicef.prop (P x1) => + H2 : cupicef.Term cupicef.prop (P x2) => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.exist X P x1 H1) + (Coq__Init__Specif.exist X P x2 H2)) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.exist X P x1 H1) + (y : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) X P) => + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) X + P)) + (Coq__Init__Specif.exist X P x1 H1) y) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 (Coq__Init__Specif.proj1__sig X P y)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + (Coq__Init__Specif.proj1__sig X P (Coq__Init__Specif.exist X P x1 H1))) + (Coq__Init__Specif.exist X P x2 H2) H. + +def let__s3 : + X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + cupicef.Univ cupicef.prop) -> + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + H1 : cupicef.Term cupicef.prop (P x1) -> + H2 : cupicef.Term cupicef.prop (P x2) -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.exist X P x1 H1) + (Coq__Init__Specif.exist X P x2 H2)) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) X P) + +:= X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + cupicef.Univ cupicef.prop) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + H1 : cupicef.Term cupicef.prop (P x1) => + H2 : cupicef.Term cupicef.prop (P x2) => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.exist X P x1 H1) + (Coq__Init__Specif.exist X P x2 H2)) => + Coq__Init__Specif.exist X P x2 H2. + +def eq__sig__snd : + X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + cupicef.Univ cupicef.prop) -> + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + H1 : cupicef.Term cupicef.prop (P x1) -> + H2 : cupicef.Term cupicef.prop (P x2) -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.exist X P x1 H1) + (Coq__Init__Specif.exist X P x2 H2)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x2)) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I X + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I X + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + H1 x2 (eq__sig__fst X P x1 x2 H1 H2 H)) + H2) + +:= X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + cupicef.Univ cupicef.prop) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + x2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + H1 : cupicef.Term cupicef.prop (P x1) => + H2 : cupicef.Term cupicef.prop (P x2) => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.exist X P x1 H1) + (Coq__Init__Specif.exist X P x2 H2)) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (Coq__Init__Specif.proj1__sig X P + (Coq__Init__Specif.exist X P x2 H2)))) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I X + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I X + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + H1 + (Coq__Init__Specif.proj1__sig X P + (Coq__Init__Specif.exist X P x2 H2)) + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) + X P)) + (Coq__Init__Specif.exist X P x1 H1) + (y : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Specif.sig + (cupicef.type (cupicef.s cupicef.z)) X P) => + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.exist X P x1 H1) y) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 (Coq__Init__Specif.proj1__sig X P y)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + (Coq__Init__Specif.proj1__sig X P + (Coq__Init__Specif.exist X P x1 H1))) + (Coq__Init__Specif.exist X P x2 H2) H)) + (Coq__Init__Specif.proj2__sig X P (Coq__Init__Specif.exist X P x2 H2))) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x2)) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I X + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I X + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + H1 x2 (eq__sig__fst X P x1 x2 H1 H2 H)) + H2) + cupicef.I + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.exist X P x1 H1) + (y : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Specif.sig (cupicef.type (cupicef.s cupicef.z)) X P) => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.exist X P x1 H1) y) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P (Coq__Init__Specif.proj1__sig X P y))) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I X + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I X + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + H1 (Coq__Init__Specif.proj1__sig X P y) + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.exist X P x1 H1) + (y0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Specif.sig + (cupicef.type (cupicef.s cupicef.z)) X P) => + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sig + (cupicef.type (cupicef.s cupicef.z)) X P)) + (Coq__Init__Specif.exist X P x1 H1) y0) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x1 (Coq__Init__Specif.proj1__sig X P y0)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + (Coq__Init__Specif.proj1__sig X P + (Coq__Init__Specif.exist X P x1 H1))) + y e)) + (Coq__Init__Specif.proj2__sig X P y)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (P + (Coq__Init__Specif.proj1__sig X P + (Coq__Init__Specif.exist X P x1 H1)))) + (Coq__Init__Specif.proj2__sig X P + (Coq__Init__Specif.exist X P x1 H1))) + (Coq__Init__Specif.exist X P x2 H2) H). + +def Eq__rect__eq__on : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q p) -> + cupicef.Univ cupicef.prop + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q p) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + p p) + (h : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + p p) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q p)) + x + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + p Q x p h)). + +def Eq__rect__eq : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Univ cupicef.prop + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop))) + cupicef.I U + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop)) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + (Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I (Q p) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q p) => + Eq__rect__eq__on U p Q x))). + +def Eq__dep__eq__on : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + cupicef.Univ cupicef.prop + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (P p) + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eq__dep U P p x p y) + (__ : cupicef.Term cupicef.prop (eq__dep U P p x p y) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x y)). + +def Eq__dep__eq : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Univ cupicef.prop + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + (P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop)) + cupicef.I U + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I (P p) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + Eq__dep__eq__on U P p x))). + +def UIP__on__ : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + p1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x y) -> + cupicef.Univ cupicef.prop + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + p1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x y) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x y) + (p2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x y) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x y)) + p1 p2). + +def UIP__ : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Univ cupicef.prop + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I U + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I U + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x y) + (p1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x y) => + UIP__on__ U x y p1))). + +def UIP__refl__on__ : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ cupicef.prop + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x x) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x x) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x x)) + p + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x)). + +def UIP__refl__ : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Univ cupicef.prop + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I U + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + UIP__refl__on__ U x). + +def Streicher__K__on__ : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + P : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x x) -> + cupicef.Univ cupicef.prop) -> + cupicef.Univ cupicef.prop + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + P : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x x) -> + cupicef.Univ cupicef.prop) => + cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (P + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x)) + (__ : + cupicef.Term cupicef.prop + (P + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x x) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x x) => + P p)). + +def Streicher__K__ : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Univ cupicef.prop + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop)) + cupicef.I U + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.prod + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop + (cupicef.rule + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x x) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x x) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (P : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x x) -> + cupicef.Univ cupicef.prop) => + Streicher__K__on__ U x P)). + +thm eq__rect__eq__on____eq__dep1__eq__on : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + __ : cupicef.Term cupicef.prop (Eq__rect__eq__on U p P y) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + __0 : cupicef.Term cupicef.prop (eq__dep1 U P p x p y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x y) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + eq__rect__eq : cupicef.Term cupicef.prop (Eq__rect__eq__on U p P y) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + H : cupicef.Term cupicef.prop (eq__dep1 U P p x p y) => + match____eq__dep1 cupicef.prop U P p x p y + (__ : cupicef.Term cupicef.prop (eq__dep1 U P p x p y) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x y) + (h : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + p p) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + y + (p0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x p0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x p0) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x y)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x y) => + H0) + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + p P y p h) + (eq__rect__eq h)) + H. + +thm eq__rect__eq____eq__dep1__eq : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + __ : cupicef.Term cupicef.prop (Eq__rect__eq U) -> + P : + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + __0 : cupicef.Term cupicef.prop (eq__dep1 U P p x p y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x y) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eq__rect__eq : cupicef.Term cupicef.prop (Eq__rect__eq U) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + eq__rect__eq__on____eq__dep1__eq__on U p P x (eq__rect__eq p P x) y. + +def let__H : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + eq__rect__eq : cupicef.Term cupicef.prop (Eq__rect__eq__on U p P x) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + H : cupicef.Term cupicef.prop (eq__dep U P p x p y) -> + cupicef.Term cupicef.prop (eq__dep U P p y p x) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + eq__rect__eq : cupicef.Term cupicef.prop (Eq__rect__eq__on U p P x) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + H : cupicef.Term cupicef.prop (eq__dep U P p x p y) => + eq__dep__sym U P p p x y H. + +thm eq__rect__eq__on____eq__dep__eq__on : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + __ : cupicef.Term cupicef.prop (Eq__rect__eq__on U p P x) -> + cupicef.Term cupicef.prop (Eq__dep__eq__on U P p x) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + eq__rect__eq : cupicef.Term cupicef.prop (Eq__rect__eq__on U p P x) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (P p) + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (eq__dep U P p x p y) + (H : cupicef.Term cupicef.prop (eq__dep U P p x p y) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x y))) + (Eq__dep__eq__on U P p x) cupicef.I + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + H : cupicef.Term cupicef.prop (eq__dep U P p x p y) => + Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + y x + (eq__rect__eq__on____eq__dep1__eq__on U p P x eq__rect__eq y + (eq__dep__dep1 U P p p y x (let__H U p P x eq__rect__eq y H)))). + +thm eq__rect__eq____eq__dep__eq : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + __ : cupicef.Term cupicef.prop (Eq__rect__eq U) -> + cupicef.Term cupicef.prop (Eq__dep__eq U) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eq__rect__eq : cupicef.Term cupicef.prop (Eq__rect__eq U) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + eq__rect__eq__on____eq__dep__eq__on U p P x (eq__rect__eq p P x) y. + +thm eq__dep__eq__on____UIP__on : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + p1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x y) -> + __ : + cupicef.Term cupicef.prop + (Eq__dep__eq__on U + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I U + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x y0)) + x + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x)) -> + cupicef.Term cupicef.prop (UIP__on__ U x y p1) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + p1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x y) => + eq__dep__eq : + cupicef.Term cupicef.prop + (Eq__dep__eq__on U + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I U + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x y0)) + x + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x)) => + cupicef.cast cupicef.prop cupicef.prop + ((y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + p2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x y0) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x y0) + (p3 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x y0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x y0)) + p2 p3)) + y p1) + (UIP__on__ U x y p1) cupicef.I + (eq__indd + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + p2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x y0) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x y0) + (p3 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x y0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x y0)) + p2 p3)) + (p2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x x) => + eq__dep__eq p2 + (eq__indd + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x + (eq__dep U + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I U + (y0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I U + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x y0)) + x + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x)) + (eq__dep__intro U + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I U + (y0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I U + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x y0)) + x + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x)) + x p2)) + y p1). + +thm eq__dep__eq____UIP : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + __ : cupicef.Term cupicef.prop (Eq__dep__eq U) -> + cupicef.Term cupicef.prop (UIP__ U) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eq__dep__eq : cupicef.Term cupicef.prop (Eq__dep__eq U) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + p1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x y) => + eq__dep__eq__on____UIP__on U x y p1 + (eq__dep__eq + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I U + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x y0)) + x + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x)). + +thm UIP__on____UIP__refl__on : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + __ : + cupicef.Term cupicef.prop + (UIP__on__ U x x + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x)) -> + cupicef.Term cupicef.prop (UIP__refl__on__ U x) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + UIP : + cupicef.Term cupicef.prop + (UIP__on__ U x x + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x)) => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x x) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x x) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x x)) + p + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x))) + (UIP__refl__on__ U x) cupicef.I + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x x) => + Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x) + p (UIP p)). + +thm UIP____UIP__refl : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + __ : cupicef.Term cupicef.prop (UIP__ U) -> + cupicef.Term cupicef.prop (UIP__refl__ U) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + UIP : cupicef.Term cupicef.prop (UIP__ U) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x x) => + UIP__on____UIP__refl__on U x + (UIP x x + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x)) + p. + +thm UIP__refl__on____Streicher__K__on : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + P : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x x) -> + cupicef.Univ cupicef.prop) -> + __ : cupicef.Term cupicef.prop (UIP__refl__on__ U x) -> + cupicef.Term cupicef.prop (Streicher__K__on__ U x P) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + P : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x x) -> + cupicef.Univ cupicef.prop) => + UIP__refl : cupicef.Term cupicef.prop (UIP__refl__on__ U x) => + cupicef.cast + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.prop + (cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (P + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x)) + (H : + cupicef.Term cupicef.prop + (P + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x x) + (p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x x) => + P p))) + (Streicher__K__on__ U x P) cupicef.I + (H : + cupicef.Term cupicef.prop + (P + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x)) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x x) => + Coq__Init__Logic.eq__ind__r + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x) + P H p (UIP__refl p)). + +thm UIP__refl____Streicher__K : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + __ : cupicef.Term cupicef.prop (UIP__refl__ U) -> + cupicef.Term cupicef.prop (Streicher__K__ U) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + UIP__refl : cupicef.Term cupicef.prop (UIP__refl__ U) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + P : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x x) -> + cupicef.Univ cupicef.prop) => + UIP__refl__on____Streicher__K__on U x P (UIP__refl x). + +thm Streicher__K__on____eq__rect__eq__on : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + __ : + cupicef.Term cupicef.prop + (Streicher__K__on__ U p + (h : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + p p) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + p P x p h))) -> + cupicef.Term cupicef.prop (Eq__rect__eq__on U p P x) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + Streicher__K : + cupicef.Term cupicef.prop + (Streicher__K__on__ U p + (h : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + p p) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + p P x p h))) => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + p p) + (h : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + p p) => + (h0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + p p) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + p P x p h0)) + h)) + (Eq__rect__eq__on U p P x) cupicef.I + (h : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + p p) => + Streicher__K + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + p P x p + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + p))) + h). + +thm Streicher__K____eq__rect__eq : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + __ : cupicef.Term cupicef.prop (Streicher__K__ U) -> + cupicef.Term cupicef.prop (Eq__rect__eq U) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + Streicher__K : cupicef.Term cupicef.prop (Streicher__K__ U) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + Streicher__K__on____eq__rect__eq__on U p P x + (Streicher__K p + (h : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + p p) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + p P x p h))). + +def let__UIP : + X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + UIP__refl : cupicef.Term cupicef.prop (UIP__refl__on__ X x) -> + y : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x) -> + z : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x)) -> + y_27_ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x) -> + y_27__27_ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + y_27_ y_27__27_) + +:= X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + UIP__refl : cupicef.Term cupicef.prop (UIP__refl__on__ X x) => + y : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x) => + z : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x)) => + y_27_ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x) => + y_27__27_ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x) => + Coq__Init__Logic.eq__trans + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + y_27_ + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + y_27__27_ (UIP__refl y_27_) + (Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + y_27__27_ + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (UIP__refl y_27__27_)). + +def let__e : + X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + UIP__refl : cupicef.Term cupicef.prop (UIP__refl__on__ X x) -> + y : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x) -> + z : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x)) + +:= X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + UIP__refl : cupicef.Term cupicef.prop (UIP__refl__on__ X x) => + y : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x) => + z : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x)) => + let__UIP X x UIP__refl y z + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x). + +def let__e0 : + X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + UIP__refl : cupicef.Term cupicef.prop (UIP__refl__on__ X x) -> + y : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x) -> + z : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x)) + +:= X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + UIP__refl : cupicef.Term cupicef.prop (UIP__refl__on__ X x) => + y : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x) => + z : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x)) => + let__UIP X x UIP__refl y z + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x). + +thm UIP__shift__on : + X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X -> + __ : cupicef.Term cupicef.prop (UIP__refl__on__ X x) -> + y : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x) -> + cupicef.Term cupicef.prop + (UIP__refl__on__ + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + y) + +:= X : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + UIP__refl : cupicef.Term cupicef.prop (UIP__refl__on__ X x) => + y : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x) => + Coq__Init__Logic.eq__ind__r + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (y0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x) => + UIP__refl__on__ + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + y0) + (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x)) + (z : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + 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(cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x))) + z + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x)))) + (UIP__refl__on__ + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x)) + cupicef.I + (z : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x)) => + Coq__Init__Logic.eq__trans + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s 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cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x))) + z + (Coq__Init__Logic.eq__trans + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__trans + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (let__UIP X x UIP__refl y z + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x)) + z) + (Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (let__UIP X x UIP__refl y z + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x)))) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x)) + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (y0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x) => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + y0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + y0)) + e + (Coq__Init__Logic.eq__trans + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + y0 y0 + (Coq__Init__Logic.eq__trans + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + y0 + (let__UIP X x UIP__refl y z + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x)) + e) + (Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + y0 y0 (let__UIP X x UIP__refl y z y0 y0)))) + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (y0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x) => + e0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + y0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x))) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x)) + (Coq__Init__Logic.eq__trans + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + y0 + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__trans + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + y0 y0 e0 + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + y0)) + (Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + 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(cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (y0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x) => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + y0) => + Coq__Init__Logic.match____eq + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x + (x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) X => + y_27_ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x_27_) => + cupicef.prod cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x_27_)) + y_27_ y_27_) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x_27_)) + y_27_ y_27_) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (z0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x))) + z0 + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x))) + x y0 + (Coq__Init__Logic.eq__trans + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + y0 y0 y0 + (Coq__Init__Logic.eq__trans + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + y0 + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + y0 + (let__UIP X x UIP__refl y z y0 + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x)) + e) + (Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + y0 y0 (let__UIP X x UIP__refl y z y0 y0)))) + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (y0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x) => + e0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + y0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x))) + (Coq__Init__Logic.eq__trans + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + y0 + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__trans + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + y0 y0 e0 + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + y0)) + (Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + y0 e0)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x))) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x))) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x))) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + (let__e0 X x UIP__refl y z)) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I X) + x) + z))) + y (UIP__refl y). + +thm UIP__shift : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + __ : cupicef.Term cupicef.prop (UIP__refl__ U) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Term cupicef.prop + (UIP__refl__ + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I U) + x x))) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + UIP__refl : cupicef.Term cupicef.prop (UIP__refl__ U) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + UIP__shift__on U x (UIP__refl x). + +def Inj__dep__pair__on : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + cupicef.Univ cupicef.prop + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (P p) + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.existT U P p x) + (Coq__Init__Specif.existT U P p y)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.existT U P p x) + (Coq__Init__Specif.existT U P p y)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x y)). + +def Inj__dep__pair : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Univ cupicef.prop + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + (P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop)) + cupicef.I U + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I (P p) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + Inj__dep__pair__on U P p x))). + +thm eq__dep__eq__on____inj__pair2__on : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + __ : cupicef.Term cupicef.prop (Eq__dep__eq__on U P p x) -> + cupicef.Term cupicef.prop (Inj__dep__pair__on U P p x) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + eq__dep__eq : cupicef.Term cupicef.prop (Eq__dep__eq__on U P p x) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (P p) + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.existT U P p x) + (Coq__Init__Specif.existT U P p y)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.existT U P p x) + (Coq__Init__Specif.existT U P p y)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x y))) + (Inj__dep__pair__on U P p x) cupicef.I + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) U P)) + (Coq__Init__Specif.existT U P p x) + (Coq__Init__Specif.existT U P p y)) => + eq__dep__eq y (eq__sigT__eq__dep U P p p x y H)). + +thm eq__dep__eq____inj__pair2 : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + __ : cupicef.Term cupicef.prop (Eq__dep__eq U) -> + cupicef.Term cupicef.prop (Inj__dep__pair U) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eq__dep__eq : cupicef.Term cupicef.prop (Eq__dep__eq U) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + eq__dep__eq__on____inj__pair2__on U P p x (eq__dep__eq P p x). + +thm f__eq__dep : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + R : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + f : + (p0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p0) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (R p0)) -> + __ : cupicef.Term cupicef.prop (eq__dep U P p x q y) -> + cupicef.Term cupicef.prop (eq__dep U R p (f p x) q (f q y)) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + R : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) => + f : + (p0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p0) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (R p0)) => + H : cupicef.Term cupicef.prop (eq__dep U P p x q y) => + match____eq__dep cupicef.prop U P p x + (q0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q0) => + __ : cupicef.Term cupicef.prop (eq__dep U P p x q0 y0) => + eq__dep U R p (f p x) q0 (f q0 y0)) + (eq__dep__intro U R p (f p x)) q y H. + +thm eq__dep__non__dep : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) P -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) P -> + __ : + cupicef.Term cupicef.prop + (eq__dep U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => P) p x + q y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I P) + x y) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) P => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) P => + H : + cupicef.Term cupicef.prop + (eq__dep U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => P) p x + q y) => + match____eq__dep cupicef.prop U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => P) p x + (q0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) P => + __ : + cupicef.Term cupicef.prop + (eq__dep U + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => P) + p x q0 y0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I P) + x y0) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I P) + x) + q y H. + +thm f__eq__dep__non__dep : + U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + R : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) -> + f : + (p0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p0) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) R) -> + __ : cupicef.Term cupicef.prop (eq__dep U P p x q y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I R) + (f p x) (f q y)) + +:= U : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + R : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + q : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q) => + f : + (p0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p0) -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) R) => + H : cupicef.Term cupicef.prop (eq__dep U P p x q y) => + match____eq__dep cupicef.prop U P p x + (q0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) U => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P q0) => + __ : cupicef.Term cupicef.prop (eq__dep U P p x q0 y0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I R) + (f p x) (f q0 y0)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I R) + (f p x)) + q y H. + +(; End of translation. ;) diff --git a/out/Coq__Logic__Eqdep__dec.dk b/out/Coq__Logic__Eqdep__dec.dk new file mode 100644 index 000000000..804f10062 --- /dev/null +++ b/out/Coq__Logic__Eqdep__dec.dk @@ -0,0 +1,13414 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def let__comp : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + eq1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + eq2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y_27_) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y y_27_) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + eq2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y_27_) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a y_27_) + eq2 y eq1. + +thm trans__sym__eq : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + u : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y y)) + ((x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y0) => + eq2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y_27_) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a y_27_) + eq2 y0 eq1) + x y y u u) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + u : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0 y0)) + (let__comp A x y0 y0 e e) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + y u. + +def let__comp0 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + eq1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + eq2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y_27_) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y y_27_) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + eq2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y_27_) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a y_27_) + eq2 y eq1. + +def let__nu : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + u : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + u : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (eqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + eqxy) + (neqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (neqxy u)) + (eq__dec y). + +def let__nu__inv : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + v : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + v : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + let__comp0 A x x y + (let__nu A x eq__dec x + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + v. + +thm nu__left__inv__on : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + u : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + ((y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + v : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) => + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y1) => + eq2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y_27_) => + Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a y_27_) + eq2 y1 eq1) + x x y0 + ((y1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + u0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y1) => + Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y1) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y1)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y1) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y1))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y1) + (eqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y1) => + eqxy) + (neqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y1)) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y1) + (neqxy u0)) + (eq__dec y1)) + x + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + v) + y + ((y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + u0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) => + Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) + (eqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) => + eqxy) + (neqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0)) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) + (neqxy u0)) + (eq__dec y0)) + y u)) + u) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + u : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0)) + (let__nu__inv A x eq__dec y0 (let__nu A x eq__dec y0 e)) e) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + ((x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y0) => + eq2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y_27_) => + Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a y_27_) + eq2 y0 eq1) + x x x + (let__nu A x eq__dec x + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + (let__nu A x eq__dec x + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x))) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (let__nu__inv A x eq__dec x + (let__nu A x eq__dec x + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x))) + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + cupicef.I + (trans__sym__eq A x x + (let__nu A x eq__dec x + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)))) + y u. + +def let__comp1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + eq1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + eq2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y_27_) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y y_27_) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + eq2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y_27_) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a y_27_) + eq2 y eq1. + +def let__nu0 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + u : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + u : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (eqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + eqxy) + (neqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (neqxy u)) + (eq__dec y). + +def let__o : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + u : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + v : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + u : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + v : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + eq__dec y. + +def let__nu__constant : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + u : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + v : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (let__nu0 A x eq__dec y u) (let__nu0 A x eq__dec y v)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + u : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + v : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (eqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + eqxy) + (neqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (neqxy u)) + (eq__dec y)) + (Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (eqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + eqxy) + (neqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (neqxy v)) + (eq__dec y))) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (let__nu0 A x eq__dec y u) (let__nu0 A x eq__dec y v)) + cupicef.I + (Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (o0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (eqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + eqxy) + (neqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (neqxy u)) + o0) + (Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (eqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + eqxy) + (neqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (neqxy v)) + o0)) + (Heq : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + Heq) + (Hneq : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) => + Coq__Init__Logic.match____False cupicef.prop + (f : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + f) + (Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Hneq v))) + (Hneq u)) + (let__o A x eq__dec y u v)). + +def let__nu__inv0 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + v : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + v : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + let__comp1 A x x y + (let__nu0 A x eq__dec x + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + v. + +thm eq__proofs__unicity__on : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + p2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + p1 p2) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + p1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + p2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (let__nu__inv0 A x eq__dec y (let__nu0 A x eq__dec y p1)) + (p3 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + p3 p2) + (Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (let__nu__inv0 A x eq__dec y (let__nu0 A x eq__dec y p2)) + (p3 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (let__nu__inv0 A x eq__dec y (let__nu0 A x eq__dec y p1)) p3) + (Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (let__nu0 A x eq__dec y p1) + (e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (let__nu__inv0 A x eq__dec y (let__nu0 A x eq__dec y p1)) + (let__nu__inv0 A x eq__dec y e)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (let__nu__inv0 A x eq__dec y (let__nu0 A x eq__dec y p1))) + (let__nu0 A x eq__dec y p2) + (let__nu__constant A x eq__dec y p1 p2)) + p2 (nu__left__inv__on A x eq__dec y p2)) + p1 (nu__left__inv__on A x eq__dec y p1). + +def let__comp2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + eq1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + eq2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y_27_) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y y_27_) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + eq2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y_27_) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a y_27_) + eq2 y eq1. + +def let__nu1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + u : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + u : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (eqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + eqxy) + (neqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (neqxy u)) + (eq__dec y). + +def let__o0 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + u : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + v : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + u : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + v : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + eq__dec y. + +def let__nu__constant0 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + u : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + v : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (let__nu1 A x eq__dec y u) (let__nu1 A x eq__dec y v)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + u : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + v : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (eqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + eqxy) + (neqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (neqxy u)) + (eq__dec y)) + (Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (eqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + eqxy) + (neqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (neqxy v)) + (eq__dec y))) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (let__nu1 A x eq__dec y u) (let__nu1 A x eq__dec y v)) + cupicef.I + (Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (o0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (eqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + eqxy) + (neqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (neqxy u)) + o0) + (Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (eqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + eqxy) + (neqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (neqxy v)) + o0)) + (Heq : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + Heq) + (Hneq : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) => + Coq__Init__Logic.match____False cupicef.prop + (f : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + f) + (Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Hneq v))) + (Hneq u)) + (let__o0 A x eq__dec y u v)). + +def let__nu__inv1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + v : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + v : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + let__comp2 A x x y + (let__nu1 A x eq__dec x + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + v. + +thm K__dec__on : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + P : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) -> + cupicef.Univ cupicef.prop) -> + __ : + cupicef.Term cupicef.prop + (P + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) -> + cupicef.Term cupicef.prop (P p) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + P : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) -> + cupicef.Univ cupicef.prop) => + H : + cupicef.Term cupicef.prop + (P + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x) + (p0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) => + P p0) + H p + (eq__proofs__unicity__on A x eq__dec x + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x) + p). + +def let__comp3 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + eq1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + eq2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y_27_) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y y_27_) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + eq2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y_27_) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a y_27_) + eq2 y eq1. + +def let__nu2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + u : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + u : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (eqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + eqxy) + (neqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (neqxy u)) + (eq__dec y). + +def let__o1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + u : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + v : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + u : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + v : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + eq__dec y. + +def let__nu__constant1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + u : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + v : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (let__nu2 A x eq__dec y u) (let__nu2 A x eq__dec y v)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + u : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + v : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (eqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + eqxy) + (neqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (neqxy u)) + (eq__dec y)) + (Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (eqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + eqxy) + (neqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (neqxy v)) + (eq__dec y))) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (let__nu2 A x eq__dec y u) (let__nu2 A x eq__dec y v)) + cupicef.I + (Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (o0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (eqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + eqxy) + (neqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (neqxy u)) + o0) + (Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y))) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (eqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + eqxy) + (neqxy : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) => + Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (neqxy v)) + o0)) + (Heq : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + Heq) + (Hneq : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) => + Coq__Init__Logic.match____False cupicef.prop + (f : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + f) + (Coq__Init__Logic.False__ind + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Hneq v))) + (Hneq u)) + (let__o1 A x eq__dec y u v)). + +def let__nu__inv2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + v : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + v : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + let__comp3 A x x y + (let__nu2 A x eq__dec x + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + v. + +def let__proj : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + exP : cupicef.Term cupicef.prop (Coq__Init__Logic.ex A P) -> + def__ : cupicef.Term cupicef.prop (P x) -> + cupicef.Term cupicef.prop (P x) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + exP : cupicef.Term cupicef.prop (Coq__Init__Logic.ex A P) => + def__ : cupicef.Term cupicef.prop (P x) => + Coq__Init__Logic.match____ex cupicef.prop A P + (exP0 : cupicef.Term cupicef.prop (Coq__Init__Logic.ex A P) => P x) + (x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + prf : cupicef.Term cupicef.prop (P x_27_) => + Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_))) => + P x) + (eqprf : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x_27_ P prf x + (Coq__Init__Logic.eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_ eqprf)) + (n : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_)) => + def__) + (eq__dec x_27_)) + exP. + +def let__H0 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + y : cupicef.Term cupicef.prop (P x) -> + y_27_ : cupicef.Term cupicef.prop (P x) -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Logic.ex A P)) + (Coq__Init__Logic.ex__intro A P x y) + (Coq__Init__Logic.ex__intro A P x y_27_)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (let__proj A x eq__dec P (Coq__Init__Logic.ex__intro A P x y) y) + (let__proj A x eq__dec P (Coq__Init__Logic.ex__intro A P x y_27_) y)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + y : cupicef.Term cupicef.prop (P x) => + y_27_ : cupicef.Term cupicef.prop (P x) => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Logic.ex A P)) + (Coq__Init__Logic.ex__intro A P x y) + (Coq__Init__Logic.ex__intro A P x y_27_)) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Logic.ex A P)) + (Coq__Init__Logic.ex__intro A P x y) + (y0 : cupicef.Term cupicef.prop (Coq__Init__Logic.ex A P) => + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Logic.ex A P)) + (Coq__Init__Logic.ex__intro A P x y) y0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (let__proj A x eq__dec P (Coq__Init__Logic.ex__intro A P x y) y) + (let__proj A x eq__dec P y0 y)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (let__proj A x eq__dec P (Coq__Init__Logic.ex__intro A P x y) y)) + (Coq__Init__Logic.ex__intro A P x y_27_) H. + +def let__o2 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + y : cupicef.Term cupicef.prop (P x) -> + y_27_ : cupicef.Term cupicef.prop (P x) -> + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Logic.ex A P)) + (Coq__Init__Logic.ex__intro A P x y) + (Coq__Init__Logic.ex__intro A P x y_27_)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + y : cupicef.Term cupicef.prop (P x) => + y_27_ : cupicef.Term cupicef.prop (P x) => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Logic.ex A P)) + (Coq__Init__Logic.ex__intro A P x y) + (Coq__Init__Logic.ex__intro A P x y_27_)) => + eq__dec x. + +thm inj__right__pair__on : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + y : cupicef.Term cupicef.prop (P x) -> + y_27_ : cupicef.Term cupicef.prop (P x) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Logic.ex A P)) + (Coq__Init__Logic.ex__intro A P x y) + (Coq__Init__Logic.ex__intro A P x y_27_)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + y y_27_) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq__dec : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + y : cupicef.Term cupicef.prop (P x) => + y_27_ : cupicef.Term cupicef.prop (P x) => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Logic.ex A P)) + (Coq__Init__Logic.ex__intro A P x y) + (Coq__Init__Logic.ex__intro A P x y_27_)) => + cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x))) => + P x) + (eqprf : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) => + Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P y x + (Coq__Init__Logic.eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x eqprf)) + (n : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) => + y) + (eq__dec x)) + (Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x))) => + P x) + (eqprf : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) => + Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P y_27_ x + (Coq__Init__Logic.eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x eqprf)) + (n : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) => + y) + (eq__dec x))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x))) => + P x) + (eqprf : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) => + Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P y x + (Coq__Init__Logic.eq__sym + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x eqprf)) + (n : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) => + y) + (eq__dec x)) + (Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x))) => + P x) + (eqprf : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) => + Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P y_27_ x + (Coq__Init__Logic.eq__sym + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x eqprf)) + (n : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) => + y) + (eq__dec x))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + y y_27_)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (let__proj A x eq__dec P (Coq__Init__Logic.ex__intro A P x y) y) + (let__proj A x eq__dec P (Coq__Init__Logic.ex__intro A P x y_27_) + y)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (let__proj A x eq__dec P (Coq__Init__Logic.ex__intro A P x y) + y) + (let__proj A x eq__dec P + (Coq__Init__Logic.ex__intro A P x y_27_) y)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + y y_27_)) + cupicef.I + (Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (o0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x))) => + P x) + (eqprf : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) => + Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P y x + (Coq__Init__Logic.eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x eqprf)) + (n : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) => + y) + o0) + (Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x))) => + P x) + (eqprf : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) => + Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P y_27_ x + (Coq__Init__Logic.eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x eqprf)) + (n : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) => + y) + o0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x))) => + P x) + (eqprf : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) => + Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P y x + (Coq__Init__Logic.eq__sym + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x eqprf)) + (n : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) => + y) + o0) + (Coq__Init__Logic.match____or cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x))) => + P x) + (eqprf : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) => + Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P y_27_ x + (Coq__Init__Logic.eq__sym + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x eqprf)) + (n : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) => + y) + o0)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + y y_27_)) + (Heq : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) => + K__dec__on A x eq__dec + (Heq0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P y x + (Coq__Init__Logic.eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x Heq0)) + (Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P y_27_ x + (Coq__Init__Logic.eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x Heq0))) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P y x + (Coq__Init__Logic.eq__sym + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x Heq0)) + (Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P y_27_ x + (Coq__Init__Logic.eq__sym + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x Heq0))) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + y y_27_)) + (H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P y x + (Coq__Init__Logic.eq__sym + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x))) + (Coq__Init__Logic.eq__ind + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P y_27_ x + (Coq__Init__Logic.eq__sym + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)))) => + H1) + Heq) + (Hneq : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + y y) => + Coq__Init__Logic.match____False cupicef.prop + (__ : cupicef.Term cupicef.prop Coq__Init__Logic.False => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + y y_27_) + (Hneq + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x))) + (let__o2 A x eq__dec P y y_27_ H)) + (let__H0 A x eq__dec P y y_27_ H). + +thm eq__proofs__unicity : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eq__dec : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + p2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + p1 p2) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eq__dec : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + eq__proofs__unicity__on A x (eq__dec x). + +thm K__dec : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eq__dec : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) -> + cupicef.Univ cupicef.prop) -> + __ : + cupicef.Term cupicef.prop + (P + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) -> + cupicef.Term cupicef.prop (P p) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eq__dec : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + K__dec__on A x (eq__dec x). + +thm inj__right__pair : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + eq__dec : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + y : cupicef.Term cupicef.prop (P x) -> + y_27_ : cupicef.Term cupicef.prop (P x) -> + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Coq__Init__Logic.ex A P)) + (Coq__Init__Logic.ex__intro A P x y) + (Coq__Init__Logic.ex__intro A P x y_27_)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + y y_27_) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eq__dec : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + inj__right__pair__on A x (eq__dec x). + +thm K__dec__type : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + __ : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) -> + cupicef.Univ cupicef.prop) -> + __0 : + cupicef.Term cupicef.prop + (P + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) -> + cupicef.Term cupicef.prop (P p) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eq__dec : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P : + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) -> + cupicef.Univ cupicef.prop) => + H : + cupicef.Term cupicef.prop + (P + (Coq__Init__Logic.eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) => + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) => + K__dec A + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Specif.match____sumbool cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y)) + (s : + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y))) => + Coq__Init__Logic.or + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y))) + (e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y) => + Coq__Init__Logic.or__introl + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y)) + e) + (n : + cupicef.Term cupicef.prop + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y)) => + Coq__Init__Logic.or__intror + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y)) + n) + (eq__dec x0 y)) + x + (p0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) => + P p0) + H p. + +thm K__dec__set : + A : cupicef.Univ cupicef.set -> + __ : + (x : cupicef.Term cupicef.set A -> + y : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + x : cupicef.Term cupicef.set A -> + P : + (__0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) -> + cupicef.Univ cupicef.prop) -> + __0 : + cupicef.Term cupicef.prop + (P + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) -> + p : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) -> + cupicef.Term cupicef.prop (P p) + +:= A : cupicef.Univ cupicef.set => + K__dec__type + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I A). + +thm eq__rect__eq__dec : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + __ : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Q : + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q p) -> + h : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + p p) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q p)) + x + (Coq__Init__Logic.eq__rect + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + p Q x p h)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eq__dec : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + Coq__Logic__EqdepFacts.Streicher__K____eq__rect__eq A + (K__dec__type A eq__dec). + +thm eq__dep__eq__dec : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + __ : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + P : + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + __0 : + cupicef.Term cupicef.prop (Coq__Logic__EqdepFacts.eq__dep A P p x p y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eq__dec : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + Coq__Logic__EqdepFacts.eq__rect__eq____eq__dep__eq A + (eq__rect__eq__dec A eq__dec). + +thm UIP__dec : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + __ : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + p2 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + p1 p2) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eq__dec : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + Coq__Logic__EqdepFacts.eq__dep__eq____UIP A (eq__dep__eq__dec A eq__dec). + +thm inj__pair2__eq__dec : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + __ : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) -> + P : + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P p) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A P)) + (Coq__Init__Specif.existT A P p x) + (Coq__Init__Specif.existT A P p y)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P p)) + x y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + eq__dec : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set + (Coq__Init__Specif.sumbool + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (Coq__Init__Logic.not + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))) => + Coq__Logic__EqdepFacts.eq__dep__eq____inj__pair2 A + (Coq__Logic__EqdepFacts.eq__rect__eq____eq__dep__eq A + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I A + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I)) + (Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (Q p) + (x : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Q p) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + p p) + (h : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + p p) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q p)) + x + (Coq__Init__Logic.eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + p Q x p h)))))) + (Coq__Logic__EqdepFacts.Eq__rect__eq A) cupicef.I + (p : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Q : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (Q p) => + h : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + p p) => + eq__rect__eq__dec A eq__dec p Q x h))). + +def UIP__refl__unit : + x : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.unit) + Coq__Init__Datatypes.tt Coq__Init__Datatypes.tt) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.unit) + Coq__Init__Datatypes.tt Coq__Init__Datatypes.tt)) + x + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.unit) + Coq__Init__Datatypes.tt)) + +:= x : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.unit) + Coq__Init__Datatypes.tt Coq__Init__Datatypes.tt) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.unit) + Coq__Init__Datatypes.tt + (y : cupicef.Term cupicef.set Coq__Init__Datatypes.unit => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.unit) + Coq__Init__Datatypes.tt y) => + Coq__Init__Datatypes.match____unit (cupicef.type (cupicef.s cupicef.z)) + (b : cupicef.Term cupicef.set Coq__Init__Datatypes.unit => + cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.unit) + Coq__Init__Datatypes.tt b) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.unit) + Coq__Init__Datatypes.tt b) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.unit) + Coq__Init__Datatypes.tt Coq__Init__Datatypes.tt) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.unit) + Coq__Init__Datatypes.tt Coq__Init__Datatypes.tt)) + x0 + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.unit) + Coq__Init__Datatypes.tt)) + y e) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.unit) + Coq__Init__Datatypes.tt Coq__Init__Datatypes.tt)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.unit) + Coq__Init__Datatypes.tt)) + Coq__Init__Datatypes.tt x. + +def UIP__refl__bool : + b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool -> + x : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b b) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b b)) + x + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b)) + +:= b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + x : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b b) => + Coq__Init__Datatypes.match____bool cupicef.prop + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 b0) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 b0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0 b0)) + x0 + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + b0))) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true + (y : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true y) => + Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true b0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true b0) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (x1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + x1 + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => + Coq__Init__Logic.True) + y e) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true Coq__Init__Datatypes.true)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.true)) + Coq__Init__Datatypes.true x0) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false + (y : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false y) => + Coq__Init__Datatypes.match____bool + (cupicef.type (cupicef.s cupicef.z)) + (b0 : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => + cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false b0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false b0) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.true) => + Coq__Init__Logic.True) + (x1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + x1 + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + y e) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false Coq__Init__Datatypes.false)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.bool) + Coq__Init__Datatypes.false)) + Coq__Init__Datatypes.false x0) + b x. + +def let__IHn : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + x : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n n) -> + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + IHn : + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 n0)) + x0 + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0))) -> + x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) (Coq__Init__Datatypes.S n0)) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 n0)) + (Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Nat.pred (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S n0) x0) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + x : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n n) => + n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 n0)) + x0 + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0))) => + x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) (Coq__Init__Datatypes.S n0)) => + IHn + (Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Nat.pred (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S n0) x0). + +def UIP__refl__nat : + n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> + x : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n n) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n n)) + x + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n)) + +:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + x : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n n) => + Coq__Init__Datatypes.nat__ind + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 n0) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 n0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 n0)) + x0 + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0))) + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O) => + Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O + (y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O y) => + Coq__Init__Datatypes.match____nat + (cupicef.type (cupicef.s cupicef.z)) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O n0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O n0) => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (x1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O)) + x1 + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O)) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + __ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n0)) => + Coq__Init__Logic.True) + y e) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O Coq__Init__Datatypes.O)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.O)) + Coq__Init__Datatypes.O x0) + (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + IHn : + (x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 n0) -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 n0)) + x0 + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0))) => + x0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) (Coq__Init__Datatypes.S n0)) => + Coq__Init__Logic.eq__ind + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 n0)) + (Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Nat.pred (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S n0) x0) + (e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0 n0) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) (Coq__Init__Datatypes.S n0))) + x0 + (Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.S n0 n0 e)) + (cupicef.cast cupicef.prop cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) (Coq__Init__Datatypes.S n0))) + x0 + (Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.S + (Coq__Init__Nat.pred (Coq__Init__Datatypes.S n0)) + (Coq__Init__Nat.pred (Coq__Init__Datatypes.S n0)) + (Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Nat.pred (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S n0) x0))) + (Coq__Init__Datatypes.match____nat + (cupicef.type (cupicef.s cupicef.z)) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) n_27_) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) n_27_) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) => + Coq__Init__Logic.True) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + x1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S n_27_)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S n_27_))) + x1 + (Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.S + (Coq__Init__Nat.pred (Coq__Init__Datatypes.S n0)) + (Coq__Init__Nat.pred (Coq__Init__Datatypes.S n_27_)) + (Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Nat.pred (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S n_27_) x1))) + (Coq__Init__Datatypes.S n0) x0) + cupicef.I + (Coq__Init__Logic.match____eq cupicef.prop + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) + (y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + e : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) y) => + Coq__Init__Datatypes.match____nat + (cupicef.type (cupicef.s cupicef.z)) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + cupicef.prod cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) n_27_) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) n_27_) => + cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) => + Coq__Init__Logic.True) + (n_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => + x1 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S n_27_)) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s 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(cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.S + (Coq__Init__Nat.pred (Coq__Init__Datatypes.S n0)) + (Coq__Init__Nat.pred (Coq__Init__Datatypes.S n_27_)) + (Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Nat.pred (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S n_27_) x1))) + y e) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S n0))) + (Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Datatypes.S + (Coq__Init__Nat.pred (Coq__Init__Datatypes.S n0)) + (Coq__Init__Nat.pred (Coq__Init__Datatypes.S n0)) + (Coq__Init__Logic.f__equal + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + Coq__Init__Nat.pred (Coq__Init__Datatypes.S n0) + (Coq__Init__Datatypes.S n0) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + (Coq__Init__Datatypes.S n0))))) + (Coq__Init__Datatypes.S n0) x0)) + (Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I Coq__Init__Datatypes.nat) + n0) + (let__IHn n x n0 IHn x0)) + n x. + +(; End of translation. ;) diff --git a/out/Coq__Logic__ProofIrrelevanceFacts.dk b/out/Coq__Logic__ProofIrrelevanceFacts.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Logic__ProofIrrelevanceFacts.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__NatInt__NZAdd.dk b/out/Coq__Numbers__NatInt__NZAdd.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__NatInt__NZAdd.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__NatInt__NZAddOrder.dk b/out/Coq__Numbers__NatInt__NZAddOrder.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__NatInt__NZAddOrder.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__NatInt__NZAxioms.dk b/out/Coq__Numbers__NatInt__NZAxioms.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__NatInt__NZAxioms.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__NatInt__NZBase.dk b/out/Coq__Numbers__NatInt__NZBase.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__NatInt__NZBase.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__NatInt__NZBits.dk b/out/Coq__Numbers__NatInt__NZBits.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__NatInt__NZBits.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__NatInt__NZDiv.dk b/out/Coq__Numbers__NatInt__NZDiv.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__NatInt__NZDiv.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__NatInt__NZGcd.dk b/out/Coq__Numbers__NatInt__NZGcd.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__NatInt__NZGcd.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__NatInt__NZLog.dk b/out/Coq__Numbers__NatInt__NZLog.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__NatInt__NZLog.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__NatInt__NZMul.dk b/out/Coq__Numbers__NatInt__NZMul.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__NatInt__NZMul.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__NatInt__NZMulOrder.dk b/out/Coq__Numbers__NatInt__NZMulOrder.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__NatInt__NZMulOrder.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__NatInt__NZOrder.dk b/out/Coq__Numbers__NatInt__NZOrder.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__NatInt__NZOrder.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__NatInt__NZParity.dk b/out/Coq__Numbers__NatInt__NZParity.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__NatInt__NZParity.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__NatInt__NZPow.dk b/out/Coq__Numbers__NatInt__NZPow.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__NatInt__NZPow.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__NatInt__NZProperties.dk b/out/Coq__Numbers__NatInt__NZProperties.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__NatInt__NZProperties.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__NatInt__NZSqrt.dk b/out/Coq__Numbers__NatInt__NZSqrt.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__NatInt__NZSqrt.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__Natural__Abstract__NAdd.dk b/out/Coq__Numbers__Natural__Abstract__NAdd.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__Natural__Abstract__NAdd.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__Natural__Abstract__NAddOrder.dk b/out/Coq__Numbers__Natural__Abstract__NAddOrder.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__Natural__Abstract__NAddOrder.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__Natural__Abstract__NAxioms.dk b/out/Coq__Numbers__Natural__Abstract__NAxioms.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__Natural__Abstract__NAxioms.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__Natural__Abstract__NBase.dk b/out/Coq__Numbers__Natural__Abstract__NBase.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__Natural__Abstract__NBase.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__Natural__Abstract__NBits.dk b/out/Coq__Numbers__Natural__Abstract__NBits.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__Natural__Abstract__NBits.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__Natural__Abstract__NDiv.dk b/out/Coq__Numbers__Natural__Abstract__NDiv.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__Natural__Abstract__NDiv.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__Natural__Abstract__NGcd.dk b/out/Coq__Numbers__Natural__Abstract__NGcd.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__Natural__Abstract__NGcd.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__Natural__Abstract__NLcm.dk b/out/Coq__Numbers__Natural__Abstract__NLcm.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__Natural__Abstract__NLcm.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__Natural__Abstract__NLog.dk b/out/Coq__Numbers__Natural__Abstract__NLog.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__Natural__Abstract__NLog.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__Natural__Abstract__NMaxMin.dk b/out/Coq__Numbers__Natural__Abstract__NMaxMin.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__Natural__Abstract__NMaxMin.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__Natural__Abstract__NMulOrder.dk b/out/Coq__Numbers__Natural__Abstract__NMulOrder.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__Natural__Abstract__NMulOrder.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__Natural__Abstract__NOrder.dk b/out/Coq__Numbers__Natural__Abstract__NOrder.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__Natural__Abstract__NOrder.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__Natural__Abstract__NParity.dk b/out/Coq__Numbers__Natural__Abstract__NParity.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__Natural__Abstract__NParity.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__Natural__Abstract__NPow.dk b/out/Coq__Numbers__Natural__Abstract__NPow.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__Natural__Abstract__NPow.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__Natural__Abstract__NProperties.dk b/out/Coq__Numbers__Natural__Abstract__NProperties.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__Natural__Abstract__NProperties.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__Natural__Abstract__NSqrt.dk b/out/Coq__Numbers__Natural__Abstract__NSqrt.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__Natural__Abstract__NSqrt.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__Natural__Abstract__NSub.dk b/out/Coq__Numbers__Natural__Abstract__NSub.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__Natural__Abstract__NSub.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Numbers__NumPrelude.dk b/out/Coq__Numbers__NumPrelude.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Numbers__NumPrelude.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Program__Basics.dk b/out/Coq__Program__Basics.dk new file mode 100644 index 000000000..b8763fdef --- /dev/null +++ b/out/Coq__Program__Basics.dk @@ -0,0 +1,109 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def compose : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + C : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + g : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + C : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + g : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + g (f x). + +def arrow : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => B). + +def impl : + A : cupicef.Univ cupicef.prop -> + B : cupicef.Univ cupicef.prop -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ cupicef.prop => + B : cupicef.Univ cupicef.prop => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I A + (__ : cupicef.Term cupicef.prop A => B). + +def const : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + a. + +def flip : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + C : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + C : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) C) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + f y x. + +def apply : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + f x. + +(; End of translation. ;) diff --git a/out/Coq__Program__Tactics.dk b/out/Coq__Program__Tactics.dk new file mode 100644 index 000000000..a538f3dac --- /dev/null +++ b/out/Coq__Program__Tactics.dk @@ -0,0 +1,22 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def fix__proto : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + a. + +def obligation : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + a. + +(; End of translation. ;) diff --git a/out/Coq__Relations__Operators__Properties.dk b/out/Coq__Relations__Operators__Properties.dk new file mode 100644 index 000000000..e0aac1220 --- /dev/null +++ b/out/Coq__Relations__Operators__Properties.dk @@ -0,0 +1,2889 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +thm clos__rt__is__preorder : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.preorder A + (Coq__Relations__Relation__Operators.clos__refl__trans A R)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + Coq__Relations__Relation__Definitions.Build__preorder A + (Coq__Relations__Relation__Operators.clos__refl__trans A R) + (Coq__Relations__Relation__Operators.rt__refl A R) + (Coq__Relations__Relation__Operators.rt__trans A R). + +thm clos__rt__idempotent : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.inclusion A + (Coq__Relations__Relation__Operators.clos__refl__trans A + (Coq__Relations__Relation__Operators.clos__refl__trans A R)) + (Coq__Relations__Relation__Operators.clos__refl__trans A R)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__trans A + (Coq__Relations__Relation__Operators.clos__refl__trans A R) + x y) + (H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A + (Coq__Relations__Relation__Operators.clos__refl__trans + A R) + x y) => + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.clos__refl__trans A R x0 + y0) + x y)))) + (Coq__Relations__Relation__Definitions.inclusion A + (Coq__Relations__Relation__Operators.clos__refl__trans A + (Coq__Relations__Relation__Operators.clos__refl__trans A R)) + (Coq__Relations__Relation__Operators.clos__refl__trans A R)) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A + (Coq__Relations__Relation__Operators.clos__refl__trans A R) x y) => + Coq__Relations__Relation__Operators.clos__refl__trans__ind A + (Coq__Relations__Relation__Operators.clos__refl__trans A R) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.clos__refl__trans A R x0 y0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x0 y0) => + H0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.rt__refl A R x0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A + (Coq__Relations__Relation__Operators.clos__refl__trans A R) + x0 y0) => + IHclos__refl__trans1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x0 y0) => + H1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A + (Coq__Relations__Relation__Operators.clos__refl__trans A R) + y0 z) => + IHclos__refl__trans2 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R y0 z) => + Coq__Relations__Relation__Operators.rt__trans A R x0 y0 z + IHclos__refl__trans1 IHclos__refl__trans2) + x y H). + +thm clos__rt__clos__rst : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.inclusion A + (Coq__Relations__Relation__Operators.clos__refl__trans A R) + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__trans A R x y) + (H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R + x y) => + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.clos__refl__sym__trans A + R x0 y0) + x y)))) + (Coq__Relations__Relation__Definitions.inclusion A + (Coq__Relations__Relation__Operators.clos__refl__trans A R) + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R)) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x y) => + Coq__Relations__Relation__Operators.clos__refl__trans__ind A R + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.clos__refl__sym__trans A R x0 y0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x0 y0) => + Coq__Relations__Relation__Operators.rst__step A R x0 y0 H0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.rst__refl A R x0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x0 y0) => + IHclos__refl__trans1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R + x0 y0) => + H1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R y0 z) => + IHclos__refl__trans2 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R + y0 z) => + Coq__Relations__Relation__Operators.rst__trans A R x0 y0 z + IHclos__refl__trans1 IHclos__refl__trans2) + x y H). + +thm clos__r__clos__rt : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.inclusion A + (Coq__Relations__Relation__Operators.clos__refl A R) + (Coq__Relations__Relation__Operators.clos__refl__trans A R)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__refl A R x y) + (H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl A R x y) => + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.clos__refl__trans A R x + y0) + y)))) + (Coq__Relations__Relation__Definitions.inclusion A + (Coq__Relations__Relation__Operators.clos__refl A R) + (Coq__Relations__Relation__Operators.clos__refl__trans A R)) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl A R x y) => + Coq__Relations__Relation__Operators.clos__refl__ind A R x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.clos__refl__trans A R x y0) + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x y0) => + Coq__Relations__Relation__Operators.rt__step A R x y0 H0) + (Coq__Relations__Relation__Operators.rt__refl A R x) y H). + +thm clos__rt__t : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x y) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R y z) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R x z) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x y) => + Coq__Relations__Relation__Operators.clos__refl__trans__ind A R + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__trans A R y0 z) + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R y0 z) => + Coq__Relations__Relation__Operators.clos__trans A R x0 z)) + (b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + d : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H1 : cupicef.Term cupicef.prop (R b d) => + H0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R d z) => + Coq__Relations__Relation__Operators.t__trans A R b d z + (Coq__Relations__Relation__Operators.t__step A R b d H1) H0) + (b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R b z) => + H0) + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + b : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + d : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R a b) => + H2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R b z) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R a z)) => + IH1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R b d) => + IH2 : + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R d z) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R b z)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R d z) => + H2 (IH2 H0)) + x y H. + +thm clos__rst__is__equiv : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.equivalence A + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + Coq__Relations__Relation__Definitions.Build__equivalence A + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R) + (Coq__Relations__Relation__Operators.rst__refl A R) + (Coq__Relations__Relation__Operators.rst__trans A R) + (Coq__Relations__Relation__Operators.rst__sym A R). + +thm clos__rst__idempotent : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.inclusion A + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R)) + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A + (Coq__Relations__Relation__Operators.clos__refl__sym__trans + A R) + x y) + (H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans + A + (Coq__Relations__Relation__Operators.clos__refl__sym__trans + A R) + x y) => + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.clos__refl__sym__trans A + R x0 y0) + x y)))) + (Coq__Relations__Relation__Definitions.inclusion A + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R)) + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R)) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R) + x y) => + Coq__Relations__Relation__Operators.clos__refl__sym__trans__ind A + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.clos__refl__sym__trans A R x0 y0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R + x0 y0) => + H0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.rst__refl A R x0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A + R) + x0 y0) => + IHclos__refl__sym__trans : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R + x0 y0) => + Coq__Relations__Relation__Operators.rst__sym A R x0 y0 + IHclos__refl__sym__trans) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A + R) + x0 y0) => + IHclos__refl__sym__trans1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R + x0 y0) => + H1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A + R) + y0 z) => + IHclos__refl__sym__trans2 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R + y0 z) => + Coq__Relations__Relation__Operators.rst__trans A R x0 y0 z + IHclos__refl__sym__trans1 IHclos__refl__sym__trans2) + x y H). + +thm clos__t1n__trans : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans__1n A R x y) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R x y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans__1n A R x y) => + Coq__Relations__Relation__Operators.clos__trans__1n__ind A R + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.clos__trans A R x0 y0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x0 y0) => + Coq__Relations__Relation__Operators.t__step A R x0 y0 H0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x0 y0) => + H1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans__1n A R y0 z) => + IHclos__trans__1n : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R y0 z) => + Coq__Relations__Relation__Operators.t__trans A R x0 y0 z + (Coq__Relations__Relation__Operators.t__step A R x0 y0 H0) + IHclos__trans__1n) + x y H. + +thm clos__trans__t1n : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R x y) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans__1n A R x y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R x y) => + Coq__Relations__Relation__Operators.clos__trans__ind A R + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.clos__trans__1n A R x0 y0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x0 y0) => + Coq__Relations__Relation__Operators.t1n__step A R x0 y0 H0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R x0 y0) => + IHclos__trans1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans__1n A R x0 y0) => + H1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R y0 z) => + IHclos__trans2 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans__1n A R y0 z) => + Coq__Relations__Relation__Operators.clos__trans__1n__ind A R + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (Coq__Relations__Relation__Operators.clos__trans A R x1 y1) + (H2 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R x1 y1) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Relations__Relation__Operators.clos__trans A R y1 z) + (H3 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R y1 z) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__trans__1n A R y1 + z) + (IHclos__trans3 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans__1n A + R y1 z) => + Coq__Relations__Relation__Operators.clos__trans__1n A R x1 + z)))) + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H2 : cupicef.Term cupicef.prop (R x1 y1) => + H3 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R x1 y1) => + H4 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R y1 z) => + IHclos__trans3 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans__1n A R y1 z) => + Coq__Relations__Relation__Operators.t1n__trans A R x1 y1 z H2 + IHclos__trans3) + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H2 : cupicef.Term cupicef.prop (R x1 y1) => + IHclos__trans3 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans__1n A R y1 z0) => + IHIHclos__trans1 : + (H3 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R y1 z0) -> + H4 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R z0 z) -> + IHclos__trans4 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans__1n A R z0 z) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans__1n A R y1 z)) => + H3 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R x1 z0) => + H4 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R z0 z) => + IHclos__trans4 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans__1n A R z0 z) => + Coq__Relations__Relation__Operators.t1n__trans A R x1 y1 z H2 + (IHIHclos__trans1 (clos__t1n__trans A R y1 z0 IHclos__trans3) H4 + IHclos__trans4)) + x0 y0 IHclos__trans1 H0 H1 IHclos__trans2) + x y H. + +thm clos__trans__t1n__iff : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Relations__Relation__Operators.clos__trans A R x y) + (Coq__Relations__Relation__Operators.clos__trans__1n A R x y)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__trans A R x y) + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R x y) => + Coq__Relations__Relation__Operators.clos__trans__1n A R x y)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__trans__1n A R x y) + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans__1n A R x y) => + Coq__Relations__Relation__Operators.clos__trans A R x y)) + (clos__trans__t1n A R x y) (clos__t1n__trans A R x y). + +thm clos__tn1__trans : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans__n1 A R x y) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R x y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans__n1 A R x y) => + Coq__Relations__Relation__Operators.clos__trans__n1__ind A R x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.clos__trans A R x y0) + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x y0) => + Coq__Relations__Relation__Operators.t__step A R x y0 H0) + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R y0 z) => + H1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans__n1 A R x y0) => + IHclos__trans__n1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R x y0) => + Coq__Relations__Relation__Operators.t__trans A R x y0 z + IHclos__trans__n1 + (Coq__Relations__Relation__Operators.t__step A R y0 z H0)) + y H. + +thm clos__trans__tn1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R x y) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans__n1 A R x y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R x y) => + Coq__Relations__Relation__Operators.clos__trans__ind A R + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.clos__trans__n1 A R x0 y0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x0 y0) => + Coq__Relations__Relation__Operators.tn1__step A R x0 y0 H0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R x0 y0) => + IHclos__trans1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans__n1 A R x0 y0) => + H1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R y0 z) => + IHclos__trans2 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans__n1 A R y0 z) => + Coq__Relations__Relation__Operators.clos__trans__n1__ind A R y0 + (z0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.clos__trans__n1 A R x0 z0) + (y1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H2 : cupicef.Term cupicef.prop (R y0 y1) => + Coq__Relations__Relation__Operators.tn1__trans A R x0 y0 y1 H2 + IHclos__trans1) + (y1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H2 : cupicef.Term cupicef.prop (R y1 z0) => + H3 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans__n1 A R y0 y1) => + H4 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans__n1 A R x0 y1) => + Coq__Relations__Relation__Operators.tn1__trans A R x0 y1 z0 H2 H4) + z IHclos__trans2) + x y H. + +thm clos__trans__tn1__iff : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Relations__Relation__Operators.clos__trans A R x y) + (Coq__Relations__Relation__Operators.clos__trans__n1 A R x y)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__trans A R x y) + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R x y) => + Coq__Relations__Relation__Operators.clos__trans__n1 A R x y)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__trans__n1 A R x y) + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans__n1 A R x y) => + Coq__Relations__Relation__Operators.clos__trans A R x y)) + (clos__trans__tn1 A R x y) (clos__tn1__trans A R x y). + +thm clos__rt1n__step : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R x y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : cupicef.Term cupicef.prop (R x y) => + Coq__Relations__Relation__Operators.rt1n__trans A R x y y H + (Coq__Relations__Relation__Operators.rt1n__refl A R y). + +thm clos__rtn1__step : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__n1 A R x y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : cupicef.Term cupicef.prop (R x y) => + Coq__Relations__Relation__Operators.rtn1__trans A R x x y H + (Coq__Relations__Relation__Operators.rtn1__refl A R x). + +thm clos__rt1n__rt : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R x y) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R x y) => + Coq__Relations__Relation__Operators.clos__refl__trans__1n__ind A R + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.clos__refl__trans A R x0 y0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.rt__refl A R x0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x0 y0) => + H1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R y0 z) => + IHclos__refl__trans__1n : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R y0 z) => + Coq__Relations__Relation__Operators.rt__trans A R x0 y0 z + (Coq__Relations__Relation__Operators.rt__step A R x0 y0 H0) + IHclos__refl__trans__1n) + x y H. + +thm clos__rt__rt1n : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x y) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R x y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x y) => + Coq__Relations__Relation__Operators.clos__refl__trans__ind A R + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.clos__refl__trans__1n A R x0 y0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x0 y0) => + clos__rt1n__step A R x0 y0 H0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.rt1n__refl A R x0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x0 y0) => + IHclos__refl__trans1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R x0 + y0) => + H1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R y0 z) => + IHclos__refl__trans2 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R y0 z) => + Coq__Relations__Relation__Operators.clos__refl__trans__1n__ind A R + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__trans A R x1 y1) + (H2 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x1 + y1) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__trans A R y1 z) + (H3 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R + y1 z) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A + R y1 z) + (IHclos__refl__trans3 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n + A R y1 z) => + Coq__Relations__Relation__Operators.clos__refl__trans__1n A + R x1 z)))) + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H2 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x1 x1) => + H3 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x1 z) => + IHclos__refl__trans3 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R + x1 z) => + IHclos__refl__trans3) + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H2 : cupicef.Term cupicef.prop (R x1 y1) => + IHclos__refl__trans3 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R + y1 z0) => + IHIHclos__refl__trans1 : + (H3 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R y1 + z0) -> + H4 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R z0 + z) -> + IHclos__refl__trans4 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A + R z0 z) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R + y1 z)) => + H3 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x1 z0) => + H4 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R z0 z) => + IHclos__refl__trans4 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R + z0 z) => + Coq__Relations__Relation__Operators.rt1n__trans A R x1 y1 z H2 + (IHIHclos__refl__trans1 + (clos__rt1n__rt A R y1 z0 IHclos__refl__trans3) H4 + IHclos__refl__trans4)) + x0 y0 IHclos__refl__trans1 H0 H1 IHclos__refl__trans2) + x y H. + +thm clos__rt__rt1n__iff : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Relations__Relation__Operators.clos__refl__trans A R x y) + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R x y)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__trans A R x y) + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x y) => + Coq__Relations__Relation__Operators.clos__refl__trans__1n A R x y)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R x y) + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R x + y) => + Coq__Relations__Relation__Operators.clos__refl__trans A R x y)) + (clos__rt__rt1n A R x y) (clos__rt1n__rt A R x y). + +thm clos__rtn1__rt : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__n1 A R x y) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__n1 A R x y) => + Coq__Relations__Relation__Operators.clos__refl__trans__n1__ind A R x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.clos__refl__trans A R x y0) + (Coq__Relations__Relation__Operators.rt__refl A R x) + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R y0 z) => + H1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__n1 A R x y0) => + IHclos__refl__trans__n1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x y0) => + Coq__Relations__Relation__Operators.rt__trans A R x y0 z + IHclos__refl__trans__n1 + (Coq__Relations__Relation__Operators.rt__step A R y0 z H0)) + y H. + +thm clos__rt__rtn1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x y) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__n1 A R x y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x y) => + Coq__Relations__Relation__Operators.clos__refl__trans__ind A R + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.clos__refl__trans__n1 A R x0 y0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x0 y0) => + clos__rtn1__step A R x0 y0 H0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.rtn1__refl A R x0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x0 y0) => + IHclos__refl__trans1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__n1 A R x0 + y0) => + H1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R y0 z) => + IHclos__refl__trans2 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__n1 A R y0 z) => + Coq__Relations__Relation__Operators.clos__refl__trans__n1__ind A R y0 + (z0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.clos__refl__trans__n1 A R x0 z0) + IHclos__refl__trans1 + (y1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H2 : cupicef.Term cupicef.prop (R y1 z0) => + H3 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__n1 A R + y0 y1) => + H4 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__n1 A R + x0 y1) => + Coq__Relations__Relation__Operators.rtn1__trans A R x0 y1 z0 H2 H4) + z IHclos__refl__trans2) + x y H. + +thm clos__rt__rtn1__iff : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Relations__Relation__Operators.clos__refl__trans A R x y) + (Coq__Relations__Relation__Operators.clos__refl__trans__n1 A R x y)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__trans A R x y) + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x y) => + Coq__Relations__Relation__Operators.clos__refl__trans__n1 A R x y)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__trans__n1 A R x y) + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__n1 A R x + y) => + Coq__Relations__Relation__Operators.clos__refl__trans A R x y)) + (clos__rt__rtn1 A R x y) (clos__rtn1__rt A R x y). + +thm clos__refl__trans__ind__left : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + __ : cupicef.Term cupicef.prop (P x) -> + __0 : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x y) -> + __1 : cupicef.Term cupicef.prop (P y) -> + __2 : cupicef.Term cupicef.prop (R y z) -> + cupicef.Term cupicef.prop (P z)) -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x z) -> + cupicef.Term cupicef.prop (P z) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + H : cupicef.Term cupicef.prop (P x) => + H0 : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x y) -> + __0 : cupicef.Term cupicef.prop (P y) -> + __1 : cupicef.Term cupicef.prop (R y z) -> + cupicef.Term cupicef.prop (P z)) => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x z) => + Coq__Relations__Relation__Operators.clos__refl__trans__ind A R + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.prop)) + cupicef.I (P x0) + (H2 : cupicef.Term cupicef.prop (P x0) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I A + (z1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__trans A + R x0 y) + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans + A R x0 y) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (P y) + (__0 : cupicef.Term cupicef.prop (P y) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (R y z1) + (__1 : cupicef.Term cupicef.prop (R y z1) => P z1)))))) + (H3 : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R + x0 y) -> + __0 : cupicef.Term cupicef.prop (P y) -> + __1 : cupicef.Term cupicef.prop (R y z1) -> + cupicef.Term cupicef.prop (P z1)) => + P z0))) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H2 : cupicef.Term cupicef.prop (R x0 y) => + H3 : cupicef.Term cupicef.prop (P x0) => + H4 : + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x0 y0) -> + __0 : cupicef.Term cupicef.prop (P y0) -> + __1 : cupicef.Term cupicef.prop (R y0 z0) -> + cupicef.Term cupicef.prop (P z0)) => + H4 x0 y (Coq__Relations__Relation__Operators.rt__refl A R x0) H3 H2) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H2 : cupicef.Term cupicef.prop (P x0) => + H3 : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x0 y) -> + __0 : cupicef.Term cupicef.prop (P y) -> + __1 : cupicef.Term cupicef.prop (R y z0) -> + cupicef.Term cupicef.prop (P z0)) => + H2) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H1__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x0 y) => + IHclos__refl__trans1 : + (H2 : cupicef.Term cupicef.prop (P x0) -> + H3 : + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x0 + y0) -> + __0 : cupicef.Term cupicef.prop (P y0) -> + __1 : cupicef.Term cupicef.prop (R y0 z1) -> + cupicef.Term cupicef.prop (P z1)) -> + cupicef.Term cupicef.prop (P y)) => + H1__0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R y z0) => + IHclos__refl__trans2 : + (H2 : cupicef.Term cupicef.prop (P y) -> + H3 : + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R y + y0) -> + __0 : cupicef.Term cupicef.prop (P y0) -> + __1 : cupicef.Term cupicef.prop (R y0 z1) -> + cupicef.Term cupicef.prop (P z1)) -> + cupicef.Term cupicef.prop (P z0)) => + H2 : cupicef.Term cupicef.prop (P x0) => + H3 : + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x0 y0) -> + __0 : cupicef.Term cupicef.prop (P y0) -> + __1 : cupicef.Term cupicef.prop (R y0 z1) -> + cupicef.Term cupicef.prop (P z1)) => + IHclos__refl__trans2 (IHclos__refl__trans1 H2 H3) + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H4 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R y y0) => + H5 : cupicef.Term cupicef.prop (P y0) => + H6 : cupicef.Term cupicef.prop (R y0 z1) => + H3 y0 z1 + (Coq__Relations__Relation__Operators.rt__trans A R x0 y y0 H1__ H4) + H5 H6)) + x z H1 H H0. + +thm rt1n__ind__right : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (P z) -> + __0 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term cupicef.prop (R x y) -> + __1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R y z) -> + __2 : cupicef.Term cupicef.prop (P y) -> + cupicef.Term cupicef.prop (P x)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R x z) -> + cupicef.Term cupicef.prop (P x) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : cupicef.Term cupicef.prop (P z) => + H0 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R y z) -> + __1 : cupicef.Term cupicef.prop (P y) -> + cupicef.Term cupicef.prop (P x)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R x z) => + Coq__Relations__Relation__Operators.clos__refl__trans__1n__ind A R + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.prop)) + cupicef.I (P z0) + (H2 : cupicef.Term cupicef.prop (P z0) => + cupicef.prod + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.prop + (cupicef.rule + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.I A + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I (R x1 y) + (__ : cupicef.Term cupicef.prop (R x1 y) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__trans__1n + A R y z0) + (__0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n + A R y z0) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (P y) + (__1 : cupicef.Term cupicef.prop (P y) => P x1)))))) + (H3 : + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x1 y) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n + A R y z0) -> + __1 : cupicef.Term cupicef.prop (P y) -> + cupicef.Term cupicef.prop (P x1)) => + P x0))) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H2 : cupicef.Term cupicef.prop (P x0) => + H3 : + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x1 y) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R y + x0) -> + __1 : cupicef.Term cupicef.prop (P y) -> + cupicef.Term cupicef.prop (P x1)) => + H2) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H2 : cupicef.Term cupicef.prop (R x0 y) => + H3 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R y z0) => + IHclos__refl__trans__1n : + (H4 : cupicef.Term cupicef.prop (P z0) -> + H5 : + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x1 y0) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A + R y0 z0) -> + __1 : cupicef.Term cupicef.prop (P y0) -> + cupicef.Term cupicef.prop (P x1)) -> + cupicef.Term cupicef.prop (P y)) => + H4 : cupicef.Term cupicef.prop (P z0) => + H5 : + (x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x1 y0) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R + y0 z0) -> + __1 : cupicef.Term cupicef.prop (P y0) -> + cupicef.Term cupicef.prop (P x1)) => + H5 x0 y H2 H3 (IHclos__refl__trans__1n H4 H5)) + x z H1 H H0. + +def let__H : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hz : cupicef.Term cupicef.prop (P z) -> + IH : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + __0 : cupicef.Term cupicef.prop (P y) -> + __1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R y z) -> + cupicef.Term cupicef.prop (P x)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hxz : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x z) -> + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x0 y) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R x0 y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hz : cupicef.Term cupicef.prop (P z) => + IH : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + __0 : cupicef.Term cupicef.prop (P y) -> + __1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R y z) -> + cupicef.Term cupicef.prop (P x)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hxz : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x z) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__trans A R x0 y) + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x0 y) => + Coq__Relations__Relation__Operators.clos__refl__trans__1n A R x0 y)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R x0 y) + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R + x0 y) => + Coq__Relations__Relation__Operators.clos__refl__trans A R x0 y)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__trans A R x0 + y) + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A + R x0 y) => + Coq__Relations__Relation__Operators.clos__refl__trans__1n A + R x0 y)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A + R x0 y) + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n + A R x0 y) => + Coq__Relations__Relation__Operators.clos__refl__trans A R x0 + y))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__trans A R x0 y) + (__0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x0 y) => + Coq__Relations__Relation__Operators.clos__refl__trans__1n A R x0 y)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x0 y) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R x0 + y)) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R + x0 y) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x0 y)) => + __) + (clos__rt__rt1n__iff A R x0 y). + +def let__Hxz : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hz : cupicef.Term cupicef.prop (P z) -> + IH : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + __0 : cupicef.Term cupicef.prop (P y) -> + __1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R y z) -> + cupicef.Term cupicef.prop (P x)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hxz : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x z) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R x z) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hz : cupicef.Term cupicef.prop (P z) => + IH : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + __0 : cupicef.Term cupicef.prop (P y) -> + __1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R y z) -> + cupicef.Term cupicef.prop (P x)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hxz : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x z) => + let__H A R P z Hz IH x Hxz x z Hxz. + +def let__H0 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hz : cupicef.Term cupicef.prop (P z) -> + IH : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + __0 : cupicef.Term cupicef.prop (P y) -> + __1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R y z) -> + cupicef.Term cupicef.prop (P x)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hxz : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x z) -> + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hxy : cupicef.Term cupicef.prop (R x0 y) -> + Hyz : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R y z) -> + Hy : cupicef.Term cupicef.prop (P y) -> + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R x1 y0) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x1 y0) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hz : cupicef.Term cupicef.prop (P z) => + IH : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + __0 : cupicef.Term cupicef.prop (P y) -> + __1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R y z) -> + cupicef.Term cupicef.prop (P x)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hxz : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x z) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hxy : cupicef.Term cupicef.prop (R x0 y) => + Hyz : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R y z) => + Hy : cupicef.Term cupicef.prop (P y) => + x1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__trans A R x1 y0) + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x1 y0) => + Coq__Relations__Relation__Operators.clos__refl__trans__1n A R x1 y0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R x1 y0) + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R + x1 y0) => + Coq__Relations__Relation__Operators.clos__refl__trans A R x1 y0)) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__trans A R x1 + y0) + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A + R x1 y0) => + Coq__Relations__Relation__Operators.clos__refl__trans__1n A + R x1 y0)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A + R x1 y0) + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n + A R x1 y0) => + Coq__Relations__Relation__Operators.clos__refl__trans A R x1 + y0))) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R x1 y0) + (__0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R + x1 y0) => + Coq__Relations__Relation__Operators.clos__refl__trans A R x1 y0)) + (__ : + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x1 y0) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R x1 + y0)) => + __0 : + (__0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R + x1 y0) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x1 y0)) => + __0) + (clos__rt__rt1n__iff A R x1 y0). + +def let__Hyz : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hz : cupicef.Term cupicef.prop (P z) -> + IH : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + __0 : cupicef.Term cupicef.prop (P y) -> + __1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R y z) -> + cupicef.Term cupicef.prop (P x)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hxz : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x z) -> + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + Hxy : cupicef.Term cupicef.prop (R x0 y) -> + Hyz : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R y z) -> + Hy : cupicef.Term cupicef.prop (P y) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R y z) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hz : cupicef.Term cupicef.prop (P z) => + IH : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + __0 : cupicef.Term cupicef.prop (P y) -> + __1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R y z) -> + cupicef.Term cupicef.prop (P x)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hxz : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x z) => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hxy : cupicef.Term cupicef.prop (R x0 y) => + Hyz : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R y z) => + Hy : cupicef.Term cupicef.prop (P y) => + let__H0 A R P z Hz IH x Hxz x0 y Hxy Hyz Hy y z Hyz. + +thm clos__refl__trans__ind__right : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (P z) -> + __0 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term cupicef.prop (R x y) -> + __1 : cupicef.Term cupicef.prop (P y) -> + __2 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R y z) -> + cupicef.Term cupicef.prop (P x)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x z) -> + cupicef.Term cupicef.prop (P x) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hz : cupicef.Term cupicef.prop (P z) => + IH : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + __0 : cupicef.Term cupicef.prop (P y) -> + __1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R y z) -> + cupicef.Term cupicef.prop (P x)) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hxz : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans A R x z) => + rt1n__ind__right A R + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => P x0) z Hz + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Hxy : cupicef.Term cupicef.prop (R x0 y) => + Hyz : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__trans__1n A R y z) => + Hy : cupicef.Term cupicef.prop (P y) => + IH x0 y Hxy Hy (let__Hyz A R P z Hz IH x Hxz x0 y Hxy Hyz Hy)) + x (let__Hxz A R P z Hz IH x Hxz). + +thm clos__rst1n__rst : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A R x y) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R x y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A R x + y) => + Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n__ind A R + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.clos__refl__sym__trans A R x0 y0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.rst__refl A R x0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop (Coq__Init__Logic.or (R x0 y0) (R y0 x0)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A R + y0 z) => + IHclos__refl__sym__trans__1n : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R y0 + z) => + Coq__Relations__Relation__Operators.rst__trans A R x0 y0 z + (Coq__Init__Logic.match____or cupicef.prop (R x0 y0) (R y0 x0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (R x0 y0) (R y0 x0)) => + Coq__Relations__Relation__Operators.clos__refl__sym__trans A R x0 + y0) + (H2 : cupicef.Term cupicef.prop (R x0 y0) => + Coq__Relations__Relation__Operators.rst__step A R x0 y0 H2) + (H2 : cupicef.Term cupicef.prop (R y0 x0) => + Coq__Relations__Relation__Operators.rst__sym A R y0 x0 + (Coq__Relations__Relation__Operators.rst__step A R y0 x0 H2)) + H0) + IHclos__refl__sym__trans__1n) + x y H. + +thm clos__rst1n__trans : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A R x y) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A R y z) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A R x z) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A R x + y) => + Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n__ind A R + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A R + y0 z) + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n + A R y0 z) => + Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A R + x0 z)) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A R + x0 z) => + H0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop (Coq__Init__Logic.or (R x0 y0) (R y0 x0)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A R + y0 z0) => + IHclos__refl__sym__trans__1n : + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n + A R z0 z) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A + R y0 z)) => + H2 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A R + z0 z) => + Coq__Relations__Relation__Operators.rst1n__trans A R x0 y0 z H0 + (IHclos__refl__sym__trans__1n H2)) + x y H. + +thm clos__rst1n__sym : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A R x y) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A R y x) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A R x + y) => + Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n__ind A R + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A R y0 + x0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.rst1n__refl A R x0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + D : cupicef.Term cupicef.prop (Coq__Init__Logic.or (R x0 y0) (R y0 x0)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A R + y0 z) => + H1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A R + z y0) => + clos__rst1n__trans A R z y0 x0 H1 + (Coq__Relations__Relation__Operators.rst1n__trans A R y0 x0 x0 + (Coq__Init__Logic.or__ind (R x0 y0) (R y0 x0) + (Coq__Init__Logic.or (R y0 x0) (R x0 y0)) + (H2 : cupicef.Term cupicef.prop (R x0 y0) => + Coq__Init__Logic.or__intror (R y0 x0) (R x0 y0) H2) + (H2 : cupicef.Term cupicef.prop (R y0 x0) => + Coq__Init__Logic.or__introl (R y0 x0) (R x0 y0) H2) + D) + (Coq__Relations__Relation__Operators.rst1n__refl A R x0))) + x y H. + +thm clos__rst__rst1n : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R x y) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A R x y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R x y) => + Coq__Relations__Relation__Operators.clos__refl__sym__trans__ind A R + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A R x0 + y0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x0 y0) => + Coq__Relations__Relation__Operators.rst1n__trans A R x0 y0 y0 + (Coq__Init__Logic.or__introl (R x0 y0) (R y0 x0) H0) + (Coq__Relations__Relation__Operators.rst1n__refl A R y0)) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.rst1n__refl A R x0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R x0 + y0) => + IHclos__refl__sym__trans : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A R + x0 y0) => + clos__rst1n__sym A R x0 y0 IHclos__refl__sym__trans) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R x0 + y0) => + IHclos__refl__sym__trans1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A R + x0 y0) => + H1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R y0 + z) => + IHclos__refl__sym__trans2 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A R + y0 z) => + clos__rst1n__trans A R x0 y0 z IHclos__refl__sym__trans1 + IHclos__refl__sym__trans2) + x y H. + +thm clos__rst__rst1n__iff : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R x y) + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A R x + y)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R x y) + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R + x y) => + Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A R x + y)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n A R x + y) + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__1n + A R x y) => + Coq__Relations__Relation__Operators.clos__refl__sym__trans A R x y)) + (clos__rst__rst1n A R x y) (clos__rst1n__rst A R x y). + +thm clos__rstn1__rst : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1 A R x y) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R x y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1 A R x + y) => + Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1__ind A R x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.clos__refl__sym__trans A R x y0) + (Coq__Relations__Relation__Operators.rst__refl A R x) + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (Coq__Init__Logic.or (R y0 z) (R z y0)) => + H1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1 A R + x y0) => + IHclos__refl__sym__trans__n1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R x + y0) => + Coq__Relations__Relation__Operators.rst__trans A R x y0 z + IHclos__refl__sym__trans__n1 + (Coq__Init__Logic.match____or cupicef.prop (R y0 z) (R z y0) + (__ : + cupicef.Term cupicef.prop + (Coq__Init__Logic.or (R y0 z) (R z y0)) => + Coq__Relations__Relation__Operators.clos__refl__sym__trans A R y0 + z) + (H2 : cupicef.Term cupicef.prop (R y0 z) => + Coq__Relations__Relation__Operators.rst__step A R y0 z H2) + (H2 : cupicef.Term cupicef.prop (R z y0) => + Coq__Relations__Relation__Operators.rst__sym A R z y0 + (Coq__Relations__Relation__Operators.rst__step A R z y0 H2)) + H0)) + y H. + +thm clos__rstn1__trans : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1 A R x y) -> + __0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1 A R y z) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1 A R x z) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1 A R x + y) => + H2 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1 A R y + z) => + Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1__ind A R y + (z0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1 A R x z0) + H1 + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : cupicef.Term cupicef.prop (Coq__Init__Logic.or (R y0 z0) (R z0 y0)) => + H3 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1 A R + y y0) => + IHclos__refl__sym__trans__n1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1 A R + x y0) => + Coq__Relations__Relation__Operators.rstn1__trans A R x y0 z0 H + IHclos__refl__sym__trans__n1) + z H2. + +thm clos__rstn1__sym : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1 A R x y) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1 A R y x) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1 A R x + y) => + Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1__ind A R x + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1 A R y0 x) + (Coq__Relations__Relation__Operators.rstn1__refl A R x) + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + D : cupicef.Term cupicef.prop (Coq__Init__Logic.or (R y0 z) (R z y0)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1 A R + x y0) => + H1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1 A R + y0 x) => + clos__rstn1__trans A R z y0 x + (Coq__Relations__Relation__Operators.rstn1__trans A R z z y0 + (Coq__Init__Logic.or__ind (R y0 z) (R z y0) + (Coq__Init__Logic.or (R z y0) (R y0 z)) + (H2 : cupicef.Term cupicef.prop (R y0 z) => + Coq__Init__Logic.or__intror (R z y0) (R y0 z) H2) + (H2 : cupicef.Term cupicef.prop (R z y0) => + Coq__Init__Logic.or__introl (R z y0) (R y0 z) H2) + D) + (Coq__Relations__Relation__Operators.rstn1__refl A R z)) + H1) + y H. + +thm clos__rst__rstn1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R x y) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1 A R x y) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R x y) => + Coq__Relations__Relation__Operators.clos__refl__sym__trans__ind A R + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1 A R x0 + y0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x0 y0) => + Coq__Relations__Relation__Operators.rstn1__trans A R x0 x0 y0 + (Coq__Init__Logic.or__introl (R x0 y0) (R y0 x0) H0) + (Coq__Relations__Relation__Operators.rstn1__refl A R x0)) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.rstn1__refl A R x0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R x0 + y0) => + IHclos__refl__sym__trans : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1 A R + x0 y0) => + clos__rstn1__sym A R x0 y0 IHclos__refl__sym__trans) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R x0 + y0) => + IHclos__refl__sym__trans1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1 A R + x0 y0) => + H1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R y0 + z) => + IHclos__refl__sym__trans2 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1 A R + y0 z) => + clos__rstn1__trans A R x0 y0 z IHclos__refl__sym__trans1 + IHclos__refl__sym__trans2) + x y H. + +thm clos__rst__rstn1__iff : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R x y) + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1 A R x + y)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R x y) + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans A R + x y) => + Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1 A R x + y)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1 A R x + y) + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__refl__sym__trans__n1 + A R x y) => + Coq__Relations__Relation__Operators.clos__refl__sym__trans A R x y)) + (clos__rst__rstn1 A R x y) (clos__rstn1__rst A R x y). + +thm clos__trans__transp__permute : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (Coq__Init__Logic.iff + (Coq__Relations__Relation__Operators.transp A + (Coq__Relations__Relation__Operators.clos__trans A R) x y) + (Coq__Relations__Relation__Operators.clos__trans A + (Coq__Relations__Relation__Operators.transp A R) x y)) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.transp A + (Coq__Relations__Relation__Operators.clos__trans A R) x y) + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.transp A + (Coq__Relations__Relation__Operators.clos__trans A R) x y) => + Coq__Relations__Relation__Operators.clos__trans A + (Coq__Relations__Relation__Operators.transp A R) x y)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Operators.clos__trans A + (Coq__Relations__Relation__Operators.transp A R) x y) + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A + (Coq__Relations__Relation__Operators.transp A R) x y) => + Coq__Relations__Relation__Operators.transp A + (Coq__Relations__Relation__Operators.clos__trans A R) x y)) + (H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.transp A + (Coq__Relations__Relation__Operators.clos__trans A R) x y) => + Coq__Relations__Relation__Operators.clos__trans__ind A R + (y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.clos__trans A + (Coq__Relations__Relation__Operators.transp A R) x0 y0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (R x0 y0) => + Coq__Relations__Relation__Operators.t__step A + (Coq__Relations__Relation__Operators.transp A R) y0 x0 H0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R x0 y0) => + IHclos__trans1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A + (Coq__Relations__Relation__Operators.transp A R) y0 x0) => + H1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A R y0 z) => + IHclos__trans2 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A + (Coq__Relations__Relation__Operators.transp A R) z y0) => + Coq__Relations__Relation__Operators.t__trans A + (Coq__Relations__Relation__Operators.transp A R) z y0 x0 + IHclos__trans2 IHclos__trans1) + y x H) + (H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A + (Coq__Relations__Relation__Operators.transp A R) x y) => + Coq__Relations__Relation__Operators.clos__trans__ind A + (Coq__Relations__Relation__Operators.transp A R) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Relations__Relation__Operators.transp A + (Coq__Relations__Relation__Operators.clos__trans A R) x0 y0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.transp A R x0 y0) => + Coq__Relations__Relation__Operators.t__step A R y0 x0 H0) + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A + (Coq__Relations__Relation__Operators.transp A R) x0 y0) => + IHclos__trans1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.transp A + (Coq__Relations__Relation__Operators.clos__trans A R) x0 y0) => + H1 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.clos__trans A + (Coq__Relations__Relation__Operators.transp A R) y0 z) => + IHclos__trans2 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Operators.transp A + (Coq__Relations__Relation__Operators.clos__trans A R) y0 z) => + Coq__Relations__Relation__Operators.t__trans A R z y0 x0 + IHclos__trans2 IHclos__trans1) + x y H). + +(; End of translation. ;) diff --git a/out/Coq__Relations__Relation__Definitions.dk b/out/Coq__Relations__Relation__Definitions.dk new file mode 100644 index 000000000..d64cec17f --- /dev/null +++ b/out/Coq__Relations__Relation__Definitions.dk @@ -0,0 +1,804 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def relation : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cupicef.Univ + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I A + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I A + (__0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)). + +def reflexive : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => R x x). + +def transitive : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (R x y) + (__ : cupicef.Term cupicef.prop (R x y) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + R y z) (__0 : cupicef.Term cupicef.prop (R y z) => R x z))))). + +def symmetric : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R x y) + (__ : cupicef.Term cupicef.prop (R x y) => R y x))). + +def antisymmetric : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (R x y) + (__ : cupicef.Term cupicef.prop (R x y) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R y x) + (__0 : cupicef.Term cupicef.prop (R y x) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))). + +def equiv : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) => + Coq__Init__Logic.and (reflexive A R) + (Coq__Init__Logic.and (transitive A R) (symmetric A R)). + +def preorder : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + cupicef.Univ cupicef.prop. + +def Build__preorder : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + preord__refl0 : cupicef.Term cupicef.prop (reflexive A R) -> + preord__trans0 : cupicef.Term cupicef.prop (transitive A R) -> + cupicef.Term cupicef.prop (preorder A R). + +[ ] cupicef.code_guarded? _ (Build__preorder _ _ _ _) --> cupicef.guarded. + +def match____preorder : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + P : (cupicef.Term cupicef.prop (preorder A R) -> cupicef.Univ s) -> + case__Build__preorder : + (preord__refl0 : cupicef.Term cupicef.prop (reflexive A R) -> + preord__trans0 : cupicef.Term cupicef.prop (transitive A R) -> + cupicef.Term s (P (Build__preorder A R preord__refl0 preord__trans0))) -> + x : cupicef.Term cupicef.prop (preorder A R) -> + cupicef.Term s (P x). + +[ s, A, R, P, case__Build__preorder, preord__refl0, preord__trans0] + match____preorder s A R P case__Build__preorder + (Build__preorder A R preord__refl0 preord__trans0) + --> + case__Build__preorder preord__refl0 preord__trans0. + +[ s, A, R, P, s'] + match____preorder s A R + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____preorder s' A R (x => P x). + +def preord__refl : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + p : cupicef.Term cupicef.prop (preorder A R) -> + cupicef.Term cupicef.prop (reflexive A R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) => + p : cupicef.Term cupicef.prop (preorder A R) => + match____preorder cupicef.prop A R + (p0 : cupicef.Term cupicef.prop (preorder A R) => reflexive A R) + (preord__refl0 : cupicef.Term cupicef.prop (reflexive A R) => + preord__trans0 : cupicef.Term cupicef.prop (transitive A R) => + preord__refl0) + p. + +def preord__trans : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + p : cupicef.Term cupicef.prop (preorder A R) -> + cupicef.Term cupicef.prop (transitive A R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) => + p : cupicef.Term cupicef.prop (preorder A R) => + match____preorder cupicef.prop A R + (p0 : cupicef.Term cupicef.prop (preorder A R) => transitive A R) + (preord__refl0 : cupicef.Term cupicef.prop (reflexive A R) => + preord__trans0 : cupicef.Term cupicef.prop (transitive A R) => + preord__trans0) + p. + +def order : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + cupicef.Univ cupicef.prop. + +def Build__order : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + ord__refl0 : cupicef.Term cupicef.prop (reflexive A R) -> + ord__trans0 : cupicef.Term cupicef.prop (transitive A R) -> + ord__antisym0 : cupicef.Term cupicef.prop (antisymmetric A R) -> + cupicef.Term cupicef.prop (order A R). + +[ ] cupicef.code_guarded? _ (Build__order _ _ _ _ _) --> cupicef.guarded. + +def match____order : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + P : (cupicef.Term cupicef.prop (order A R) -> cupicef.Univ s) -> + case__Build__order : + (ord__refl0 : cupicef.Term cupicef.prop (reflexive A R) -> + ord__trans0 : cupicef.Term cupicef.prop (transitive A R) -> + ord__antisym0 : cupicef.Term cupicef.prop (antisymmetric A R) -> + cupicef.Term s + (P (Build__order A R ord__refl0 ord__trans0 ord__antisym0))) -> + x : cupicef.Term cupicef.prop (order A R) -> + cupicef.Term s (P x). + +[ s, A, R, P, case__Build__order, ord__refl0, ord__trans0, ord__antisym0] + match____order s A R P case__Build__order + (Build__order A R ord__refl0 ord__trans0 ord__antisym0) + --> + case__Build__order ord__refl0 ord__trans0 ord__antisym0. + +[ s, A, R, P, s'] + match____order s A R + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____order s' A R (x => P x). + +def ord__refl : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + o : cupicef.Term cupicef.prop (order A R) -> + cupicef.Term cupicef.prop (reflexive A R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) => + o : cupicef.Term cupicef.prop (order A R) => + match____order cupicef.prop A R + (o0 : cupicef.Term cupicef.prop (order A R) => reflexive A R) + (ord__refl0 : cupicef.Term cupicef.prop (reflexive A R) => + ord__trans0 : cupicef.Term cupicef.prop (transitive A R) => + ord__antisym0 : cupicef.Term cupicef.prop (antisymmetric A R) => + ord__refl0) + o. + +def ord__trans : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + o : cupicef.Term cupicef.prop (order A R) -> + cupicef.Term cupicef.prop (transitive A R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) => + o : cupicef.Term cupicef.prop (order A R) => + match____order cupicef.prop A R + (o0 : cupicef.Term cupicef.prop (order A R) => transitive A R) + (ord__refl0 : cupicef.Term cupicef.prop (reflexive A R) => + ord__trans0 : cupicef.Term cupicef.prop (transitive A R) => + ord__antisym0 : cupicef.Term cupicef.prop (antisymmetric A R) => + ord__trans0) + o. + +def ord__antisym : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + o : cupicef.Term cupicef.prop (order A R) -> + cupicef.Term cupicef.prop (antisymmetric A R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) => + o : cupicef.Term cupicef.prop (order A R) => + match____order cupicef.prop A R + (o0 : cupicef.Term cupicef.prop (order A R) => antisymmetric A R) + (ord__refl0 : cupicef.Term cupicef.prop (reflexive A R) => + ord__trans0 : cupicef.Term cupicef.prop (transitive A R) => + ord__antisym0 : cupicef.Term cupicef.prop (antisymmetric A R) => + ord__antisym0) + o. + +def equivalence : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + cupicef.Univ cupicef.prop. + +def Build__equivalence : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + equiv__refl0 : cupicef.Term cupicef.prop (reflexive A R) -> + equiv__trans0 : cupicef.Term cupicef.prop (transitive A R) -> + equiv__sym0 : cupicef.Term cupicef.prop (symmetric A R) -> + cupicef.Term cupicef.prop (equivalence A R). + +[ ] cupicef.code_guarded? _ (Build__equivalence _ _ _ _ _) --> cupicef.guarded. + +def match____equivalence : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + P : (cupicef.Term cupicef.prop (equivalence A R) -> cupicef.Univ s) -> + case__Build__equivalence : + (equiv__refl0 : cupicef.Term cupicef.prop (reflexive A R) -> + equiv__trans0 : cupicef.Term cupicef.prop (transitive A R) -> + equiv__sym0 : cupicef.Term cupicef.prop (symmetric A R) -> + cupicef.Term s + (P (Build__equivalence A R equiv__refl0 equiv__trans0 equiv__sym0))) -> + x : cupicef.Term cupicef.prop (equivalence A R) -> + cupicef.Term s (P x). + +[ s, A, R, P, case__Build__equivalence, equiv__refl0, equiv__trans0, equiv__sym0] + match____equivalence s A R P case__Build__equivalence + (Build__equivalence A R equiv__refl0 equiv__trans0 equiv__sym0) + --> + case__Build__equivalence equiv__refl0 equiv__trans0 equiv__sym0. + +[ s, A, R, P, s'] + match____equivalence s A R + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____equivalence s' A R (x => P x). + +def equiv__refl : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + e : cupicef.Term cupicef.prop (equivalence A R) -> + cupicef.Term cupicef.prop (reflexive A R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) => + e : cupicef.Term cupicef.prop (equivalence A R) => + match____equivalence cupicef.prop A R + (e0 : cupicef.Term cupicef.prop (equivalence A R) => reflexive A R) + (equiv__refl0 : cupicef.Term cupicef.prop (reflexive A R) => + equiv__trans0 : cupicef.Term cupicef.prop (transitive A R) => + equiv__sym0 : cupicef.Term cupicef.prop (symmetric A R) => + equiv__refl0) + e. + +def equiv__trans : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + e : cupicef.Term cupicef.prop (equivalence A R) -> + cupicef.Term cupicef.prop (transitive A R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) => + e : cupicef.Term cupicef.prop (equivalence A R) => + match____equivalence cupicef.prop A R + (e0 : cupicef.Term cupicef.prop (equivalence A R) => transitive A R) + (equiv__refl0 : cupicef.Term cupicef.prop (reflexive A R) => + equiv__trans0 : cupicef.Term cupicef.prop (transitive A R) => + equiv__sym0 : cupicef.Term cupicef.prop (symmetric A R) => + equiv__trans0) + e. + +def equiv__sym : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + e : cupicef.Term cupicef.prop (equivalence A R) -> + cupicef.Term cupicef.prop (symmetric A R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) => + e : cupicef.Term cupicef.prop (equivalence A R) => + match____equivalence cupicef.prop A R + (e0 : cupicef.Term cupicef.prop (equivalence A R) => symmetric A R) + (equiv__refl0 : cupicef.Term cupicef.prop (reflexive A R) => + equiv__trans0 : cupicef.Term cupicef.prop (transitive A R) => + equiv__sym0 : cupicef.Term cupicef.prop (symmetric A R) => + equiv__sym0) + e. + +def PER : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + cupicef.Univ cupicef.prop. + +def Build__PER : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + per__sym0 : cupicef.Term cupicef.prop (symmetric A R) -> + per__trans0 : cupicef.Term cupicef.prop (transitive A R) -> + cupicef.Term cupicef.prop (PER A R). + +[ ] cupicef.code_guarded? _ (Build__PER _ _ _ _) --> cupicef.guarded. + +def match____PER : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + P : (cupicef.Term cupicef.prop (PER A R) -> cupicef.Univ s) -> + case__Build__PER : + (per__sym0 : cupicef.Term cupicef.prop (symmetric A R) -> + per__trans0 : cupicef.Term cupicef.prop (transitive A R) -> + cupicef.Term s (P (Build__PER A R per__sym0 per__trans0))) -> + x : cupicef.Term cupicef.prop (PER A R) -> + cupicef.Term s (P x). + +[ s, A, R, P, case__Build__PER, per__sym0, per__trans0] + match____PER s A R P case__Build__PER + (Build__PER A R per__sym0 per__trans0) + --> + case__Build__PER per__sym0 per__trans0. + +[ s, A, R, P, s'] + match____PER s A R + (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____PER s' A R (x => P x). + +def per__sym : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + p : cupicef.Term cupicef.prop (PER A R) -> + cupicef.Term cupicef.prop (symmetric A R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) => + p : cupicef.Term cupicef.prop (PER A R) => + match____PER cupicef.prop A R + (p0 : cupicef.Term cupicef.prop (PER A R) => symmetric A R) + (per__sym0 : cupicef.Term cupicef.prop (symmetric A R) => + per__trans0 : cupicef.Term cupicef.prop (transitive A R) => + per__sym0) + p. + +def per__trans : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + p : cupicef.Term cupicef.prop (PER A R) -> + cupicef.Term cupicef.prop (transitive A R) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) => + p : cupicef.Term cupicef.prop (PER A R) => + match____PER cupicef.prop A R + (p0 : cupicef.Term cupicef.prop (PER A R) => transitive A R) + (per__sym0 : cupicef.Term cupicef.prop (symmetric A R) => + per__trans0 : cupicef.Term cupicef.prop (transitive A R) => + per__trans0) + p. + +def inclusion : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R1 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + R2 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R1 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) => + R2 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (R1 x y) + (__ : cupicef.Term cupicef.prop (R1 x y) => R2 x y))). + +def same__relation : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R1 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + R2 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R1 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) => + R2 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) => + Coq__Init__Logic.and (inclusion A R1 R2) (inclusion A R2 R1). + +def commut : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R1 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + R2 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R1 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) => + R2 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (relation A) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I (R1 y x) + (__ : cupicef.Term cupicef.prop (R1 y x) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( + R2 z y) + (__0 : cupicef.Term cupicef.prop (R2 z y) => + Coq__Init__Logic.ex2 A + (y_27_ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + R2 y_27_ x) + (y_27_ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + R1 z y_27_)))))). + +(; End of translation. ;) diff --git a/out/Coq__Relations__Relation__Operators.dk b/out/Coq__Relations__Relation__Operators.dk new file mode 100644 index 000000000..ce8e0a4c3 --- /dev/null +++ b/out/Coq__Relations__Relation__Operators.dk @@ -0,0 +1,4526 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def clos__trans : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop. + +def t__step : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (clos__trans A R x y). + +def t__trans : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (clos__trans A R x y) -> + __0 : cupicef.Term cupicef.prop (clos__trans A R y z) -> + cupicef.Term cupicef.prop (clos__trans A R x z). + +[ ] cupicef.code_guarded? _ (t__step _ _ _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (t__trans _ _ _ _ _ _ _) --> cupicef.guarded. + +def match____clos__trans : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (clos__trans A R x __) -> + cupicef.Univ s) -> + case__t__step : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term s (P y (t__step A R x y __))) -> + case__t__trans : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (clos__trans A R x y) -> + __0 : cupicef.Term cupicef.prop (clos__trans A R y z) -> + cupicef.Term s (P z (t__trans A R x y z __ __0))) -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x0 : cupicef.Term cupicef.prop (clos__trans A R x __) -> + cupicef.Term s (P __ x0). + +[ s, A, R, x, P, case__t__step, case__t__trans, y, __] + match____clos__trans s A R x P case__t__step case__t__trans {y} + (t__step A R x y __) + --> + case__t__step y __. + +[ s, A, R, x, P, case__t__step, case__t__trans, y, z, __, __0] + match____clos__trans s A R x P case__t__step case__t__trans {z} + (t__trans A R x y z __ __0) + --> + case__t__trans y z __ __0. + +[ s, A, R, x, P, s'] + match____clos__trans s A R x + (__ => x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P __ x))) + --> + match____clos__trans s' A R x (__ => x => P __ x). + +def clos__trans__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (P x y)) -> + f0 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (clos__trans A R x y) -> + __0 : cupicef.Term cupicef.prop (P x y) -> + __1 : cupicef.Term cupicef.prop (clos__trans A R y z) -> + __2 : cupicef.Term cupicef.prop (P y z) -> + cupicef.Term cupicef.prop (P x z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + c : cupicef.Term cupicef.prop (clos__trans A R x a) -> + cupicef.Term cupicef.prop (P x a) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (P x y)) => + f0 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (clos__trans A R x y) -> + __0 : cupicef.Term cupicef.prop (P x y) -> + __1 : cupicef.Term cupicef.prop (clos__trans A R y z) -> + __2 : cupicef.Term cupicef.prop (P y z) -> + cupicef.Term cupicef.prop (P x z)) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.2 cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (clos__trans A R x a) + (c : cupicef.Term cupicef.prop (clos__trans A R x a) => + P x a)))))) + (c => + c + (F => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + c : cupicef.Term cupicef.prop (clos__trans A R x a) => + match____clos__trans cupicef.prop A R x + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + __ : cupicef.Term cupicef.prop (clos__trans A R x a0) => + P x a0) + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + r : cupicef.Term cupicef.prop (R x y) => + f x y r) + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + c0 : cupicef.Term cupicef.prop (clos__trans A R x y) => + c1 : cupicef.Term cupicef.prop (clos__trans A R y z) => + f0 x y z c0 (F x y c0) c1 (F y z c1)) + a c)) + cupicef.0. + +def clos__trans__1n : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop. + +def t1n__step : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (clos__trans__1n A R x y). + +def t1n__trans : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + __0 : cupicef.Term cupicef.prop (clos__trans__1n A R y z) -> + cupicef.Term cupicef.prop (clos__trans__1n A R x z). + +[ ] cupicef.code_guarded? _ (t1n__step _ _ _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (t1n__trans _ _ _ _ _ _ _) --> cupicef.guarded. + +def match____clos__trans__1n : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (clos__trans__1n A R x __) -> + cupicef.Univ s) -> + case__t1n__step : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term s (P y (t1n__step A R x y __))) -> + case__t1n__trans : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + __0 : cupicef.Term cupicef.prop (clos__trans__1n A R y z) -> + cupicef.Term s (P z (t1n__trans A R x y z __ __0))) -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x0 : cupicef.Term cupicef.prop (clos__trans__1n A R x __) -> + cupicef.Term s (P __ x0). + +[ s, A, R, x, P, case__t1n__step, case__t1n__trans, y, __] + match____clos__trans__1n s A R x P case__t1n__step case__t1n__trans { + y} (t1n__step A R x y __) + --> + case__t1n__step y __. + +[ s, A, R, x, P, case__t1n__step, case__t1n__trans, y, z, __, __0] + match____clos__trans__1n s A R x P case__t1n__step case__t1n__trans { + z} (t1n__trans A R x y z __ __0) + --> + case__t1n__trans y z __ __0. + +[ s, A, R, x, P, s'] + match____clos__trans__1n s A R x + (__ => x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P __ x))) + --> + match____clos__trans__1n s' A R x (__ => x => P __ x). + +def clos__trans__1n__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (P x y)) -> + f0 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + __0 : cupicef.Term cupicef.prop (clos__trans__1n A R y z) -> + __1 : cupicef.Term cupicef.prop (P y z) -> + cupicef.Term cupicef.prop (P x z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + c : cupicef.Term cupicef.prop (clos__trans__1n A R x a) -> + cupicef.Term cupicef.prop (P x a) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (P x y)) => + f0 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + __0 : cupicef.Term cupicef.prop (clos__trans__1n A R y z) -> + __1 : cupicef.Term cupicef.prop (P y z) -> + cupicef.Term cupicef.prop (P x z)) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.2 cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (clos__trans__1n A R x a) + (c : cupicef.Term cupicef.prop (clos__trans__1n A R x a) => + P x a)))))) + (c => + c + (F => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + c : cupicef.Term cupicef.prop (clos__trans__1n A R x a) => + match____clos__trans__1n cupicef.prop A R x + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + __ : cupicef.Term cupicef.prop (clos__trans__1n A R x a0) => + P x a0) + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + r : cupicef.Term cupicef.prop (R x y) => + f x y r) + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + r : cupicef.Term cupicef.prop (R x y) => + c0 : cupicef.Term cupicef.prop (clos__trans__1n A R y z) => + f0 x y z r c0 (F y z c0)) + a c)) + cupicef.0. + +def clos__trans__n1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop. + +def tn1__step : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (clos__trans__n1 A R x y). + +def tn1__trans : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y z) -> + __0 : cupicef.Term cupicef.prop (clos__trans__n1 A R x y) -> + cupicef.Term cupicef.prop (clos__trans__n1 A R x z). + +[ ] cupicef.code_guarded? _ (tn1__step _ _ _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (tn1__trans _ _ _ _ _ _ _) --> cupicef.guarded. + +def match____clos__trans__n1 : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (clos__trans__n1 A R x __) -> + cupicef.Univ s) -> + case__tn1__step : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term s (P y (tn1__step A R x y __))) -> + case__tn1__trans : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y z) -> + __0 : cupicef.Term cupicef.prop (clos__trans__n1 A R x y) -> + cupicef.Term s (P z (tn1__trans A R x y z __ __0))) -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x0 : cupicef.Term cupicef.prop (clos__trans__n1 A R x __) -> + cupicef.Term s (P __ x0). + +[ s, A, R, x, P, case__tn1__step, case__tn1__trans, y, __] + match____clos__trans__n1 s A R x P case__tn1__step case__tn1__trans { + y} (tn1__step A R x y __) + --> + case__tn1__step y __. + +[ s, A, R, x, P, case__tn1__step, case__tn1__trans, y, z, __, __0] + match____clos__trans__n1 s A R x P case__tn1__step case__tn1__trans { + z} (tn1__trans A R x y z __ __0) + --> + case__tn1__trans y z __ __0. + +[ s, A, R, x, P, s'] + match____clos__trans__n1 s A R x + (__ => x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P __ x))) + --> + match____clos__trans__n1 s' A R x (__ => x => P __ x). + +def clos__trans__n1__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + f : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (P y)) -> + f0 : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y z) -> + __0 : cupicef.Term cupicef.prop (clos__trans__n1 A R x y) -> + __1 : cupicef.Term cupicef.prop (P y) -> + cupicef.Term cupicef.prop (P z)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + c : cupicef.Term cupicef.prop (clos__trans__n1 A R x a) -> + cupicef.Term cupicef.prop (P a) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + f : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (P y)) => + f0 : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y z) -> + __0 : cupicef.Term cupicef.prop (clos__trans__n1 A R x y) -> + __1 : cupicef.Term cupicef.prop (P y) -> + cupicef.Term cupicef.prop (P z)) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (clos__trans__n1 A R x a) + (c : cupicef.Term cupicef.prop (clos__trans__n1 A R x a) => + P a))))) + (c => + c + (F => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + c : cupicef.Term cupicef.prop (clos__trans__n1 A R x a) => + match____clos__trans__n1 cupicef.prop A R x + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + __ : cupicef.Term cupicef.prop (clos__trans__n1 A R x a0) => + P a0) + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + r : cupicef.Term cupicef.prop (R x y) => + f y r) + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + r : cupicef.Term cupicef.prop (R y z) => + c0 : cupicef.Term cupicef.prop (clos__trans__n1 A R x y) => + f0 y z r c0 (F y c0)) + a c)) + cupicef.0. + +def clos__refl : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop. + +def r__step : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (clos__refl A R x y). + +def r__refl : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (clos__refl A R x x). + +[ ] cupicef.code_guarded? _ (r__step _ _ _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (r__refl _ _ _) --> cupicef.guarded. + +def match____clos__refl : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (clos__refl A R x __) -> + cupicef.Univ s) -> + case__r__step : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term s (P y (r__step A R x y __))) -> + case__r__refl : cupicef.Term s (P x (r__refl A R x)) -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x0 : cupicef.Term cupicef.prop (clos__refl A R x __) -> + cupicef.Term s (P __ x0). + +[ s, A, R, x, P, case__r__step, case__r__refl, y, __] + match____clos__refl s A R x P case__r__step case__r__refl {y} + (r__step A R x y __) + --> + case__r__step y __. + +[ s, A, R, x, P, case__r__step, case__r__refl] + match____clos__refl s A R x P case__r__step case__r__refl {x} + (r__refl A R x) + --> + case__r__refl. + +[ s, A, R, x, P, s'] + match____clos__refl s A R x + (__ => x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P __ x))) + --> + match____clos__refl s' A R x (__ => x => P __ x). + +def clos__refl__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + f : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (P y)) -> + f0 : cupicef.Term cupicef.prop (P x) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + c : cupicef.Term cupicef.prop (clos__refl A R x a) -> + cupicef.Term cupicef.prop (P a) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + f : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (P y)) => + f0 : cupicef.Term cupicef.prop (P x) => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + c : cupicef.Term cupicef.prop (clos__refl A R x a) => + match____clos__refl cupicef.prop A R x + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + __ : cupicef.Term cupicef.prop (clos__refl A R x a0) => + P a0) + f f0 a c. + +def clos__refl__trans : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop. + +def rt__step : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (clos__refl__trans A R x y). + +def rt__refl : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (clos__refl__trans A R x x). + +def rt__trans : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (clos__refl__trans A R x y) -> + __0 : cupicef.Term cupicef.prop (clos__refl__trans A R y z) -> + cupicef.Term cupicef.prop (clos__refl__trans A R x z). + +[ ] cupicef.code_guarded? _ (rt__step _ _ _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (rt__refl _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (rt__trans _ _ _ _ _ _ _) --> cupicef.guarded. + +def match____clos__refl__trans : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (clos__refl__trans A R x __) -> + cupicef.Univ s) -> + case__rt__step : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term s (P y (rt__step A R x y __))) -> + case__rt__refl : cupicef.Term s (P x (rt__refl A R x)) -> + case__rt__trans : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (clos__refl__trans A R x y) -> + __0 : cupicef.Term cupicef.prop (clos__refl__trans A R y z) -> + cupicef.Term s (P z (rt__trans A R x y z __ __0))) -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x0 : cupicef.Term cupicef.prop (clos__refl__trans A R x __) -> + cupicef.Term s (P __ x0). + +[ s, A, R, x, P, case__rt__step, case__rt__refl, case__rt__trans, y, __] + match____clos__refl__trans s A R x P case__rt__step case__rt__refl + case__rt__trans {y} (rt__step A R x y __) + --> + case__rt__step y __. + +[ s, A, R, x, P, case__rt__step, case__rt__refl, case__rt__trans] + match____clos__refl__trans s A R x P case__rt__step case__rt__refl + case__rt__trans {x} (rt__refl A R x) + --> + case__rt__refl. + +[ s, A, R, x, P, case__rt__step, case__rt__refl, case__rt__trans, y, z, __, __0] + match____clos__refl__trans s A R x P case__rt__step case__rt__refl + case__rt__trans {z} (rt__trans A R x y z __ __0) + --> + case__rt__trans y z __ __0. + +[ s, A, R, x, P, s'] + match____clos__refl__trans s A R x + (__ => x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P __ x))) + --> + match____clos__refl__trans s' A R x (__ => x => P __ x). + +def clos__refl__trans__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (P x y)) -> + f0 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (P x x)) -> + f1 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (clos__refl__trans A R x y) -> + __0 : cupicef.Term cupicef.prop (P x y) -> + __1 : cupicef.Term cupicef.prop (clos__refl__trans A R y z) -> + __2 : cupicef.Term cupicef.prop (P y z) -> + cupicef.Term cupicef.prop (P x z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + c : cupicef.Term cupicef.prop (clos__refl__trans A R x a) -> + cupicef.Term cupicef.prop (P x a) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (P x y)) => + f0 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (P x x)) => + f1 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (clos__refl__trans A R x y) -> + __0 : cupicef.Term cupicef.prop (P x y) -> + __1 : cupicef.Term cupicef.prop (clos__refl__trans A R y z) -> + __2 : cupicef.Term cupicef.prop (P y z) -> + cupicef.Term cupicef.prop (P x z)) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.2 cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (clos__refl__trans A R x a) + (c : + cupicef.Term cupicef.prop (clos__refl__trans A R x a) => + P x a)))))) + (c => + c + (F => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + c : cupicef.Term cupicef.prop (clos__refl__trans A R x a) => + match____clos__refl__trans cupicef.prop A R x + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + __ : cupicef.Term cupicef.prop (clos__refl__trans A R x a0) => + P x a0) + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + r : cupicef.Term cupicef.prop (R x y) => + f x y r) + (f0 x) + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + c0 : cupicef.Term cupicef.prop (clos__refl__trans A R x y) => + c1 : cupicef.Term cupicef.prop (clos__refl__trans A R y z) => + f1 x y z c0 (F x y c0) c1 (F y z c1)) + a c)) + cupicef.0. + +def clos__refl__trans__1n : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop. + +def rt1n__refl : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (clos__refl__trans__1n A R x x). + +def rt1n__trans : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + __0 : cupicef.Term cupicef.prop (clos__refl__trans__1n A R y z) -> + cupicef.Term cupicef.prop (clos__refl__trans__1n A R x z). + +[ ] cupicef.code_guarded? _ (rt1n__refl _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (rt1n__trans _ _ _ _ _ _ _) --> cupicef.guarded. + +def match____clos__refl__trans__1n : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (clos__refl__trans__1n A R x __) -> + cupicef.Univ s) -> + case__rt1n__refl : cupicef.Term s (P x (rt1n__refl A R x)) -> + case__rt1n__trans : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + __0 : cupicef.Term cupicef.prop (clos__refl__trans__1n A R y z) -> + cupicef.Term s (P z (rt1n__trans A R x y z __ __0))) -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x0 : cupicef.Term cupicef.prop (clos__refl__trans__1n A R x __) -> + cupicef.Term s (P __ x0). + +[ s, A, R, x, P, case__rt1n__refl, case__rt1n__trans] + match____clos__refl__trans__1n s A R x P case__rt1n__refl case__rt1n__trans + {x} (rt1n__refl A R x) + --> + case__rt1n__refl. + +[ s, A, R, x, P, case__rt1n__refl, case__rt1n__trans, y, z, __, __0] + match____clos__refl__trans__1n s A R x P case__rt1n__refl case__rt1n__trans + {z} (rt1n__trans A R x y z __ __0) + --> + case__rt1n__trans y z __ __0. + +[ s, A, R, x, P, s'] + match____clos__refl__trans__1n s A R x + (__ => x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P __ x))) + --> + match____clos__refl__trans__1n s' A R x (__ => x => P __ x). + +def clos__refl__trans__1n__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (P x x)) -> + f0 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + __0 : cupicef.Term cupicef.prop (clos__refl__trans__1n A R y z) -> + __1 : cupicef.Term cupicef.prop (P y z) -> + cupicef.Term cupicef.prop (P x z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + c : cupicef.Term cupicef.prop (clos__refl__trans__1n A R x a) -> + cupicef.Term cupicef.prop (P x a) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (P x x)) => + f0 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + __0 : cupicef.Term cupicef.prop (clos__refl__trans__1n A R y z) -> + __1 : cupicef.Term cupicef.prop (P y z) -> + cupicef.Term cupicef.prop (P x z)) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.2 cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (clos__refl__trans__1n A R x a) + (c : + cupicef.Term cupicef.prop + (clos__refl__trans__1n A R x a) => + P x a)))))) + (c => + c + (F => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + c : cupicef.Term cupicef.prop (clos__refl__trans__1n A R x a) => + match____clos__refl__trans__1n cupicef.prop A R x + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + __ : cupicef.Term cupicef.prop (clos__refl__trans__1n A R x a0) => + P x a0) + (f x) + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + r : cupicef.Term cupicef.prop (R x y) => + c0 : cupicef.Term cupicef.prop (clos__refl__trans__1n A R y z) => + f0 x y z r c0 (F y z c0)) + a c)) + cupicef.0. + +def clos__refl__trans__n1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop. + +def rtn1__refl : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (clos__refl__trans__n1 A R x x). + +def rtn1__trans : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y z) -> + __0 : cupicef.Term cupicef.prop (clos__refl__trans__n1 A R x y) -> + cupicef.Term cupicef.prop (clos__refl__trans__n1 A R x z). + +[ ] cupicef.code_guarded? _ (rtn1__refl _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (rtn1__trans _ _ _ _ _ _ _) --> cupicef.guarded. + +def match____clos__refl__trans__n1 : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (clos__refl__trans__n1 A R x __) -> + cupicef.Univ s) -> + case__rtn1__refl : cupicef.Term s (P x (rtn1__refl A R x)) -> + case__rtn1__trans : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y z) -> + __0 : cupicef.Term cupicef.prop (clos__refl__trans__n1 A R x y) -> + cupicef.Term s (P z (rtn1__trans A R x y z __ __0))) -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x0 : cupicef.Term cupicef.prop (clos__refl__trans__n1 A R x __) -> + cupicef.Term s (P __ x0). + +[ s, A, R, x, P, case__rtn1__refl, case__rtn1__trans] + match____clos__refl__trans__n1 s A R x P case__rtn1__refl case__rtn1__trans + {x} (rtn1__refl A R x) + --> + case__rtn1__refl. + +[ s, A, R, x, P, case__rtn1__refl, case__rtn1__trans, y, z, __, __0] + match____clos__refl__trans__n1 s A R x P case__rtn1__refl case__rtn1__trans + {z} (rtn1__trans A R x y z __ __0) + --> + case__rtn1__trans y z __ __0. + +[ s, A, R, x, P, s'] + match____clos__refl__trans__n1 s A R x + (__ => x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P __ x))) + --> + match____clos__refl__trans__n1 s' A R x (__ => x => P __ x). + +def clos__refl__trans__n1__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + f : cupicef.Term cupicef.prop (P x) -> + f0 : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y z) -> + __0 : cupicef.Term cupicef.prop (clos__refl__trans__n1 A R x y) -> + __1 : cupicef.Term cupicef.prop (P y) -> + cupicef.Term cupicef.prop (P z)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + c : cupicef.Term cupicef.prop (clos__refl__trans__n1 A R x a) -> + cupicef.Term cupicef.prop (P a) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + f : cupicef.Term cupicef.prop (P x) => + f0 : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R y z) -> + __0 : cupicef.Term cupicef.prop (clos__refl__trans__n1 A R x y) -> + __1 : cupicef.Term cupicef.prop (P y) -> + cupicef.Term cupicef.prop (P z)) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (clos__refl__trans__n1 A R x a) + (c : + cupicef.Term cupicef.prop (clos__refl__trans__n1 A R x a) => + P a))))) + (c => + c + (F => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + c : cupicef.Term cupicef.prop (clos__refl__trans__n1 A R x a) => + match____clos__refl__trans__n1 cupicef.prop A R x + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + __ : cupicef.Term cupicef.prop (clos__refl__trans__n1 A R x a0) => + P a0) + f + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + r : cupicef.Term cupicef.prop (R y z) => + c0 : cupicef.Term cupicef.prop (clos__refl__trans__n1 A R x y) => + f0 y z r c0 (F y c0)) + a c)) + cupicef.0. + +def clos__refl__sym__trans : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop. + +def rst__step : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (clos__refl__sym__trans A R x y). + +def rst__refl : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (clos__refl__sym__trans A R x x). + +def rst__sym : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (clos__refl__sym__trans A R x y) -> + cupicef.Term cupicef.prop (clos__refl__sym__trans A R y x). + +def rst__trans : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (clos__refl__sym__trans A R x y) -> + __0 : cupicef.Term cupicef.prop (clos__refl__sym__trans A R y z) -> + cupicef.Term cupicef.prop (clos__refl__sym__trans A R x z). + +[ ] cupicef.code_guarded? _ (rst__step _ _ _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (rst__refl _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (rst__sym _ _ _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (rst__trans _ _ _ _ _ _ _) --> cupicef.guarded. + +def match____clos__refl__sym__trans : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (clos__refl__sym__trans A R __ __0) -> + cupicef.Univ s) -> + case__rst__step : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x0 y) -> + cupicef.Term s (P x0 y (rst__step A R x0 y __))) -> + case__rst__refl : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term s (P x0 x0 (rst__refl A R x0))) -> + case__rst__sym : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (clos__refl__sym__trans A R x0 y) -> + cupicef.Term s (P y x0 (rst__sym A R x0 y __))) -> + case__rst__trans : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (clos__refl__sym__trans A R x0 y) -> + __0 : cupicef.Term cupicef.prop (clos__refl__sym__trans A R y z) -> + cupicef.Term s (P x0 z (rst__trans A R x0 y z __ __0))) -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x : cupicef.Term cupicef.prop (clos__refl__sym__trans A R __ __0) -> + cupicef.Term s (P __ __0 x). + +[ s, A, R, P, case__rst__step, case__rst__refl, case__rst__sym, case__rst__trans, x0, y, __] + match____clos__refl__sym__trans s A R P case__rst__step case__rst__refl + case__rst__sym case__rst__trans {x0} {y} (rst__step A R x0 y __) + --> + case__rst__step x0 y __. + +[ s, A, R, P, case__rst__step, case__rst__refl, case__rst__sym, case__rst__trans, x0] + match____clos__refl__sym__trans s A R P case__rst__step case__rst__refl + case__rst__sym case__rst__trans {x0} {x0} (rst__refl A R x0) + --> + case__rst__refl x0. + +[ s, A, R, P, case__rst__step, case__rst__refl, case__rst__sym, case__rst__trans, x0, y, __] + match____clos__refl__sym__trans s A R P case__rst__step case__rst__refl + case__rst__sym case__rst__trans {y} {x0} (rst__sym A R x0 y __) + --> + case__rst__sym x0 y __. + +[ s, A, R, P, case__rst__step, case__rst__refl, case__rst__sym, case__rst__trans, x0, y, z, __, __0] + match____clos__refl__sym__trans s A R P case__rst__step case__rst__refl + case__rst__sym case__rst__trans {x0} {z} (rst__trans A R x0 y z __ __0) + --> + case__rst__trans x0 y z __ __0. + +[ s, A, R, P, s'] + match____clos__refl__sym__trans s A R + (__ => + __0 => + x => + cupicef.uncode _ (cupicef.code (cupicef.cU s') (P __ __0 x))) + --> + match____clos__refl__sym__trans s' A R (__ => __0 => x => P __ __0 x). + +def clos__refl__sym__trans__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (P x y)) -> + f0 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (P x x)) -> + f1 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (clos__refl__sym__trans A R x y) -> + __0 : cupicef.Term cupicef.prop (P x y) -> + cupicef.Term cupicef.prop (P y x)) -> + f2 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (clos__refl__sym__trans A R x y) -> + __0 : cupicef.Term cupicef.prop (P x y) -> + __1 : cupicef.Term cupicef.prop (clos__refl__sym__trans A R y z) -> + __2 : cupicef.Term cupicef.prop (P y z) -> + cupicef.Term cupicef.prop (P x z)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + c : cupicef.Term cupicef.prop (clos__refl__sym__trans A R a a0) -> + cupicef.Term cupicef.prop (P a a0) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (R x y) -> + cupicef.Term cupicef.prop (P x y)) => + f0 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (P x x)) => + f1 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (clos__refl__sym__trans A R x y) -> + __0 : cupicef.Term cupicef.prop (P x y) -> + cupicef.Term cupicef.prop (P y x)) => + f2 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (clos__refl__sym__trans A R x y) -> + __0 : cupicef.Term cupicef.prop (P x y) -> + __1 : cupicef.Term cupicef.prop (clos__refl__sym__trans A R y z) -> + __2 : cupicef.Term cupicef.prop (P y z) -> + cupicef.Term cupicef.prop (P x z)) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.2 cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (clos__refl__sym__trans A R a a0) + (c : + cupicef.Term cupicef.prop + (clos__refl__sym__trans A R a a0) => + P a a0)))))) + (c => + c + (F => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + c : cupicef.Term cupicef.prop (clos__refl__sym__trans A R a a0) => + match____clos__refl__sym__trans cupicef.prop A R + (a1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + a2 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + __ : cupicef.Term cupicef.prop (clos__refl__sym__trans A R a1 a2) => + P a1 a2) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + r : cupicef.Term cupicef.prop (R x y) => + f x y r) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => f0 x) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + c0 : cupicef.Term cupicef.prop (clos__refl__sym__trans A R x y) => + f1 x y c0 (F x y c0)) + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + c0 : cupicef.Term cupicef.prop (clos__refl__sym__trans A R x y) => + c1 : cupicef.Term cupicef.prop (clos__refl__sym__trans A R y z) => + f2 x y z c0 (F x y c0) c1 (F y z c1)) + a a0 c)) + cupicef.0. + +def clos__refl__sym__trans__1n : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop. + +def rst1n__refl : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (clos__refl__sym__trans__1n A R x x). + +def rst1n__trans : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (Coq__Init__Logic.or (R x y) (R y x)) -> + __0 : cupicef.Term cupicef.prop (clos__refl__sym__trans__1n A R y z) -> + cupicef.Term cupicef.prop (clos__refl__sym__trans__1n A R x z). + +[ ] cupicef.code_guarded? _ (rst1n__refl _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (rst1n__trans _ _ _ _ _ _ _) --> cupicef.guarded. + +def match____clos__refl__sym__trans__1n : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (clos__refl__sym__trans__1n A R x __) -> + cupicef.Univ s) -> + case__rst1n__refl : cupicef.Term s (P x (rst1n__refl A R x)) -> + case__rst1n__trans : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (Coq__Init__Logic.or (R x y) (R y x)) -> + __0 : cupicef.Term cupicef.prop (clos__refl__sym__trans__1n A R y z) -> + cupicef.Term s (P z (rst1n__trans A R x y z __ __0))) -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x0 : cupicef.Term cupicef.prop (clos__refl__sym__trans__1n A R x __) -> + cupicef.Term s (P __ x0). + +[ s, A, R, x, P, case__rst1n__refl, case__rst1n__trans] + match____clos__refl__sym__trans__1n s A R x P case__rst1n__refl + case__rst1n__trans {x} (rst1n__refl A R x) + --> + case__rst1n__refl. + +[ s, A, R, x, P, case__rst1n__refl, case__rst1n__trans, y, z, __, __0] + match____clos__refl__sym__trans__1n s A R x P case__rst1n__refl + case__rst1n__trans {z} (rst1n__trans A R x y z __ __0) + --> + case__rst1n__trans y z __ __0. + +[ s, A, R, x, P, s'] + match____clos__refl__sym__trans__1n s A R x + (__ => x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P __ x))) + --> + match____clos__refl__sym__trans__1n s' A R x (__ => x => P __ x). + +def clos__refl__sym__trans__1n__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (P x x)) -> + f0 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (Coq__Init__Logic.or (R x y) (R y x)) -> + __0 : cupicef.Term cupicef.prop (clos__refl__sym__trans__1n A R y z) -> + __1 : cupicef.Term cupicef.prop (P y z) -> + cupicef.Term cupicef.prop (P x z)) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + c : cupicef.Term cupicef.prop (clos__refl__sym__trans__1n A R x a) -> + cupicef.Term cupicef.prop (P x a) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + P : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (P x x)) => + f0 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (Coq__Init__Logic.or (R x y) (R y x)) -> + __0 : cupicef.Term cupicef.prop (clos__refl__sym__trans__1n A R y z) -> + __1 : cupicef.Term cupicef.prop (P y z) -> + cupicef.Term cupicef.prop (P x z)) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.2 cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (clos__refl__sym__trans__1n A R x a) + (c : + cupicef.Term cupicef.prop + (clos__refl__sym__trans__1n A R x a) => + P x a)))))) + (c => + c + (F => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + c : cupicef.Term cupicef.prop (clos__refl__sym__trans__1n A R x a) => + match____clos__refl__sym__trans__1n cupicef.prop A R x + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + __ : + cupicef.Term cupicef.prop (clos__refl__sym__trans__1n A R x a0) => + P x a0) + (f x) + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + o : + cupicef.Term cupicef.prop (Coq__Init__Logic.or (R x y) (R y x)) => + c0 : + cupicef.Term cupicef.prop (clos__refl__sym__trans__1n A R y z) => + f0 x y z o c0 (F y z c0)) + a c)) + cupicef.0. + +def clos__refl__sym__trans__n1 : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop. + +def rstn1__refl : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (clos__refl__sym__trans__n1 A R x x). + +def rstn1__trans : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (Coq__Init__Logic.or (R y z) (R z y)) -> + __0 : cupicef.Term cupicef.prop (clos__refl__sym__trans__n1 A R x y) -> + cupicef.Term cupicef.prop (clos__refl__sym__trans__n1 A R x z). + +[ ] cupicef.code_guarded? _ (rstn1__refl _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (rstn1__trans _ _ _ _ _ _ _) --> cupicef.guarded. + +def match____clos__refl__sym__trans__n1 : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (clos__refl__sym__trans__n1 A R x __) -> + cupicef.Univ s) -> + case__rstn1__refl : cupicef.Term s (P x (rstn1__refl A R x)) -> + case__rstn1__trans : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (Coq__Init__Logic.or (R y z) (R z y)) -> + __0 : cupicef.Term cupicef.prop (clos__refl__sym__trans__n1 A R x y) -> + cupicef.Term s (P z (rstn1__trans A R x y z __ __0))) -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x0 : cupicef.Term cupicef.prop (clos__refl__sym__trans__n1 A R x __) -> + cupicef.Term s (P __ x0). + +[ s, A, R, x, P, case__rstn1__refl, case__rstn1__trans] + match____clos__refl__sym__trans__n1 s A R x P case__rstn1__refl + case__rstn1__trans {x} (rstn1__refl A R x) + --> + case__rstn1__refl. + +[ s, A, R, x, P, case__rstn1__refl, case__rstn1__trans, y, z, __, __0] + match____clos__refl__sym__trans__n1 s A R x P case__rstn1__refl + case__rstn1__trans {z} (rstn1__trans A R x y z __ __0) + --> + case__rstn1__trans y z __ __0. + +[ s, A, R, x, P, s'] + match____clos__refl__sym__trans__n1 s A R x + (__ => x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P __ x))) + --> + match____clos__refl__sym__trans__n1 s' A R x (__ => x => P __ x). + +def clos__refl__sym__trans__n1__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + f : cupicef.Term cupicef.prop (P x) -> + f0 : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (Coq__Init__Logic.or (R y z) (R z y)) -> + __0 : cupicef.Term cupicef.prop (clos__refl__sym__trans__n1 A R x y) -> + __1 : cupicef.Term cupicef.prop (P y) -> + cupicef.Term cupicef.prop (P z)) -> + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + c : cupicef.Term cupicef.prop (clos__refl__sym__trans__n1 A R x a) -> + cupicef.Term cupicef.prop (P a) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + P : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + f : cupicef.Term cupicef.prop (P x) => + f0 : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (Coq__Init__Logic.or (R y z) (R z y)) -> + __0 : cupicef.Term cupicef.prop (clos__refl__sym__trans__n1 A R x y) -> + __1 : cupicef.Term cupicef.prop (P y) -> + cupicef.Term cupicef.prop (P z)) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (clos__refl__sym__trans__n1 A R x a) + (c : + cupicef.Term cupicef.prop + (clos__refl__sym__trans__n1 A R x a) => + P a))))) + (c => + c + (F => + a : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + c : cupicef.Term cupicef.prop (clos__refl__sym__trans__n1 A R x a) => + match____clos__refl__sym__trans__n1 cupicef.prop A R x + (a0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + __ : + cupicef.Term cupicef.prop (clos__refl__sym__trans__n1 A R x a0) => + P a0) + f + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + o : + cupicef.Term cupicef.prop (Coq__Init__Logic.or (R y z) (R z y)) => + c0 : + cupicef.Term cupicef.prop (clos__refl__sym__trans__n1 A R x y) => + f0 y z o c0 (F y c0)) + a c)) + cupicef.0. + +def transp : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + R y x. + +def union : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R1 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + R2 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R1 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + R2 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.or (R1 x y) (R2 x y). + +def le__AsB : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + leA : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + leB : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) -> + __ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) cupicef.set) + (Coq__Init__Datatypes.sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + __0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) cupicef.set) + (Coq__Init__Datatypes.sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ cupicef.prop. + +def le__aa : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + leA : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + leB : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (leA x y) -> + cupicef.Term cupicef.prop + (le__AsB A B leA leB (Coq__Init__Datatypes.inl A B x) + (Coq__Init__Datatypes.inl A B y)). + +def le__ab : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + leA : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + leB : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term cupicef.prop + (le__AsB A B leA leB (Coq__Init__Datatypes.inl A B x) + (Coq__Init__Datatypes.inr A B y)). + +def le__bb : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + leA : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + leB : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __ : cupicef.Term cupicef.prop (leB x y) -> + cupicef.Term cupicef.prop + (le__AsB A B leA leB (Coq__Init__Datatypes.inr A B x) + (Coq__Init__Datatypes.inr A B y)). + +[ ] cupicef.code_guarded? _ (le__aa _ _ _ _ _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (le__ab _ _ _ _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (le__bb _ _ _ _ _ _ _) --> cupicef.guarded. + +def match____le__AsB : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + leA : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + leB : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) -> + P : + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) cupicef.set) + (Coq__Init__Datatypes.sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + __0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) cupicef.set) + (Coq__Init__Datatypes.sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Term cupicef.prop (le__AsB A B leA leB __ __0) -> + cupicef.Univ s) -> + case__le__aa : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (leA x0 y) -> + cupicef.Term s + (P (Coq__Init__Datatypes.inl A B x0) (Coq__Init__Datatypes.inl A B y) + (le__aa A B leA leB x0 y __))) -> + case__le__ab : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term s + (P (Coq__Init__Datatypes.inl A B x0) (Coq__Init__Datatypes.inr A B y) + (le__ab A B leA leB x0 y))) -> + case__le__bb : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __ : cupicef.Term cupicef.prop (leB x0 y) -> + cupicef.Term s + (P (Coq__Init__Datatypes.inr A B x0) (Coq__Init__Datatypes.inr A B y) + (le__bb A B leA leB x0 y __))) -> + __ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) cupicef.set) + (Coq__Init__Datatypes.sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + __0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) cupicef.set) + (Coq__Init__Datatypes.sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + x : cupicef.Term cupicef.prop (le__AsB A B leA leB __ __0) -> + cupicef.Term s (P __ __0 x). + +[ s, A, B, leA, leB, P, case__le__aa, case__le__ab, case__le__bb, x0, y, __] + match____le__AsB s A B leA leB P case__le__aa case__le__ab case__le__bb + {Coq__Init__Datatypes.inl A B x0} {Coq__Init__Datatypes.inl A B y} + (le__aa A B leA leB x0 y __) + --> + case__le__aa x0 y __. + +[ s, A, B, leA, leB, P, case__le__aa, case__le__ab, case__le__bb, x0, y] + match____le__AsB s A B leA leB P case__le__aa case__le__ab case__le__bb + {Coq__Init__Datatypes.inl A B x0} {Coq__Init__Datatypes.inr A B y} + (le__ab A B leA leB x0 y) + --> + case__le__ab x0 y. + +[ s, A, B, leA, leB, P, case__le__aa, case__le__ab, case__le__bb, x0, y, __] + match____le__AsB s A B leA leB P case__le__aa case__le__ab case__le__bb + {Coq__Init__Datatypes.inr A B x0} {Coq__Init__Datatypes.inr A B y} + (le__bb A B leA leB x0 y __) + --> + case__le__bb x0 y __. + +[ s, A, B, leA, leB, P, s'] + match____le__AsB s A B leA leB + (__ => + __0 => + x => + cupicef.uncode _ (cupicef.code (cupicef.cU s') (P __ __0 x))) + --> + match____le__AsB s' A B leA leB (__ => __0 => x => P __ __0 x). + +def le__AsB__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + leA : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + leB : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) -> + P : + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) cupicef.set) + (Coq__Init__Datatypes.sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + __0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) cupicef.set) + (Coq__Init__Datatypes.sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ cupicef.prop) -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (leA x y) -> + cupicef.Term cupicef.prop + (P (Coq__Init__Datatypes.inl A B x) (Coq__Init__Datatypes.inl A B y))) -> + f0 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term cupicef.prop + (P (Coq__Init__Datatypes.inl A B x) (Coq__Init__Datatypes.inr A B y))) -> + f1 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __ : cupicef.Term cupicef.prop (leB x y) -> + cupicef.Term cupicef.prop + (P (Coq__Init__Datatypes.inr A B x) (Coq__Init__Datatypes.inr A B y))) -> + s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) cupicef.set) + (Coq__Init__Datatypes.sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + s0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) cupicef.set) + (Coq__Init__Datatypes.sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + l : cupicef.Term cupicef.prop (le__AsB A B leA leB s s0) -> + cupicef.Term cupicef.prop (P s s0) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + leA : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + leB : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) => + P : + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) cupicef.set) + (Coq__Init__Datatypes.sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + __0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) cupicef.set) + (Coq__Init__Datatypes.sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ cupicef.prop) => + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (leA x y) -> + cupicef.Term cupicef.prop + (P (Coq__Init__Datatypes.inl A B x) (Coq__Init__Datatypes.inl A B y))) => + f0 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term cupicef.prop + (P (Coq__Init__Datatypes.inl A B x) (Coq__Init__Datatypes.inr A B y))) => + f1 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __ : cupicef.Term cupicef.prop (leB x y) -> + cupicef.Term cupicef.prop + (P (Coq__Init__Datatypes.inr A B x) (Coq__Init__Datatypes.inr A B y))) => + s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) cupicef.set) + (Coq__Init__Datatypes.sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + s0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) cupicef.set) + (Coq__Init__Datatypes.sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + l : cupicef.Term cupicef.prop (le__AsB A B leA leB s s0) => + match____le__AsB cupicef.prop A B leA leB + (s1 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) cupicef.set) + (Coq__Init__Datatypes.sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + s2 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) cupicef.set) + (Coq__Init__Datatypes.sum (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + __ : cupicef.Term cupicef.prop (le__AsB A B leA leB s1 s2) => + P s1 s2) + f f0 f1 s s0 l. + +def lexprod : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + leA : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + leB : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x) -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x) -> + cupicef.Univ cupicef.prop) -> + __ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + __0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ cupicef.prop. + +def left__lex : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + leA : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + leB : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x) -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x) -> + cupicef.Univ cupicef.prop) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x) -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x_27_) -> + __ : cupicef.Term cupicef.prop (leA x x_27_) -> + cupicef.Term cupicef.prop + (lexprod A B leA leB (Coq__Init__Specif.existT A B x y) + (Coq__Init__Specif.existT A B x_27_ y_27_)). + +def right__lex : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + leA : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + leB : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x) -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x) -> + cupicef.Univ cupicef.prop) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x) -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x) -> + __ : cupicef.Term cupicef.prop (leB x y y_27_) -> + cupicef.Term cupicef.prop + (lexprod A B leA leB (Coq__Init__Specif.existT A B x y) + (Coq__Init__Specif.existT A B x y_27_)). + +[ ] cupicef.code_guarded? _ (left__lex _ _ _ _ _ _ _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (right__lex _ _ _ _ _ _ _ _) --> cupicef.guarded. + +def match____lexprod : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + leA : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + leB : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x) -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x) -> + cupicef.Univ cupicef.prop) -> + P : + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + __0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Term cupicef.prop (lexprod A B leA leB __ __0) -> + cupicef.Univ s) -> + case__left__lex : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x0) -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x_27_) -> + __ : cupicef.Term cupicef.prop (leA x0 x_27_) -> + cupicef.Term s + (P (Coq__Init__Specif.existT A B x0 y) + (Coq__Init__Specif.existT A B x_27_ y_27_) + (left__lex A B leA leB x0 x_27_ y y_27_ __))) -> + case__right__lex : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x0) -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x0) -> + __ : cupicef.Term cupicef.prop (leB x0 y y_27_) -> + cupicef.Term s + (P (Coq__Init__Specif.existT A B x0 y) + (Coq__Init__Specif.existT A B x0 y_27_) + (right__lex A B leA leB x0 y y_27_ __))) -> + __ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + __0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + x : cupicef.Term cupicef.prop (lexprod A B leA leB __ __0) -> + cupicef.Term s (P __ __0 x). + +[ s, A, B, leA, leB, P, case__left__lex, case__right__lex, x0, x_27_, y, y_27_, __] + match____lexprod s A B leA leB P case__left__lex case__right__lex + {Coq__Init__Specif.existT A B x0 y} + {Coq__Init__Specif.existT A B x_27_ y_27_} + (left__lex A B leA leB x0 x_27_ y y_27_ __) + --> + case__left__lex x0 x_27_ y y_27_ __. + +[ s, A, B, leA, leB, P, case__left__lex, case__right__lex, x0, y, y_27_, __] + match____lexprod s A B leA leB P case__left__lex case__right__lex + {Coq__Init__Specif.existT A B x0 y} + {Coq__Init__Specif.existT A B x0 y_27_} + (right__lex A B leA leB x0 y y_27_ __) + --> + case__right__lex x0 y y_27_ __. + +[ s, A, B, leA, leB, P, s'] + match____lexprod s A B leA leB + (__ => + __0 => + x => + cupicef.uncode _ (cupicef.code (cupicef.cU s') (P __ __0 x))) + --> + match____lexprod s' A B leA leB (__ => __0 => x => P __ __0 x). + +def lexprod__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + leA : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + leB : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x) -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x) -> + cupicef.Univ cupicef.prop) -> + P : + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + __0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ cupicef.prop) -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x) -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x_27_) -> + __ : cupicef.Term cupicef.prop (leA x x_27_) -> + cupicef.Term cupicef.prop + (P (Coq__Init__Specif.existT A B x y) + (Coq__Init__Specif.existT A B x_27_ y_27_))) -> + f0 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x) -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x) -> + __ : cupicef.Term cupicef.prop (leB x y y_27_) -> + cupicef.Term cupicef.prop + (P (Coq__Init__Specif.existT A B x y) + (Coq__Init__Specif.existT A B x y_27_))) -> + s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + s0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + l : cupicef.Term cupicef.prop (lexprod A B leA leB s s0) -> + cupicef.Term cupicef.prop (P s s0) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + leA : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + leB : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x) -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x) -> + cupicef.Univ cupicef.prop) => + P : + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + __0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ cupicef.prop) => + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x) -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x_27_) -> + __ : cupicef.Term cupicef.prop (leA x x_27_) -> + cupicef.Term cupicef.prop + (P (Coq__Init__Specif.existT A B x y) + (Coq__Init__Specif.existT A B x_27_ y_27_))) => + f0 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x) -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (B x) -> + __ : cupicef.Term cupicef.prop (leB x y y_27_) -> + cupicef.Term cupicef.prop + (P (Coq__Init__Specif.existT A B x y) + (Coq__Init__Specif.existT A B x y_27_))) => + s : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + s0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + l : cupicef.Term cupicef.prop (lexprod A B leA leB s s0) => + match____lexprod cupicef.prop A B leA leB + (s1 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + s2 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Specif.sigT (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + __ : cupicef.Term cupicef.prop (lexprod A B leA leB s1 s2) => + P s1 s2) + f f0 s s0 l. + +def symprod : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + leA : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + leB : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) -> + __ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + __0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ cupicef.prop. + +def left__sym : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + leA : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + leB : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (leA x x_27_) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term cupicef.prop + (symprod A B leA leB (Coq__Init__Datatypes.pair A B x y) + (Coq__Init__Datatypes.pair A B x_27_ y)). + +def right__sym : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + leA : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + leB : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __ : cupicef.Term cupicef.prop (leB y y_27_) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (symprod A B leA leB (Coq__Init__Datatypes.pair A B x y) + (Coq__Init__Datatypes.pair A B x y_27_)). + +[ ] cupicef.code_guarded? _ (left__sym _ _ _ _ _ _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (right__sym _ _ _ _ _ _ _ _) --> cupicef.guarded. + +def match____symprod : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + leA : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + leB : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) -> + P : + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + __0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Term cupicef.prop (symprod A B leA leB __ __0) -> + cupicef.Univ s) -> + case__left__sym : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (leA x0 x_27_) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term s + (P (Coq__Init__Datatypes.pair A B x0 y) + (Coq__Init__Datatypes.pair A B x_27_ y) + (left__sym A B leA leB x0 x_27_ __ y))) -> + case__right__sym : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __ : cupicef.Term cupicef.prop (leB y y_27_) -> + x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term s + (P (Coq__Init__Datatypes.pair A B x0 y) + (Coq__Init__Datatypes.pair A B x0 y_27_) + (right__sym A B leA leB y y_27_ __ x0))) -> + __ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + __0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + x : cupicef.Term cupicef.prop (symprod A B leA leB __ __0) -> + cupicef.Term s (P __ __0 x). + +[ s, A, B, leA, leB, P, case__left__sym, case__right__sym, x0, x_27_, __, y] + match____symprod s A B leA leB P case__left__sym case__right__sym + {Coq__Init__Datatypes.pair A B x0 y} + {Coq__Init__Datatypes.pair A B x_27_ y} + (left__sym A B leA leB x0 x_27_ __ y) + --> + case__left__sym x0 x_27_ __ y. + +[ s, A, B, leA, leB, P, case__left__sym, case__right__sym, y, y_27_, __, x0] + match____symprod s A B leA leB P case__left__sym case__right__sym + {Coq__Init__Datatypes.pair A B x0 y} + {Coq__Init__Datatypes.pair A B x0 y_27_} + (right__sym A B leA leB y y_27_ __ x0) + --> + case__right__sym y y_27_ __ x0. + +[ s, A, B, leA, leB, P, s'] + match____symprod s A B leA leB + (__ => + __0 => + x => + cupicef.uncode _ (cupicef.code (cupicef.cU s') (P __ __0 x))) + --> + match____symprod s' A B leA leB (__ => __0 => x => P __ __0 x). + +def symprod__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + leA : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + leB : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) -> + P : + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + __0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ cupicef.prop) -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (leA x x_27_) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term cupicef.prop + (P (Coq__Init__Datatypes.pair A B x y) + (Coq__Init__Datatypes.pair A B x_27_ y))) -> + f0 : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __ : cupicef.Term cupicef.prop (leB y y_27_) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (P (Coq__Init__Datatypes.pair A B x y) + (Coq__Init__Datatypes.pair A B x y_27_))) -> + p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + p0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + s : cupicef.Term cupicef.prop (symprod A B leA leB p p0) -> + cupicef.Term cupicef.prop (P p p0) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + leA : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + leB : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Univ cupicef.prop) => + P : + (__ : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + __0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) -> + cupicef.Univ cupicef.prop) => + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + x_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (leA x x_27_) -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + cupicef.Term cupicef.prop + (P (Coq__Init__Datatypes.pair A B x y) + (Coq__Init__Datatypes.pair A B x_27_ y))) => + f0 : + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + y_27_ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B -> + __ : cupicef.Term cupicef.prop (leB y y_27_) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop + (P (Coq__Init__Datatypes.pair A B x y) + (Coq__Init__Datatypes.pair A B x y_27_))) => + p : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + p0 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + s : cupicef.Term cupicef.prop (symprod A B leA leB p p0) => + match____symprod cupicef.prop A B leA leB + (p1 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + p2 : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A B) => + __ : cupicef.Term cupicef.prop (symprod A B leA leB p1 p2) => + P p1 p2) + f f0 p p0 s. + +def swapprod : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + __ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A A) -> + __0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A A) -> + cupicef.Univ cupicef.prop. + +def sp__noswap : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A A) -> + __ : + cupicef.Term cupicef.prop + (symprod A A R R (Coq__Init__Datatypes.pair A A x y) p) -> + cupicef.Term cupicef.prop + (swapprod A R (Coq__Init__Datatypes.pair A A x y) p). + +def sp__swap : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A A) -> + __ : + cupicef.Term cupicef.prop + (symprod A A R R (Coq__Init__Datatypes.pair A A x y) p) -> + cupicef.Term cupicef.prop + (swapprod A R (Coq__Init__Datatypes.pair A A y x) p). + +[ ] cupicef.code_guarded? _ (sp__noswap _ _ _ _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (sp__swap _ _ _ _ _ _) --> cupicef.guarded. + +def match____swapprod : + s : cupicef.Sort -> + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + P : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A A) -> + __0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A A) -> + cupicef.Term cupicef.prop (swapprod A R __ __0) -> + cupicef.Univ s) -> + case__sp__noswap : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A A) -> + __ : + cupicef.Term cupicef.prop + (symprod A A R R (Coq__Init__Datatypes.pair A A x0 y) p) -> + cupicef.Term s + (P (Coq__Init__Datatypes.pair A A x0 y) p (sp__noswap A R x0 y p __))) -> + case__sp__swap : + (x0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A A) -> + __ : + cupicef.Term cupicef.prop + (symprod A A R R (Coq__Init__Datatypes.pair A A x0 y) p) -> + cupicef.Term s + (P (Coq__Init__Datatypes.pair A A y x0) p (sp__swap A R x0 y p __))) -> + __ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A A) -> + __0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A A) -> + x : cupicef.Term cupicef.prop (swapprod A R __ __0) -> + cupicef.Term s (P __ __0 x). + +[ s, A, R, P, case__sp__noswap, case__sp__swap, x0, y, p, __] + match____swapprod s A R P case__sp__noswap case__sp__swap + {Coq__Init__Datatypes.pair A A x0 y} {p} (sp__noswap A R x0 y p __) + --> + case__sp__noswap x0 y p __. + +[ s, A, R, P, case__sp__noswap, case__sp__swap, x0, y, p, __] + match____swapprod s A R P case__sp__noswap case__sp__swap + {Coq__Init__Datatypes.pair A A y x0} {p} (sp__swap A R x0 y p __) + --> + case__sp__swap x0 y p __. + +[ s, A, R, P, s'] + match____swapprod s A R + (__ => + __0 => + x => + cupicef.uncode _ (cupicef.code (cupicef.cU s') (P __ __0 x))) + --> + match____swapprod s' A R (__ => __0 => x => P __ __0 x). + +def swapprod__ind : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) -> + P : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A A) -> + __0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A A) -> + cupicef.Univ cupicef.prop) -> + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A A) -> + __ : + cupicef.Term cupicef.prop + (symprod A A R R (Coq__Init__Datatypes.pair A A x y) p) -> + cupicef.Term cupicef.prop (P (Coq__Init__Datatypes.pair A A x y) p)) -> + f0 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A A) -> + __ : + cupicef.Term cupicef.prop + (symprod A A R R (Coq__Init__Datatypes.pair A A x y) p) -> + cupicef.Term cupicef.prop (P (Coq__Init__Datatypes.pair A A y x) p)) -> + p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A A) -> + p0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A A) -> + s : cupicef.Term cupicef.prop (swapprod A R p p0) -> + cupicef.Term cupicef.prop (P p p0) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + R : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Univ cupicef.prop) => + P : + (__ : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A A) -> + __0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A A) -> + cupicef.Univ cupicef.prop) => + f : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A A) -> + __ : + cupicef.Term cupicef.prop + (symprod A A R R (Coq__Init__Datatypes.pair A A x y) p) -> + cupicef.Term cupicef.prop (P (Coq__Init__Datatypes.pair A A x y) p)) => + f0 : + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A A) -> + __ : + cupicef.Term cupicef.prop + (symprod A A R R (Coq__Init__Datatypes.pair A A x y) p) -> + cupicef.Term cupicef.prop (P (Coq__Init__Datatypes.pair A A y x) p)) => + p : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A A) => + p0 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A A) => + s : cupicef.Term cupicef.prop (swapprod A R p p0) => + match____swapprod cupicef.prop A R + (p1 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A A) => + p2 : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) + (Coq__Init__Datatypes.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z)) A A) => + __ : cupicef.Term cupicef.prop (swapprod A R p1 p2) => + P p1 p2) + f f0 p p0 s. + +def Ltl : + A : cupicef.Univ cupicef.set -> + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) -> + __ : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + __0 : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + cupicef.Univ cupicef.prop. + +def Lt__nil : + A : cupicef.Univ cupicef.set -> + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) -> + a : cupicef.Term cupicef.set A -> + x : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + cupicef.Term cupicef.prop + (Ltl A leA + (Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A)) + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + a x)). + +def Lt__hd : + A : cupicef.Univ cupicef.set -> + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) -> + a : cupicef.Term cupicef.set A -> + b : cupicef.Term cupicef.set A -> + __ : cupicef.Term cupicef.prop (leA a b) -> + x : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + y : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + cupicef.Term cupicef.prop + (Ltl A leA + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + a x) + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + b y)). + +def Lt__tl : + A : cupicef.Univ cupicef.set -> + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) -> + a : cupicef.Term cupicef.set A -> + x : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + y : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + __ : cupicef.Term cupicef.prop (Ltl A leA x y) -> + cupicef.Term cupicef.prop + (Ltl A leA + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + a x) + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + a y)). + +[ ] cupicef.code_guarded? _ (Lt__nil _ _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (Lt__hd _ _ _ _ _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (Lt__tl _ _ _ _ _ _) --> cupicef.guarded. + +def match____Ltl : + s : cupicef.Sort -> + A : cupicef.Univ cupicef.set -> + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) -> + P : + (__ : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + __0 : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + cupicef.Term cupicef.prop (Ltl A leA __ __0) -> + cupicef.Univ s) -> + case__Lt__nil : + (a : cupicef.Term cupicef.set A -> + x0 : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + cupicef.Term s + (P + (Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A)) + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + a x0) + (Lt__nil A leA a x0))) -> + case__Lt__hd : + (a : cupicef.Term cupicef.set A -> + b : cupicef.Term cupicef.set A -> + __ : cupicef.Term cupicef.prop (leA a b) -> + x0 : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + y : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + cupicef.Term s + (P + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + a x0) + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + b y) + (Lt__hd A leA a b __ x0 y))) -> + case__Lt__tl : + (a : cupicef.Term cupicef.set A -> + x0 : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + y : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + __ : cupicef.Term cupicef.prop (Ltl A leA x0 y) -> + cupicef.Term s + (P + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + a x0) + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + a y) + (Lt__tl A leA a x0 y __))) -> + __ : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + __0 : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + x : cupicef.Term cupicef.prop (Ltl A leA __ __0) -> + cupicef.Term s (P __ __0 x). + +[ s, A, leA, P, case__Lt__nil, case__Lt__hd, case__Lt__tl, a, x0] + match____Ltl s A leA P case__Lt__nil case__Lt__hd case__Lt__tl + {Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I A)} + {Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I A) + a x0} + (Lt__nil A leA a x0) + --> + case__Lt__nil a x0. + +[ s, A, leA, P, case__Lt__nil, case__Lt__hd, case__Lt__tl, a, b, __, x0, y] + match____Ltl s A leA P case__Lt__nil case__Lt__hd case__Lt__tl + {Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I A) + a x0} + {Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I A) + b y} + (Lt__hd A leA a b __ x0 y) + --> + case__Lt__hd a b __ x0 y. + +[ s, A, leA, P, case__Lt__nil, case__Lt__hd, case__Lt__tl, a, x0, y, __] + match____Ltl s A leA P case__Lt__nil case__Lt__hd case__Lt__tl + {Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I A) + a x0} + {Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I A) + a y} + (Lt__tl A leA a x0 y __) + --> + case__Lt__tl a x0 y __. + +[ s, A, leA, P, s'] + match____Ltl s A leA + (__ => + __0 => + x => + cupicef.uncode _ (cupicef.code (cupicef.cU s') (P __ __0 x))) + --> + match____Ltl s' A leA (__ => __0 => x => P __ __0 x). + +def let__Nil : + A : cupicef.Univ cupicef.set -> + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) -> + cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) + +:= A : cupicef.Univ cupicef.set => + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) => + Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I A). + +def let__List : + A : cupicef.Univ cupicef.set -> + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) -> + cupicef.Univ cupicef.set + +:= A : cupicef.Univ cupicef.set => + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) => + Coq__Init__Datatypes.list cupicef.set A. + +def Ltl__ind : + A : cupicef.Univ cupicef.set -> + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) -> + P : + (__ : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + __0 : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + cupicef.Univ cupicef.prop) -> + f : + (a : cupicef.Term cupicef.set A -> + x : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + cupicef.Term cupicef.prop + (P + (Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A)) + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + a x))) -> + f0 : + (a : cupicef.Term cupicef.set A -> + b : cupicef.Term cupicef.set A -> + __ : cupicef.Term cupicef.prop (leA a b) -> + x : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + y : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + cupicef.Term cupicef.prop + (P + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + a x) + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + b y))) -> + f1 : + (a : cupicef.Term cupicef.set A -> + x : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + y : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + __ : cupicef.Term cupicef.prop (Ltl A leA x y) -> + __0 : cupicef.Term cupicef.prop (P x y) -> + cupicef.Term cupicef.prop + (P + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + a x) + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + a y))) -> + l : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + l0 : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + l1 : cupicef.Term cupicef.prop (Ltl A leA l l0) -> + cupicef.Term cupicef.prop (P l l0) + +:= A : cupicef.Univ cupicef.set => + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) => + P : + (__ : cupicef.Term cupicef.set (let__List A leA) -> + __0 : cupicef.Term cupicef.set (let__List A leA) -> + cupicef.Univ cupicef.prop) => + f : + (a : cupicef.Term cupicef.set A -> + x : cupicef.Term cupicef.set (let__List A leA) -> + cupicef.Term cupicef.prop + (P (let__Nil A leA) + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + a x))) => + f0 : + (a : cupicef.Term cupicef.set A -> + b : cupicef.Term cupicef.set A -> + __ : cupicef.Term cupicef.prop (leA a b) -> + x : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + y : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + cupicef.Term cupicef.prop + (P + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + a x) + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + b y))) => + f1 : + (a : cupicef.Term cupicef.set A -> + x : cupicef.Term cupicef.set (let__List A leA) -> + y : cupicef.Term cupicef.set (let__List A leA) -> + __ : cupicef.Term cupicef.prop (Ltl A leA x y) -> + __0 : cupicef.Term cupicef.prop (P x y) -> + cupicef.Term cupicef.prop + (P + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + a x) + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + a y))) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.2 cupicef.prop + (cupicef.prod cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I (let__List A leA) + (l : cupicef.Term cupicef.set (let__List A leA) => + cupicef.prod cupicef.set + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (let__List A leA) + (l0 : cupicef.Term cupicef.set (let__List A leA) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Ltl A leA l l0) + (l1 : cupicef.Term cupicef.prop (Ltl A leA l l0) => + P l l0)))))) + (c => + c + (F => + l : cupicef.Term cupicef.set (let__List A leA) => + l0 : cupicef.Term cupicef.set (let__List A leA) => + l1 : cupicef.Term cupicef.prop (Ltl A leA l l0) => + match____Ltl cupicef.prop A leA + (l2 : cupicef.Term cupicef.set (let__List A leA) => + l3 : cupicef.Term cupicef.set (let__List A leA) => + __ : cupicef.Term cupicef.prop (Ltl A leA l2 l3) => + P l2 l3) + (a : cupicef.Term cupicef.set A => + x : cupicef.Term cupicef.set (let__List A leA) => + f a x) + (a : cupicef.Term cupicef.set A => + b : cupicef.Term cupicef.set A => + l2 : cupicef.Term cupicef.prop (leA a b) => + x : + cupicef.Term cupicef.set + (Coq__Init__Datatypes.list cupicef.set A) => + y : + cupicef.Term cupicef.set + (Coq__Init__Datatypes.list cupicef.set A) => + f0 a b l2 x y) + (a : cupicef.Term cupicef.set A => + x : cupicef.Term cupicef.set (let__List A leA) => + y : cupicef.Term cupicef.set (let__List A leA) => + l2 : cupicef.Term cupicef.prop (Ltl A leA x y) => + f1 a x y l2 (F x y l2)) + l l0 l1)) + cupicef.0. + +def Desc : + A : cupicef.Univ cupicef.set -> + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) -> + __ : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + cupicef.Univ cupicef.prop. + +def d__nil : + A : cupicef.Univ cupicef.set -> + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) -> + cupicef.Term cupicef.prop + (Desc A leA + (Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A))). + +def d__one : + A : cupicef.Univ cupicef.set -> + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) -> + x : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.prop + (Desc A leA + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + x + (Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A)))). + +def d__conc : + A : cupicef.Univ cupicef.set -> + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) -> + x : cupicef.Term cupicef.set A -> + y : cupicef.Term cupicef.set A -> + l : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + __ : + cupicef.Term cupicef.prop + (clos__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I A) + leA x y) -> + __0 : + cupicef.Term cupicef.prop + (Desc A leA + (Coq__Init__Datatypes.app + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + l + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + y + (Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A))))) -> + cupicef.Term cupicef.prop + (Desc A leA + (Coq__Init__Datatypes.app + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.app + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + l + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + y + (Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A)))) + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + x + (Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A))))). + +[ ] cupicef.code_guarded? _ (d__nil _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (d__one _ _ _) --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ (d__conc _ _ _ _ _ _ _) --> cupicef.guarded. + +def match____Desc : + s : cupicef.Sort -> + A : cupicef.Univ cupicef.set -> + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) -> + P : + (__ : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + cupicef.Term cupicef.prop (Desc A leA __) -> + cupicef.Univ s) -> + case__d__nil : + cupicef.Term s + (P + (Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A)) + (d__nil A leA)) -> + case__d__one : + (x0 : cupicef.Term cupicef.set A -> + cupicef.Term s + (P + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + x0 + (Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A))) + (d__one A leA x0))) -> + case__d__conc : + (x0 : cupicef.Term cupicef.set A -> + y : cupicef.Term cupicef.set A -> + l : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + __ : + cupicef.Term cupicef.prop + (clos__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + leA x0 y) -> + __0 : + cupicef.Term cupicef.prop + (Desc A leA + (Coq__Init__Datatypes.app + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + l + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + y + (Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A))))) -> + cupicef.Term s + (P + (Coq__Init__Datatypes.app + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.app + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + l + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + y + (Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A)))) + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + x0 + (Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A)))) + (d__conc A leA x0 y l __ __0))) -> + __ : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + x : cupicef.Term cupicef.prop (Desc A leA __) -> + cupicef.Term s (P __ x). + +[ s, A, leA, P, case__d__nil, case__d__one, case__d__conc] + match____Desc s A leA P case__d__nil case__d__one case__d__conc + {Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I A)} + (d__nil A leA) + --> + case__d__nil. + +[ s, A, leA, P, case__d__nil, case__d__one, case__d__conc, x0] + match____Desc s A leA P case__d__nil case__d__one case__d__conc + {Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I A) + x0 + (Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A))} + (d__one A leA x0) + --> + case__d__one x0. + +[ s, A, leA, P, case__d__nil, case__d__one, case__d__conc, x0, y, l, __, __0] + match____Desc s A leA P case__d__nil case__d__one case__d__conc + {Coq__Init__Datatypes.app + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.app + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + l + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + y + (Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A)))) + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + x0 + (Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A)))} + (d__conc A leA x0 y l __ __0) + --> + case__d__conc x0 y l __ __0. + +[ s, A, leA, P, s'] + match____Desc s A leA + (__ => x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P __ x))) + --> + match____Desc s' A leA (__ => x => P __ x). + +def let__Nil0 : + A : cupicef.Univ cupicef.set -> + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) -> + cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) + +:= A : cupicef.Univ cupicef.set => + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) => + Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I A). + +def let__List0 : + A : cupicef.Univ cupicef.set -> + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) -> + cupicef.Univ cupicef.set + +:= A : cupicef.Univ cupicef.set => + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) => + Coq__Init__Datatypes.list cupicef.set A. + +def Desc__ind : + A : cupicef.Univ cupicef.set -> + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) -> + P : + (__ : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + cupicef.Univ cupicef.prop) -> + f : + cupicef.Term cupicef.prop + (P + (Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A))) -> + f0 : + (x : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.prop + (P + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + x + (Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A))))) -> + f1 : + (x : cupicef.Term cupicef.set A -> + y : cupicef.Term cupicef.set A -> + l : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + __ : + cupicef.Term cupicef.prop + (clos__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + leA x y) -> + __0 : + cupicef.Term cupicef.prop + (Desc A leA + (Coq__Init__Datatypes.app + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + l + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + y + (Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A))))) -> + __1 : + cupicef.Term cupicef.prop + (P + (Coq__Init__Datatypes.app + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + l + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + y + (Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A))))) -> + cupicef.Term cupicef.prop + (P + (Coq__Init__Datatypes.app + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.app + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + l + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + y + (Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A)))) + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + x + (Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A)))))) -> + l : cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) -> + d : cupicef.Term cupicef.prop (Desc A leA l) -> + cupicef.Term cupicef.prop (P l) + +:= A : cupicef.Univ cupicef.set => + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) => + P : + (__ : cupicef.Term cupicef.set (let__List0 A leA) -> + cupicef.Univ cupicef.prop) => + f : cupicef.Term cupicef.prop (P (let__Nil0 A leA)) => + f0 : + (x : cupicef.Term cupicef.set A -> + cupicef.Term cupicef.prop + (P + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + x (let__Nil0 A leA)))) => + f1 : + (x : cupicef.Term cupicef.set A -> + y : cupicef.Term cupicef.set A -> + l : cupicef.Term cupicef.set (let__List0 A leA) -> + __ : + cupicef.Term cupicef.prop + (clos__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + leA x y) -> + __0 : + cupicef.Term cupicef.prop + (Desc A leA + (Coq__Init__Datatypes.app + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + l + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + y (let__Nil0 A leA)))) -> + __1 : + cupicef.Term cupicef.prop + (P + (Coq__Init__Datatypes.app + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + l + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + y (let__Nil0 A leA)))) -> + cupicef.Term cupicef.prop + (P + (Coq__Init__Datatypes.app + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + (Coq__Init__Datatypes.app + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + l + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + y (let__Nil0 A leA))) + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + x (let__Nil0 A leA))))) => + cupicef.fixproj cupicef.1 + (c => + c + (cupicef.SA cupicef.1 cupicef.prop + (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (let__List0 A leA) + (l : cupicef.Term cupicef.set (let__List0 A leA) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Desc A leA l) + (d : cupicef.Term cupicef.prop (Desc A leA l) => P l))))) + (c => + c + (F => + l : cupicef.Term cupicef.set (let__List0 A leA) => + d : cupicef.Term cupicef.prop (Desc A leA l) => + match____Desc cupicef.prop A leA + (l0 : cupicef.Term cupicef.set (let__List0 A leA) => + __ : cupicef.Term cupicef.prop (Desc A leA l0) => + P l0) + f (x : cupicef.Term cupicef.set A => f0 x) + (x : cupicef.Term cupicef.set A => + y : cupicef.Term cupicef.set A => + l0 : cupicef.Term cupicef.set (let__List0 A leA) => + c : + cupicef.Term cupicef.prop + (clos__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + leA x y) => + d0 : + cupicef.Term cupicef.prop + (Desc A leA + (Coq__Init__Datatypes.app + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + l0 + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + y (let__Nil0 A leA)))) => + f1 x y l0 c d0 + (F + (Coq__Init__Datatypes.app + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + l0 + (Coq__Init__Datatypes.cons + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + y (let__Nil0 A leA))) + d0)) + l d)) + cupicef.0. + +def let__Nil1 : + A : cupicef.Univ cupicef.set -> + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) -> + cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) + +:= A : cupicef.Univ cupicef.set => + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) => + Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I A). + +def let__List1 : + A : cupicef.Univ cupicef.set -> + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) -> + cupicef.Univ cupicef.set + +:= A : cupicef.Univ cupicef.set => + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) => + Coq__Init__Datatypes.list cupicef.set A. + +def Pow : + A : cupicef.Univ cupicef.set -> + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) -> + cupicef.Univ cupicef.set + +:= A : cupicef.Univ cupicef.set => + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) => + Coq__Init__Specif.sig cupicef.set (let__List1 A leA) (Desc A leA). + +def let__Nil2 : + A : cupicef.Univ cupicef.set -> + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) -> + cupicef.Term cupicef.set (Coq__Init__Datatypes.list cupicef.set A) + +:= A : cupicef.Univ cupicef.set => + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) => + Coq__Init__Datatypes.nil + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I A). + +def let__List2 : + A : cupicef.Univ cupicef.set -> + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) -> + cupicef.Univ cupicef.set + +:= A : cupicef.Univ cupicef.set => + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) => + Coq__Init__Datatypes.list cupicef.set A. + +def lex__exp : + A : cupicef.Univ cupicef.set -> + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) -> + a : cupicef.Term cupicef.set (Pow A leA) -> + b : cupicef.Term cupicef.set (Pow A leA) -> + cupicef.Univ cupicef.prop + +:= A : cupicef.Univ cupicef.set => + leA : + (__ : cupicef.Term cupicef.set A -> + __0 : cupicef.Term cupicef.set A -> + cupicef.Univ cupicef.prop) => + a : cupicef.Term cupicef.set (Pow A leA) => + b : cupicef.Term cupicef.set (Pow A leA) => + Ltl A leA + (Coq__Init__Specif.proj1__sig + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I (let__List2 A leA)) + (Desc A leA) a) + (Coq__Init__Specif.proj1__sig + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I (let__List2 A leA)) + (Desc A leA) b). + +(; End of translation. ;) diff --git a/out/Coq__Relations__Relations.dk b/out/Coq__Relations__Relations.dk new file mode 100644 index 000000000..030cebd4e --- /dev/null +++ b/out/Coq__Relations__Relations.dk @@ -0,0 +1,898 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +thm inverse__image__of__equivalence : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + r : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) -> + __ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.equivalence B r) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.equivalence A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + r (f x) (f y))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + r : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation B) => + H : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.equivalence B r) => + Coq__Relations__Relation__Definitions.Build__equivalence A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + r (f x) (f y)) + (Coq__Relations__Relation__Definitions.match____equivalence cupicef.prop + B r + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.equivalence B r) => + Coq__Relations__Relation__Definitions.reflexive A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + r (f x) (f y))) + (cupicef.cast + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop)))) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop))) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop)))) + cupicef.I (Coq__Relations__Relation__Definitions.reflexive B r) + (equiv__refl : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.reflexive B r) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop))) + cupicef.I + (Coq__Relations__Relation__Definitions.transitive B r) + (equiv__trans : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.transitive B r) => + cupicef.prod cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop)) + cupicef.I + (Coq__Relations__Relation__Definitions.symmetric B r) + (equiv__sym : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.symmetric B r) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (x : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + r (f x) (f x)))))) + (cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I (Coq__Relations__Relation__Definitions.reflexive B r) + (equiv__refl : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.reflexive B r) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Relations__Relation__Definitions.transitive B r) + (equiv__trans : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.transitive B r) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Definitions.symmetric B r) + (equiv__sym : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.symmetric B r) => + Coq__Relations__Relation__Definitions.reflexive A + (x : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + r (f x) (f y)))))) + cupicef.I + (equiv__refl : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.reflexive B r) => + equiv__trans : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.transitive B r) => + equiv__sym : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.symmetric B r) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + equiv__refl (f x))) + H) + (Coq__Relations__Relation__Definitions.match____equivalence cupicef.prop + B r + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.equivalence B r) => + Coq__Relations__Relation__Definitions.transitive A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + r (f x) (f y))) + (cupicef.cast + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))))))) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))))) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))))))) + cupicef.I (Coq__Relations__Relation__Definitions.reflexive B r) + (__tmp : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.reflexive B r) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))))) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))))) + cupicef.I + (Coq__Relations__Relation__Definitions.transitive B r) + (equiv__trans : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.transitive B r) => + cupicef.prod cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))))) + cupicef.I + (Coq__Relations__Relation__Definitions.symmetric B r) + (__tmp0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.symmetric B r) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.I A + (x : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I A + (y : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (z : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I (r (f x) (f y)) + (H0 : + cupicef.Term cupicef.prop (r (f x) (f y)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (r (f y) (f z)) + (H1 : + cupicef.Term cupicef.prop + (r (f y) (f z)) => + r (f x) (f z)))))))))) + (cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I (Coq__Relations__Relation__Definitions.reflexive B r) + (equiv__refl : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.reflexive B r) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Relations__Relation__Definitions.transitive B r) + (equiv__trans : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.transitive B r) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Definitions.symmetric B r) + (equiv__sym : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.symmetric B r) => + Coq__Relations__Relation__Definitions.transitive A + (x : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + r (f x) (f y)))))) + cupicef.I + (__tmp : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.reflexive B r) => + equiv__trans : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.transitive B r) => + __tmp0 : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.symmetric B r) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (r (f x) (f y)) => + H1 : cupicef.Term cupicef.prop (r (f y) (f z)) => + equiv__trans (f x) (f y) (f z) H0 H1)) + H) + (Coq__Relations__Relation__Definitions.match____equivalence cupicef.prop + B r + (__ : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.equivalence B r) => + Coq__Relations__Relation__Definitions.symmetric A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + r (f x) (f y))) + (cupicef.cast + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)))))) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))))) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)))))) + cupicef.I (Coq__Relations__Relation__Definitions.reflexive B r) + (equiv__refl : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.reflexive B r) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.I + (Coq__Relations__Relation__Definitions.transitive B r) + (equiv__trans : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.transitive B r) => + cupicef.prod cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I + (Coq__Relations__Relation__Definitions.symmetric B r) + (equiv__sym : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.symmetric B r) => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : + cupicef.Term + (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + cupicef.I (r (f x) (f y)) + (H0 : cupicef.Term cupicef.prop (r (f x) (f y)) => + r (f y) (f x)))))))) + (cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I (Coq__Relations__Relation__Definitions.reflexive B r) + (equiv__refl : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.reflexive B r) => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Relations__Relation__Definitions.transitive B r) + (equiv__trans : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.transitive B r) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Relations__Relation__Definitions.symmetric B r) + (equiv__sym : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.symmetric B r) => + Coq__Relations__Relation__Definitions.symmetric A + (x : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + r (f x) (f y)))))) + cupicef.I + (equiv__refl : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.reflexive B r) => + equiv__trans : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.transitive B r) => + equiv__sym : + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.symmetric B r) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H0 : cupicef.Term cupicef.prop (r (f x) (f y)) => + equiv__sym (f x) (f y) H0)) + H). + +thm inverse__image__of__eq : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) -> + cupicef.Term cupicef.prop + (Coq__Relations__Relation__Definitions.equivalence A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y))) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + B : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + f : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) B) => + Coq__Relations__Relation__Definitions.Build__equivalence A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) cupicef.prop + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f x))) + (Coq__Relations__Relation__Definitions.reflexive A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y))) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x))) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y)) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f y) (f z)) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f y) (f z)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f z))))))) + (Coq__Relations__Relation__Definitions.transitive A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y))) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y)) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f y) (f z)) => + Coq__Init__Logic.eq__trans + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y) (f z) H H0)) + (cupicef.cast + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y)) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y)) => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f y) (f x))))) + (Coq__Relations__Relation__Definitions.symmetric A + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y))) + cupicef.I + (x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y)) => + Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y) H)). + +(; End of translation. ;) diff --git a/out/Coq__Setoids__Setoid.dk b/out/Coq__Setoids__Setoid.dk new file mode 100644 index 000000000..252563c80 --- /dev/null +++ b/out/Coq__Setoids__Setoid.dk @@ -0,0 +1,513 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def Setoid__Theory : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + cupicef.Univ cupicef.prop + +:= Coq__Classes__RelationClasses.Equivalence. + +def Build__Setoid__Theory : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + R : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + Equivalence__Reflexive : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Reflexive A R) -> + Equivalence__Symmetric : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Symmetric A R) -> + Equivalence__Transitive : + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Transitive A R) -> + cupicef.Term cupicef.prop (Coq__Classes__RelationClasses.Equivalence A R) + +:= Coq__Classes__RelationClasses.Build__Equivalence. + +def Seq__refl : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + Aeq : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + s : cupicef.Term cupicef.prop (Setoid__Theory A Aeq) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.prop (Aeq x x) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + Aeq : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + s : cupicef.Term cupicef.prop (Setoid__Theory A Aeq) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Classes__RelationClasses.reflexivity A Aeq + (Coq__Classes__RelationClasses.Equivalence__Reflexive A Aeq s) x. + +def Seq__sym : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + Aeq : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + s : cupicef.Term cupicef.prop (Setoid__Theory A Aeq) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (Aeq x y) -> + cupicef.Term cupicef.prop (Aeq y x) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + Aeq : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + s : cupicef.Term cupicef.prop (Setoid__Theory A Aeq) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : cupicef.Term cupicef.prop (Aeq x y) => + Coq__Classes__RelationClasses.symmetry A Aeq + (Coq__Classes__RelationClasses.Equivalence__Symmetric A Aeq s) x y H. + +def Seq__trans : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + Aeq : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) -> + s : cupicef.Term cupicef.prop (Setoid__Theory A Aeq) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __ : cupicef.Term cupicef.prop (Aeq x y) -> + __0 : cupicef.Term cupicef.prop (Aeq y z) -> + cupicef.Term cupicef.prop (Aeq x z) + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + Aeq : + cupicef.Term + (cupicef.sup (cupicef.type (cupicef.s cupicef.z)) + (cupicef.type (cupicef.s cupicef.z))) + (Coq__Relations__Relation__Definitions.relation A) => + s : cupicef.Term cupicef.prop (Setoid__Theory A Aeq) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + z : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + H : cupicef.Term cupicef.prop (Aeq x y) => + H0 : cupicef.Term cupicef.prop (Aeq y z) => + Coq__Classes__RelationClasses.transitivity A Aeq + (Coq__Classes__RelationClasses.Equivalence__Transitive A Aeq s) x y z H + H0. + +thm gen__st : + A : cupicef.Univ cupicef.set -> + cupicef.Term cupicef.prop + (Setoid__Theory + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I A) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A))) + +:= A : cupicef.Univ cupicef.set => + Coq__Classes__RelationClasses.Build__Equivalence + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) + cupicef.I A) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A)) + (cupicef.cast (cupicef.rule cupicef.set cupicef.prop) cupicef.prop + (cupicef.prod cupicef.set cupicef.prop + (cupicef.rule cupicef.set cupicef.prop) cupicef.I A + (x : cupicef.Term cupicef.set A => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (Coq__Classes__RelationClasses.Reflexive + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A))) + cupicef.I + (x : cupicef.Term cupicef.set A => + Coq__Init__Logic.eq__refl + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + (cupicef.cast + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.prod cupicef.set + (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (x : cupicef.Term cupicef.set A => + cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I A + (y : cupicef.Term cupicef.set A => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y x)))) + (Coq__Classes__RelationClasses.Symmetric + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A))) + cupicef.I + (x : cupicef.Term cupicef.set A => + y : cupicef.Term cupicef.set A => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + Coq__Init__Logic.eq__sym + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y H)) + (cupicef.cast + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.prop + (cupicef.prod cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))))) + cupicef.I A + (x : cupicef.Term cupicef.set A => + cupicef.prod cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)))) + cupicef.I A + (y : cupicef.Term cupicef.set A => + cupicef.prod cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + (cupicef.rule cupicef.set + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop))) + cupicef.I A + (z : cupicef.Term cupicef.set A => + cupicef.prod cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) + (cupicef.rule cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop)) + cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + (H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + cupicef.prod cupicef.prop cupicef.prop + (cupicef.rule cupicef.prop cupicef.prop) cupicef.I + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y z) + (H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast + (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y z) => + Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.type + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x z)))))) + (Coq__Classes__RelationClasses.Transitive + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I A) + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A))) + cupicef.I + (x : cupicef.Term cupicef.set A => + y : cupicef.Term cupicef.set A => + z : cupicef.Term cupicef.set A => + H : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) => + H0 : + cupicef.Term cupicef.prop + (Coq__Init__Logic.eq + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set + (cupicef.type (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y z) => + Coq__Init__Logic.eq__trans + (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y z H H0)). + +(; End of translation. ;) diff --git a/out/Coq__Structures__Equalities.dk b/out/Coq__Structures__Equalities.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Structures__Equalities.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Structures__GenericMinMax.dk b/out/Coq__Structures__GenericMinMax.dk new file mode 100644 index 000000000..2c1b4fec2 --- /dev/null +++ b/out/Coq__Structures__GenericMinMax.dk @@ -0,0 +1,48 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def gmax : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cmp : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.comparison) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + cmp : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.comparison) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Datatypes.match____comparison + (cupicef.type (cupicef.s cupicef.z)) + (x0 : cupicef.Term cupicef.set Coq__Init__Datatypes.comparison => A) x y + x (cmp x y). + +def gmin : + A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) -> + cmp : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.comparison) -> + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A + +:= A : cupicef.Univ (cupicef.type (cupicef.s cupicef.z)) => + cmp : + (__ : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + __0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A -> + cupicef.Term cupicef.set Coq__Init__Datatypes.comparison) => + x : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + y : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) A => + Coq__Init__Datatypes.match____comparison + (cupicef.type (cupicef.s cupicef.z)) + (x0 : cupicef.Term cupicef.set Coq__Init__Datatypes.comparison => A) x x + y (cmp x y). + +(; End of translation. ;) diff --git a/out/Coq__Structures__Orders.dk b/out/Coq__Structures__Orders.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Structures__Orders.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Structures__OrdersFacts.dk b/out/Coq__Structures__OrdersFacts.dk new file mode 100644 index 000000000..0175c2437 --- /dev/null +++ b/out/Coq__Structures__OrdersFacts.dk @@ -0,0 +1,4 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +(; End of translation. ;) diff --git a/out/Coq__Structures__OrdersTac.dk b/out/Coq__Structures__OrdersTac.dk new file mode 100644 index 000000000..298db919d --- /dev/null +++ b/out/Coq__Structures__OrdersTac.dk @@ -0,0 +1,131 @@ +(; This file was automatically generated by Coqine. ;) +(; The encoding used was: "template_cic". ;) + +def ord : + cupicef.Univ cupicef.set. + +def OEQ : + cupicef.Term cupicef.set ord. + +def OLT : + cupicef.Term cupicef.set ord. + +def OLE : + cupicef.Term cupicef.set ord. + +[ ] cupicef.code_guarded? _ OEQ --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ OLT --> cupicef.guarded. + +[ ] cupicef.code_guarded? _ OLE --> cupicef.guarded. + +def match____ord : + s : cupicef.Sort -> + P : (cupicef.Term cupicef.set ord -> cupicef.Univ s) -> + case__OEQ : cupicef.Term s (P OEQ) -> + case__OLT : cupicef.Term s (P OLT) -> + case__OLE : cupicef.Term s (P OLE) -> + x : cupicef.Term cupicef.set ord -> + cupicef.Term s (P x). + +[ s, P, case__OEQ, case__OLT, case__OLE] + match____ord s P case__OEQ case__OLT case__OLE OEQ + --> + case__OEQ. + +[ s, P, case__OEQ, case__OLT, case__OLE] + match____ord s P case__OEQ case__OLT case__OLE OLT + --> + case__OLT. + +[ s, P, case__OEQ, case__OLT, case__OLE] + match____ord s P case__OEQ case__OLT case__OLE OLE + --> + case__OLE. + +[ s, P, s'] + match____ord s (x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P x))) + --> + match____ord s' (x => P x). + +def ord__rect : + P : + (o : cupicef.Term cupicef.set ord -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) -> + f : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P OEQ) -> + f0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P OLT) -> + f1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P OLE) -> + o : cupicef.Term cupicef.set ord -> + cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P o) + +:= P : + (o : cupicef.Term cupicef.set ord -> + cupicef.Univ (cupicef.type (cupicef.s cupicef.z))) => + f : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P OEQ) => + f0 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P OLT) => + f1 : cupicef.Term (cupicef.type (cupicef.s cupicef.z)) (P OLE) => + o : cupicef.Term cupicef.set ord => + match____ord (cupicef.type (cupicef.s cupicef.z)) + (o0 : cupicef.Term cupicef.set ord => P o0) f f0 f1 o. + +def ord__ind : + P : (o : cupicef.Term cupicef.set ord -> cupicef.Univ cupicef.prop) -> + f : cupicef.Term cupicef.prop (P OEQ) -> + f0 : cupicef.Term cupicef.prop (P OLT) -> + f1 : cupicef.Term cupicef.prop (P OLE) -> + o : cupicef.Term cupicef.set ord -> + cupicef.Term cupicef.prop (P o) + +:= P : (o : cupicef.Term cupicef.set ord -> cupicef.Univ cupicef.prop) => + f : cupicef.Term cupicef.prop (P OEQ) => + f0 : cupicef.Term cupicef.prop (P OLT) => + f1 : cupicef.Term cupicef.prop (P OLE) => + o : cupicef.Term cupicef.set ord => + match____ord cupicef.prop (o0 : cupicef.Term cupicef.set ord => P o0) f f0 + f1 o. + +def ord__rec : + P : (o : cupicef.Term cupicef.set ord -> cupicef.Univ cupicef.set) -> + f : cupicef.Term cupicef.set (P OEQ) -> + f0 : cupicef.Term cupicef.set (P OLT) -> + f1 : cupicef.Term cupicef.set (P OLE) -> + o : cupicef.Term cupicef.set ord -> + cupicef.Term cupicef.set (P o) + +:= P : (o : cupicef.Term cupicef.set ord -> cupicef.Univ cupicef.set) => + ord__rect + (cupicef.cast + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + (cupicef.prod cupicef.set (cupicef.type (cupicef.s cupicef.z)) + (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) + cupicef.I ord + (o : cupicef.Term cupicef.set ord => + cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) + (cupicef.rule cupicef.set + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z)))) + cupicef.I ord + (o : cupicef.Term cupicef.set ord => + cupicef.univ (cupicef.type (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I)) + cupicef.I P). + +def trans__ord : + o : cupicef.Term cupicef.set ord -> + o_27_ : cupicef.Term cupicef.set ord -> + cupicef.Term cupicef.set ord + +:= o : cupicef.Term cupicef.set ord => + o_27_ : cupicef.Term cupicef.set ord => + match____ord cupicef.set (o0 : cupicef.Term cupicef.set ord => ord) o_27_ + (match____ord cupicef.set (o_27_0 : cupicef.Term cupicef.set ord => ord) + o OLT OLT o_27_) + (match____ord cupicef.set (o_27_0 : cupicef.Term cupicef.set ord => ord) + o OLT OLE o_27_) + o. + +(; End of translation. ;) diff --git a/out/Makefile b/out/Makefile new file mode 100644 index 000000000..103ed004a --- /dev/null +++ b/out/Makefile @@ -0,0 +1,4 @@ +all: + dune exec lambdapi -- check --lib-root=. -c cupicef.lp + dune exec lambdapi -- check --lib-root=. -c Coq__Init__Logic.dk + dune exec lambdapi -- check --lib-root=. -c Coq__Init__Datatypes.dk \ No newline at end of file diff --git a/out/cupicef.lp b/out/cupicef.lp new file mode 100644 index 000000000..42d29c894 --- /dev/null +++ b/out/cupicef.lp @@ -0,0 +1,345 @@ +constant symbol Sort : TYPE; +symbol sup : Sort → Sort → Sort; +constant symbol Univ : Π (s : Sort), TYPE; +symbol Term : Π (s : Sort), Π (a : Univ s), TYPE; +constant symbol Bool : TYPE; +constant symbol eps : Bool → TYPE; +constant symbol true : Bool; +constant symbol I : eps true; +symbol Axiom : Sort → Sort → Bool; +symbol Rule : Sort → Sort → Sort → Bool; +symbol Cumul : Sort → Sort → Bool; +symbol Eq : Sort → Sort → Bool; +rule Eq $x $x ↪ true; +associative commutative symbol and : Bool → Bool → Bool; +rule and true $c ↪ $c; +symbol BoolO : TYPE; +constant symbol BoolNone : BoolO; +constant symbol BoolSome : Bool → BoolO; +symbol Multi : Bool → BoolO → TYPE; +rule Multi $A BoolNone ↪ eps $A +with Multi $A (BoolSome $B) ↪ eps $B + → Π (C : BoolO), Multi (and $A $B) C; +constant symbol pair : Π (A : BoolO), Multi true A; +symbol SortO : TYPE; +constant symbol SortNone : SortO; +constant symbol SortSome : Sort → SortO; +symbol Trans : Sort → Sort → SortO → TYPE; +rule Trans $s1 $s2 SortNone ↪ eps (Cumul $s1 $s2) +with Trans $s1 $s2 (SortSome $s3) ↪ eps (Cumul $s2 $s3) + → Π (so : SortO), Trans $s1 $s3 so; +constant symbol trans : Π (s : Sort), Π (so : SortO), Trans s s so; +constant symbol cumul_eq + : Π (s1 : Sort), Π (s2 : Sort), eps (Eq s1 s2) → eps (Cumul s1 s2); +constant symbol eq_sym + : Π (s1 : Sort), Π (s2 : Sort), eps (Eq s1 s2) → eps (Eq s2 s1); +symbol univ : Π (s : Sort), Π (s' : Sort), eps (Axiom s s') → Univ s'; +symbol prod + : Π (s1 : Sort), + Π (s2 : Sort), + Π (s3 : Sort), + eps (Rule s1 s2 s3) + → Π (a : Univ s1), Π (b : Term s1 a → Univ s2), Univ s3; +symbol SubType : Π (s : Sort), Π (s' : Sort), Univ s → Univ s' → Bool; +symbol cast + : Π (s : Sort), + Π (s' : Sort), + Π (a : Univ s), + Π (b : Univ s'), eps (SubType s s' a b) → Term s a → Term s' b; +constant symbol sinf : Sort; +symbol Code ≔ Univ sinf; +symbol Decode : Code → TYPE; +symbol uncode : Π (t : Code), Code → Decode t; +symbol code : Π (t : Code), Decode t → Code; +constant symbol cU : Π (s : Sort), Code; +constant symbol cPi : Π (A : Code), (Decode A → Code) → Code; +constant symbol cLam : Π (A : Code), (Decode A → Code) → Code; +symbol cApp : Code → Code → Code; +rule Decode (cU $s) ↪ Univ $s; +rule Decode (cPi $a $b) ↪ Π (x : Decode $a), Decode ($b x); +rule code (cU sinf) $t ↪ $t; +rule uncode (cU sinf) $t ↪ $t; +rule Term $s $t ↪ Decode (code (cU $s) $t); +rule univ $s $s' $p ↪ uncode (cU $s') (cU $s); +rule prod $s1 $s2 $s3 $p $a $b ↪ uncode (cU $s3) + (cPi (code (cU $s1) $a) + (λ x, code (cU $s2.[]) ($b.[] x))); +rule cast $s1 $s2 $a $b _ $t ↪ uncode (code (cU $s2) $b) + (code (code (cU $s1) $a) $t); +rule Cumul $s $s ↪ true; +symbol forall : (Code → Bool) → Bool; +rule forall (λ x, $B.[]) ↪ $B; +symbol ST : Code → Code → Bool; +symbol EQ : Code → Code → Bool; +rule ST (cU $s1) (cU $s2) ↪ Cumul $s1 $s2; +rule EQ (cU $s1) (cU $s2) ↪ Eq $s1 $s2; +rule ST (cPi $a $b) (cPi $a' $b') ↪ and (EQ $a $a') + (forall + (λ (x : Code), + ST ($b.[] (uncode $a.[] x)) + ($b'.[] (uncode $a'.[] x)))); +rule EQ (cPi $a $b) (cPi $a' $b') ↪ and (EQ $a $a') + (forall + (λ (x : Code), + EQ ($b.[] (uncode $a.[] x)) + ($b'.[] (uncode $a'.[] x)))); +rule EQ $A $A ↪ true; +rule ST $A $A ↪ true; +rule SubType $sa $sb $a $b ↪ ST (code (cU $sa) $a) (code (cU $sb) $b); +rule code _ (uncode _ $t) ↪ $t; +rule code (cPi $A $B) $F ↪ cLam $A (λ x, code ($B.[] x) ($F.[] x)); +rule uncode (cPi $A $B) $F $U ↪ uncode ($B $U) (cApp $F (code $A $U)); +rule uncode (cPi $A $B) (cLam $A' $F) ↪ λ x, + uncode ($B x) + ($F (uncode $A' (code $A x))); +rule cApp (code (cPi $A $B) $F) $U ↪ code ($B (uncode $A $U)) + ($F (uncode $A $U)); +rule cApp (cLam $A $F) $U ↪ $F (uncode $A $U); +rule uncode $s (code $s $t) ↪ $t; +symbol univ' (s : Sort) (s' : Sort) ≔ uncode (cU s') (cU s); +symbol lift' (s : Sort) (s' : Sort) (t : Univ s) + ≔ uncode (cU s') (code (cU s) t); +symbol prod' (s1 : Sort) (s2 : Sort) (s : Sort) (A : Univ s1) + (B : Term s1 A → Univ s2) : Univ s + ≔ uncode (cU s) (cPi (code (cU s1) A) (λ x, code (cU s2) (B x))); +constant symbol Nat : TYPE; +symbol s : Nat → Nat; +constant symbol z : Nat; +constant symbol prop : Sort; +constant symbol {|type|} : Nat → Sort; +symbol set : Sort ≔ {|type|} z; +symbol type0 : Sort ≔ {|type|} (s z); +associative commutative symbol max : Nat → Nat → Nat; +rule max $i z ↪ $i; +rule s (max $i $j) ↪ max (s $i) (s $j); +rule max $i $i ↪ $i +with max $i (max $i $j) ↪ max $i $j +with max $i (s $i) ↪ s $i +with max $i (max (s $i) $j) ↪ max (s $i) $j +with max $i (s (max $i $j)) ↪ s (max $i $j) +with max (max $i $k) (s (max $i $j)) ↪ max $k (s (max $i $j)); + +rule max z $i ↪ $i; +rule s (max $i $j) ↪ max (s $i) (s $j); +rule max $i $i ↪ $i +with max (max $i $j) $i ↪ max $i $j +with max (s $i) $i ↪ s $i +with max (max (s $i) $j) $i ↪ max (s $i) $j +with max (s (max $i $j)) $i ↪ s (max $i $j) +with max (s (max $i $j)) (max $i $k) ↪ max $k (s (max $i $j)); + +rule sup prop $s ↪ $s +with sup $s prop ↪ $s +with sup ({|type|} $i) ({|type|} $j) ↪ {|type|} (max $i $j); +symbol Leq : Nat → Nat → Bool; +rule Leq z $j ↪ true +with Leq (s $i) (s $j) ↪ Leq $i $j +with Leq (max $i $j) $k ↪ and (Leq $i $k) (Leq $j $k) +with Leq $i $i ↪ true +with Leq $i (s $i) ↪ true +with Leq $i (max $i $j) ↪ true +with Leq $i (max (s $i) $j) ↪ true; +rule Axiom prop ({|type|} _) ↪ true +with Axiom ({|type|} $i) ({|type|} $j) ↪ Leq $i $j; +rule Rule _ prop prop ↪ true +with Rule prop ({|type|} $i) ({|type|} $i) ↪ true +with Rule ({|type|} $i) ({|type|} $j) ({|type|} $k) ↪ Leq (max $i $j) $k; +rule Cumul prop prop ↪ true +with Cumul prop ({|type|} $i) ↪ true +with Cumul ({|type|} $i) ({|type|} $j) ↪ Leq $i $j; +symbol axiom : Sort → Sort; +rule axiom prop ↪ {|type|} z +with axiom ({|type|} $i) ↪ {|type|} (s $i); +rule Axiom $x (axiom $x) ↪ true; +symbol {|rule|} : Sort → Sort → Sort; +rule {|rule|} prop $s ↪ $s +with {|rule|} _ prop ↪ prop +with {|rule|} ({|type|} $i) ({|type|} $j) ↪ {|type|} (max $i $j); +rule Rule $s1 $s2 ({|rule|} $s1 $s2) ↪ true; +rule Rule $s $s $s ↪ true; +constant symbol N : TYPE; +constant symbol 0 : N; +constant symbol _S : N → N; +symbol 1 ≔ _S 0; +symbol 2 ≔ _S 1; +symbol 3 ≔ _S 2; +symbol 4 ≔ _S 3; +symbol 5 ≔ _S 4; +symbol 6 ≔ _S 5; +symbol 7 ≔ _S 6; +symbol 8 ≔ _S 7; +symbol 9 ≔ _S 8; +constant symbol SingleArity : TYPE; +constant symbol SAcons : N → Code → SingleArity; +symbol SA_code : SingleArity → Code; +rule SA_code (SAcons _ $A) ↪ $A; +constant symbol MutualArity : N → TYPE; +constant symbol MAnil : MutualArity 0; +constant symbol MAcons + : Π (n : N), SingleArity → MutualArity n → MutualArity (_S n); +symbol MA_ith_SA : Π (n : N), MutualArity n → N → SingleArity; +rule MA_ith_SA _ (MAcons _ $SA _) 0 ↪ $SA; +rule MA_ith_SA _ (MAcons $n _ $MA) (_S $i) ↪ MA_ith_SA $n $MA $i; +symbol MA_ith_code (n : N) (MA : MutualArity n) (i : N) : Code + ≔ SA_code (MA_ith_SA n MA i); +symbol MA_lift_code : Code → Π (m : N), MutualArity m → Code; +rule MA_lift_code $B _ MAnil ↪ $B; +rule MA_lift_code $B _ (MAcons $m $SA $MA) ↪ cPi (SA_code $SA) + (λ f, + MA_lift_code $B.[] $m.[] + $MA.[]); +symbol MA_code (n : N) (MA : MutualArity n) (i : N) : Code + ≔ MA_lift_code (MA_ith_code n MA i) n MA; +constant symbol MutualFixpointAux + : Π (n : N), MutualArity n → Π (i : N), MutualArity i → TYPE; +symbol MutualFixpoint (n : N) (MA : MutualArity n) : TYPE + ≔ MutualFixpointAux n MA n MA; +constant symbol MFTnil + : Π (n : N), Π (MA : MutualArity n), MutualFixpointAux n MA 0 MAnil; +constant symbol MFTcons + : Π (n : N), + Π (MAn : MutualArity n), + Π (i : N), + Π (MAi : MutualArity i), + Π (SA : SingleArity), + Decode (MA_lift_code (SA_code SA) n MAn) + → MutualFixpointAux n MAn i MAi + → MutualFixpointAux n MAn (_S i) (MAcons i SA MAi); +symbol fix_body + : Π (n : N), + Π (MA : MutualArity n), + MutualFixpoint n MA → Π (i : N), Decode (MA_code n MA i); +symbol fix_body' + : Π (n : N), + Π (MAn : MutualArity n), + Π (i : N), + Π (MAi : MutualArity i), + MutualFixpointAux n MAn i MAi + → Π (j : N), + Decode (MA_lift_code (SA_code (MA_ith_SA i MAi j)) n MAn); +rule fix_body $n $MA $MF $i ↪ fix_body' $n $MA $n $MA $MF $i; +rule fix_body' $n $MA (_S _) (MAcons _ $SA _) (MFTcons _ _ _ _ _ $t _) 0 ↪ $t; +rule fix_body' $n $MA (_S _) (MAcons $i _ $MAi) (MFTcons _ _ _ _ _ _ $MFT) + (_S $j) ↪ fix_body' $n $MA $i $MAi $MFT $j; +symbol app_body + : Π (n : N), + Π (MA : MutualArity n), + Π (A : Code), + Π (B : Decode A → Code), + Π (body : Decode (MA_lift_code (cPi A B) n MA)), + Π (a : Decode A), Decode (MA_lift_code (B a) n MA); +rule app_body _ MAnil _ _ $body $a ↪ $body $a; +rule app_body _ (MAcons $n $SA $MA) $A $B $body $a ↪ λ f, + app_body $n $MA $A + $B ($body f) $a; +constant symbol Guarded? : TYPE; +constant symbol guarded : Guarded?; +symbol code_guarded? : Π (Ind : Code), Decode Ind → Guarded?; +symbol guarded? : Π (s : Sort), Π (Ind : Univ s), Term s Ind → Guarded?; +rule guarded? $s $Ind $c ↪ code_guarded? (code (cU $s) $Ind) $c; +symbol fix_proj + : Π (n : N), + Π (MA : MutualArity n), + MutualFixpoint n MA → Π (i : N), Decode (MA_ith_code n MA i); +symbol fix_proj_1 + : Π (n : N), + Π (MA : MutualArity n), + MutualFixpoint n MA + → Π (SA : SingleArity), + Decode (MA_lift_code (SA_code SA) n MA) → Decode (SA_code SA); +rule fix_proj $n $MA $MF $i ↪ fix_proj_1 $n $MA $MF (MA_ith_SA $n $MA $i) + (fix_body $n $MA $MF $i); +rule fix_proj_1 $n $MA $MF (SAcons (_S $i) (cPi $A $B)) $body $a ↪ +fix_proj_1 $n $MA $MF (SAcons $i ($B $a)) (app_body $n $MA $A $B $body $a); +symbol fix_proj_2 + : Π (n : N), + Π (MA : MutualArity n), + MutualFixpoint n MA + → Π (A : Code), + Decode (MA_lift_code A n MA) → Guarded? → Decode A; +rule fix_proj_1 $n $MA $MF (SAcons 0 (cPi $Ind $B)) $body $a ↪ fix_proj_2 + $n $MA $MF + (cPi $Ind $B) + $body + (code_guarded? + $Ind $a) + $a; +symbol fix_proj_3 + : Π (A : Code), + Π (i : N), + Π (MAi : MutualArity i), + Π (f : Π (k : N), Decode (MA_ith_code i MAi k)), + Decode (MA_lift_code A i MAi) → Decode A; +rule fix_proj_2 $n $MA $MF $A $body guarded ↪ fix_proj_3 $A $n $MA + (λ k, + fix_proj $n.[] $MA.[] + $MF.[] k) + $body; +rule fix_proj_3 _ _ MAnil _ $body ↪ $body; +rule fix_proj_3 $A _ (MAcons $i _ $MAi) $f $body ↪ fix_proj_3 $A $i $MAi + (λ k, $f.[] (_S k)) + ($body ($f 0)); +symbol MutualArity' : N → N → TYPE; +rule MutualArity' $n 0 ↪ MutualArity $n; +rule MutualArity' $n (_S $i) ↪ SingleArity → MutualArity' $n $i; +symbol MA_magic + : Π (n : N), + Π (m : N), SingleArity → MutualArity' n m → MutualArity' (_S n) m; +rule MA_magic $n 0 $A $MA ↪ MAcons $n $A $MA; +rule MA_magic $n (_S $m) $A $MA $A' ↪ MA_magic $n $m $A ($MA $A'); +symbol make_MA : Π (n : N), MutualArity' n n; +rule make_MA 0 ↪ MAnil; +rule make_MA (_S $n) $SA ↪ MA_magic $n $n $SA (make_MA $n); +symbol MutualFixpoint' + : Π (n : N), + MutualArity n + → Π (i : N), MutualArity i → Π (j : N), MutualArity j → TYPE; +rule MutualFixpoint' $n $MAn $i $MAi _ MAnil ↪ MutualFixpointAux $n $MAn $i + $MAi; +rule MutualFixpoint' $n $MAn $i $MAi _ (MAcons $j $SA $MAj) ↪ Decode + (MA_lift_code + (SA_code $SA) + $n $MAn) + → MutualFixpoint' + $n $MAn + $i $MAi + $j $MAj; +symbol fix_magic + : Π (n : N), + Π (MAn : MutualArity n), + Π (i : N), + Π (MAi : MutualArity i), + Π (j : N), + Π (MAj : MutualArity j), + Π (SA : SingleArity), + Decode (MA_lift_code (SA_code SA) n MAn) + → MutualFixpoint' n MAn i MAi j MAj + → MutualFixpoint' n MAn (_S i) (MAcons i SA MAi) j + MAj; +rule fix_magic $n $MAn $i $MAi _ MAnil $SA $t $MFT ↪ MFTcons $n $MAn $i + $MAi $SA $t $MFT; +rule fix_magic $n $MAn $i $MAi (_S _) (MAcons $j _ $MAj) $SA $t $MFT $t' ↪ +fix_magic $n $MAn $i $MAi $j $MAj $SA $t ($MFT $t'); +symbol make_fix + : Π (n : N), + Π (MAn : MutualArity n), + Π (i : N), + Π (MAi : MutualArity i), MutualFixpoint' n MAn i MAi i MAi; +rule make_fix $n $MAn _ MAnil ↪ MFTnil $n $MAn; +rule make_fix $n $MAn (_S _) (MAcons $j $SA $MAj) $t ↪ fix_magic $n $MAn $j + $MAj $j $MAj $SA $t + (make_fix $n $MAn $j + $MAj); +symbol SA : N → Π (s : Sort), Univ s → SingleArity; +rule SA $n $s $t ↪ SAcons $n (code (cU $s) $t); +symbol fix + : Π (n : N), Π (MA : MutualArity n), MutualFixpoint' n MA n MA n MA; +rule fix $n $MAn ↪ make_fix $n $MAn $n $MAn; +symbol fixproj + : Π (n : N), + Π (MA : MutualArity' n n → MutualArity n), + (MutualFixpoint' n (MA (make_MA n)) n (MA (make_MA n)) n + (MA (make_MA n)) → MutualFixpoint n (MA (make_MA n))) + → Π (i : N), Decode (MA_ith_code n (MA (make_MA n)) i); +rule fixproj $n $MA $MF ↪ fix_proj $n ($MA (make_MA $n)) + ($MF (fix $n ($MA (make_MA $n)))); diff --git a/out/lambdap.elpi b/out/lambdap.elpi new file mode 100644 index 000000000..0f2e420ab --- /dev/null +++ b/out/lambdap.elpi @@ -0,0 +1,1298 @@ +% automatically generated + +% ---- Lambdapi datatypes ---- + +% A symbol +typeabbrev symbol (ctype "symbol"). + + + +kind term type. +type typ term. +type kin term. +type symb symbol -> term. +type appl term -> term -> term. +type abst term -> (term -> term) -> term. +type prod term -> (term -> term) -> term. + + +kind sealed-goal type. +type nabla (term -> sealed-goal) -> sealed-goal. +type seal goal -> sealed-goal. + +kind goal type. +type goal list prop -> term -> term -> goal. + +type of term -> term -> prop. + +pred msolve o:list sealed-goal. + + + +% ---- Builtin predicates ---- + +% [lp.sig S T] Gives the type of a symbol +external pred lp.sig i:symbol, o:term. + +% [lp.tc-instances L] Gives the list of type class instances +external pred lp.tc-instances o:list symbol. + +% [lp.tc? S] Tells if S is a type class +external pred lp.tc? i:symbol. + +% [lp.term->string T S] Pretty prints a term T to the string S +external pred lp.term->string i:term, o:string. + +% ---- Elpi predicates ---- + +% == Core builtins ===================================== + +% -- Logic -- + +pred true. + +true. + +pred fail. + +pred false. + +external pred (=) o:A, o:A. % unification + +typeabbrev int (ctype "int"). + + +typeabbrev string (ctype "string"). + + +typeabbrev float (ctype "float"). + + +pred (;) o:prop, o:prop. + +(A ; _) :- A. + +(_ ; B) :- B. + +type (:-) prop -> prop -> prop. + +type (:-) prop -> list prop -> prop. + +type (,) variadic prop prop. + +type uvar A. + +type (as) A -> A -> A. + +type (=>) prop -> prop -> prop. + +type (=>) list prop -> prop -> prop. + +% -- Control -- + +external pred !. % The cut operator + +pred not i:prop. + +not X :- X, !, fail. + +not _. + +% [declare_constraint C Key1 Key2...] declares C blocked +% on Key1 Key2 ... (variables, or lists thereof). +external type declare_constraint variadic any prop. + +external pred print_constraints. % prints all constraints + +% [halt ...] halts the program and print the terms +external type halt variadic any prop. + +pred stop. + +stop :- halt. + +% -- Evaluation -- + +% [calc Expr Out] unifies Out with the value of Expr. It can be used in +% tandem with spilling, eg [f {calc (N + 1)}] +external pred calc i:A, o:A. + +pred (is) o:A, i:A. + +X is Y :- calc Y X. + +type (-) A -> A -> A. + +type (i-) int -> int -> int. + +type (r-) float -> float -> float. + +type (+) int -> int -> int. + +type (+) float -> float -> float. + +type (i+) int -> int -> int. + +type (r+) float -> float -> float. + +type (*) int -> int -> int. + +type (*) float -> float -> float. + +type (/) float -> float -> float. + +type (mod) int -> int -> int. + +type (div) int -> int -> int. + +type (^) string -> string -> string. + +type (~) int -> int. + +type (~) float -> float. + +type (i~) int -> int. + +type (r~) float -> float. + +type abs int -> int. + +type abs float -> float. + +type iabs int -> int. + +type rabs float -> float. + +type max int -> int -> int. + +type max float -> float -> float. + +type min int -> int -> int. + +type min float -> float -> float. + +type sqrt float -> float. + +type sin float -> float. + +type cos float -> float. + +type arctan float -> float. + +type ln float -> float. + +type int_to_real int -> float. + +type floor float -> int. + +type ceil float -> int. + +type truncate float -> int. + +type size string -> int. + +type chr int -> string. + +type rhc string -> int. + +type string_to_int string -> int. + +type int_to_string int -> string. + +type substring string -> int -> int -> string. + +type real_to_string float -> string. + +% -- Arithmetic tests -- + +% [lt_ X Y] checks if X < Y. Works for string, int and float +external pred lt_ i:A, i:A. + +% [gt_ X Y] checks if X > Y. Works for string, int and float +external pred gt_ i:A, i:A. + +% [le_ X Y] checks if X =< Y. Works for string, int and float +external pred le_ i:A, i:A. + +% [ge_ X Y] checks if X >= Y. Works for string, int and float +external pred ge_ i:A, i:A. + +type (<), (>), (=<), (>=) A -> A -> prop. + +X > Y :- gt_ X Y. + +X < Y :- lt_ X Y. + +X =< Y :- le_ X Y. + +X >= Y :- ge_ X Y. + +type (i<), (i>), (i=<), (i>=) int -> int -> prop. + +X i< Y :- lt_ X Y. + +X i> Y :- gt_ X Y. + +X i=< Y :- le_ X Y. + +X i>= Y :- ge_ X Y. + +type (r<), (r>), (r=<), (r>=) float -> float -> prop. + +X r< Y :- lt_ X Y. + +X r> Y :- gt_ X Y. + +X r=< Y :- le_ X Y. + +X r>= Y :- ge_ X Y. + +type (s<), (s>), (s=<), (s>=) string -> string -> prop. + +X s< Y :- lt_ X Y. + +X s> Y :- gt_ X Y. + +X s=< Y :- le_ X Y. + +X s>= Y :- ge_ X Y. + +% -- Standard data types (supported in the FFI) -- + +kind list type -> type. + +type (::) X -> list X -> list X. + +type ([]) list X. + +% Boolean values: tt and ff since true and false are predicates +kind bool type. +type tt bool. +type ff bool. + +% Pair: the constructor is pr, since ',' is for conjunction +kind pair type -> type -> type. +type pr A -> B -> pair A B. + +pred fst i:pair A B, o:A. + +fst (pr A _) A. + +pred snd i:pair A B, o:B. + +snd (pr _ B) B. + +% The option type (aka Maybe) +kind option type -> type. +type none option A. +type some A -> option A. + +% Result of a comparison +kind cmp type. +type eq cmp. +type lt cmp. +type gt cmp. + +% Used in builtin variants that return Coq's error rather than failing +kind diagnostic type. +type ok diagnostic. % Success +type error string -> diagnostic. % Failure + +% == I/O builtins ===================================== + +% -- I/O -- + +typeabbrev in_stream (ctype "in_stream"). + +type std_in in_stream. + +typeabbrev out_stream (ctype "out_stream"). + +type std_out out_stream. +type std_err out_stream. + +% [open_in FileName InStream] opens FileName for input +external pred open_in i:string, o:in_stream. + +% [open_out FileName OutStream] opens FileName for output +external pred open_out i:string, o:out_stream. + +% [open_append FileName OutStream] opens FileName for output in append mode +external pred open_append i:string, o:out_stream. + +% [close_in InStream] closes input stream InStream +external pred close_in i:in_stream. + +% [close_out OutStream] closes output stream OutStream +external pred close_out i:out_stream. + +% [output OutStream Data] writes Data to OutStream +external pred output i:out_stream, i:string. + +% [flush OutStream] flush all output not yet finalized to OutStream +external pred flush i:out_stream. + +% [input InStream Bytes Data] reads Bytes from InStream +external pred input i:in_stream, i:int, o:string. + +% [input_line InStream Line] reads a full line from InStream +external pred input_line i:in_stream, o:string. + +% [eof InStream] checks if no more data can be read from InStream +external pred eof i:in_stream. + +% -- System -- + +% [gettimeofday T] sets T to the number of seconds elapsed since 1/1/1970 +external pred gettimeofday o:float. + +% [getenv VarName Value] Like Sys.getenv +external pred getenv i:string, o:option string. + +% [system Command RetVal] executes Command and sets RetVal to the exit code +external pred system i:string, o:int. + +% -- Debugging -- + +% [term_to_string T S] prints T to S +external pred term_to_string i:any, o:string. + +% == Lambda Prolog builtins ===================================== + +% -- Extra I/O -- + +% [open_string DataIn InStream] opens DataIn as an input stream +external pred open_string i:string, o:in_stream. + +% [lookahead InStream NextChar] peeks one byte from InStream +external pred lookahead i:in_stream, o:string. + +% -- Hacks -- + +% [string_to_term S T] parses a term T from S +external pred string_to_term i:string, o:any. + +% [readterm InStream T] reads T from InStream, ends with \n +external pred readterm i:in_stream, o:any. + +pred printterm i:out_stream, i:A. + +printterm S T :- term_to_string T T1, output S T1. + +pred read o:A. + +read S :- flush std_out, input_line std_in X, string_to_term X S. + +% == Elpi builtins ===================================== + +% [dprint ...] prints raw terms (debugging) +external type dprint variadic any prop. + +% [print ...] prints terms +external type print variadic any prop. + +% Deprecated, use trace.counter +pred counter i:string, o:int. +counter C N :- trace.counter C N. + +% [quote_syntax FileName QueryText QuotedProgram QuotedQuery] quotes the +% program from FileName and the QueryText. See elpi-quoted_syntax.elpi for +% the syntax tree +external pred quote_syntax i:string, i:string, o:list A, o:A. + +typeabbrev loc (ctype "Loc.t"). + + +% [loc.fields Loc File StartChar StopChar Line LineStartsAtChar] Decomposes +% a loc into its fields +external pred loc.fields i:loc, o:string, o:int, o:int, o:int, o:int. + +% == Regular Expressions ===================================== + +% [rex.match Rex Subject] checks if Subject matches Rex. Matching is based +% on OCaml's Str library +external pred rex.match i:string, i:string. + +% [rex.replace Rex Replacement Subject Out] Out is obtained by replacing all +% occurrences of Rex with Replacement in Subject. See also OCaml's +% Str.global_replace +external pred rex.replace i:string, i:string, i:string, o:string. + +% [rex.split Rex Subject Out] Out is obtained by splitting Subject at all +% occurrences of Rex. See also OCaml's Str.split +external pred rex.split i:string, i:string, o:list string. + +% Deprecated, use rex.match +pred rex_match i:string, i:string. +rex_match Rx S :- rex.match Rx S. + +% Deprecated, use rex.replace +pred rex_replace i:string, i:string, i:string, o:string. +rex_replace Rx R S O :- rex.replace Rx R S O. + +% Deprecated, use rex.split +pred rex_split i:string, i:string, o:list string. +rex_split Rx S L :- rex.split Rx S L. + +% == Elpi nonlogical builtins ===================================== + +% Opaque ML data types +kind ctyp type. +type ctype string -> ctyp. + +% [var V ...] checks if the term V is a variable. When used with tree +% arguments it relates an applied variable with its head and argument list. +external type var any -> variadic any prop. + +% [prune V L] V is pruned to L (V is unified with a variable that only sees +% the list of names L) +external pred prune o:any, i:list any. + +% [distinct_names L] checks if L is a list of distinct names. If L is the +% scope of a unification variable (its arguments, as per var predicate) then +% distinct_names L checks that such variable is in the Miller pattern +% fragment (L_\lambda) +external pred distinct_names i:list any. + +% [same_var V1 V2] checks if the two terms V1 and V2 are the same variable, +% ignoring the arguments of the variables +external pred same_var i:A, i:A. + +% [same_term T1 T2] checks if the two terms T1 and T2 are syntactically +% equal (no unification). It behaves differently than same_var since it +% recursively compares the arguments of the variables +external pred same_term i:A, i:A. + + +% Infix notation for same_term +pred (==) i:A, i:A. +X == Y :- same_term X Y. + + +% [cmp_term A B Cmp] Compares A and B. Only works if A and B are ground. +external pred cmp_term i:any, i:any, o:cmp. + +% [name T ...] checks if T is a eigenvariable. When used with tree arguments +% it relates an applied name with its head and argument list. +external type name any -> variadic any prop. + +% [constant T ...] checks if T is a (global) constant. When used with tree +% arguments it relates an applied constant with its head and argument list. +external type constant any -> variadic any prop. + +external pred names % generates the list of eigenvariable + o:list any. % list of eigenvariables in order of age (young first) + +external pred occurs % checks if the atom occurs in the term + i:any, % an atom, that is a global constant or a bound name (aka eigenvariable) + i:any. % a term + +% [closed_term T] unify T with a variable that has no eigenvariables in +% scope +external pred closed_term o:any. + +% [ground_term T] Checks if T contains unification variables +external pred ground_term i:any. + +% [is_cdata T Ctype] checks if T is primitive of type Ctype, eg (ctype +% "int") +external pred is_cdata i:any, o:ctyp. + +pred primitive? i:A, i:string. + +primitive? X S :- is_cdata X (ctype S). + +% [new_int N] unifies N with a different int every time it is called. Values +% of N are guaranteed to be incresing. +external pred new_int o:int. + +% [findall_solution P L] finds all the solved instances of P and puts them +% in L +% in the order in which they are found. Instances can contain +% eigenvariables +% and unification variables. + +external pred findall_solutions i:prop, o:list prop. + +% Holds data across bracktracking; can only contain closed terms +typeabbrev safe (ctype "safe"). + + +% [new_safe Safe] creates a safe: a store that persists across backtracking +external pred new_safe o:safe. + +% [stash_in_safe Safe Data] stores Data in the Safe +external pred stash_in_safe i:safe, i:A. + +% [open_safe Safe Data] retrieves the Data stored in Safe +external pred open_safe i:safe, o:list A. + + +% [if C T E] picks the first success of C then runs T (never E). +% if C has no success it runs E. +pred if i:prop, i:prop, i:prop. +if B T _ :- B, !, T. +if _ _ E :- E. + +% [if2 C1 B1 C2 B2 E] like if but with 2 then branches (and one else branch). +pred if2 i:prop, i:prop, i:prop, i:prop, i:prop. +if2 G1 P1 _ _ _ :- G1, !, P1. +if2 _ _ G2 P2 _ :- G2, !, P2. +if2 _ _ _ _ E :- !, E. + +% [random.init Seed] Initialize OCaml's PRNG with the given Seed +external pred random.init i:int. + +% [random.self_init] Initialize OCaml's PRNG with some seed +external pred random.self_init . + +% [random.int Bound N] unifies N with a random int between 0 and Bound +% (excluded) +external pred random.int i:int, o:int. + +#line 0 "builtin_stdlib.elpi" +% == stdlib ======================================================= + +% Conventions: +% - all predicates declare a mode with some input arguments, unless... +% - predicates whose name ends with R are relations (work in any direction, +% that is all arguments are in output mode) +% - predicates whose name ends with ! do contain a cut and generate only the +% first result +% - all errors given by this library end up calling fatal-error[-w-data], +% override it in order to handle them differently +% - all debug prints by this library end up calling debug-print, override it +% in order to handle them differently + +namespace std { + +pred fatal-error i:string. +:name "default-fatal-error" +fatal-error Msg :- halt Msg. + +pred fatal-error-w-data i:string, i:A. +:name "default-fatal-error-w-data" +fatal-error-w-data Msg Data :- halt Msg ":" Data. + +pred debug-print i:string, i:A. +:name "default-debug-print" +debug-print Msg Data :- print Msg Data. + +% -- Errors, Debugging, Hacks -- + +pred ignore-failure! i:prop. +ignore-failure! P :- P, !. +ignore-failure! _. + +% [assert! C M] takes the first success of C or fails with message M +pred assert! i:prop, i:string. +assert! Cond Msg :- (Cond ; fatal-error-w-data Msg Cond), !. + +% [assert-ok! C M] like assert! but the last argument of the predicate must +% be a diagnostic that is printed after M in case it is not ok +pred assert-ok! i:(diagnostic -> prop), i:string. +assert-ok! Cond Msg :- Cond Diagnostic, !, (Diagnostic = ok ; Diagnostic = error S, fatal-error-w-data Msg S), !. +assert-ok! _ Msg :- fatal-error-w-data Msg "no diagnostic returned". + +% [spy P] traces the call to P, printing all success and the final failure +pred spy i:prop. +spy P :- trace.counter "run" NR, if (not(NR = 0)) (debug-print "run=" NR) true, + debug-print "----<<---- enter: " P, + P, + debug-print "---->>---- exit: " P. +spy P :- debug-print "---->>---- fail: " P, fail. + +% [spy! P] traces the first call to P without leaving a choice point +pred spy! i:prop. +spy! P :- trace.counter "run" NR, if (not(NR = 0)) (debug-print "run=" NR) true, + debug-print "----<<---- enter: " P, + P, + debug-print "---->>---- exit: " P, !. +spy! P :- debug-print "---->>---- fail: " P, fail. + +% to silence the type checker +pred unsafe-cast o:A, o:B. +unsafe-cast X X. + +% -- List processing -- + +pred length i:list A, o:int. +length [_|L] N :- length L N1, N is N1 + 1. +length [] 0. + +pred rev i:list A, o:list A. +rev L RL :- rev.aux L [] RL. +rev.aux [X|XS] ACC R :- rev.aux XS [X|ACC] R. +rev.aux [] L L. + +pred last i:list A, o:A. +last [] _ :- fatal-error "last on empty list". +last [X] X :- !. +last [_|XS] R :- last XS R. + +pred append i:list A, i:list A, o:list A. +append [X|XS] L [X|L1] :- append XS L L1 . +append [] L L . + +pred appendR o:list A, o:list A, o:list A. +appendR [X|XS] L [X|L1] :- appendR XS L L1 . +appendR [] L L . + +pred take i:int, i:list A, o:list A. +take 0 _ [] :- !. +take N [X|XS] [X|L] :- !, N1 is N - 1, take N1 XS L. +take _ _ _ :- fatal-error "take run out of list items". + +pred take-last i:int, i:list A, o:list A. +take-last N L R :- + length L M, + D is M - N, + drop D L R. + +pred drop i:int, i:list A, o:list A. +drop 0 L L :- !. +drop N [_|XS] L :- !, N1 is N - 1, drop N1 XS L. +drop _ _ _ :- fatal-error "drop run out of list items". + +pred drop-last i:int, i:list A, o:list A. +drop-last N L R :- + length L M, I is M - N, take I L R. + +pred split-at i:int, i:list A, o:list A, o:list A. +split-at 0 L [] L :- !. +split-at N [X|XS] [X|LN] LM :- !, N1 is N - 1, split-at N1 XS LN LM. +split-at _ _ _ _ :- fatal-error "split-at run out of list items". + +pred fold i:list B, i:A, i:(B -> A -> A -> prop), o:A. +fold [] A _ A. +fold [X|XS] A F R :- F X A A1, fold XS A1 F R. + +pred fold2 i:list C, i:list B, i:A, i:(C -> B -> A -> A -> prop), o:A. +fold2 [] [_|_] _ _ _ :- fatal-error "fold2 on lists of different length". +fold2 [_|_] [] _ _ _ :- fatal-error "fold2 on lists of different length". +fold2 [] [] A _ A. +fold2 [X|XS] [Y|YS] A F R :- F X Y A A1, fold2 XS YS A1 F R. + +pred map i:list A, i:(A -> B -> prop), o:list B. +map [] _ []. +map [X|XS] F [Y|YS] :- F X Y, map XS F YS. + +pred map-i i:list A, i:(int -> A -> B -> prop), o:list B. +map-i L F R :- map-i.aux L 0 F R. +map-i.aux [] _ _ []. +map-i.aux [X|XS] N F [Y|YS] :- F N X Y, M is N + 1, map-i.aux XS M F YS. + +pred map-filter i:list A, i:(A -> B -> prop), o:list B. +map-filter [] _ []. +map-filter [X|XS] F [Y|YS] :- F X Y, !, map-filter XS F YS. +map-filter [_|XS] F YS :- map-filter XS F YS. + +:index(1 1) +pred map2 i:list A, i:list B, i:(A -> B -> C -> prop), o:list C. +map2 [] [_|_] _ _ :- fatal-error "map2 on lists of different length". +map2 [_|_] [] _ _ :- fatal-error "map2 on lists of different length". +map2 [] [] _ []. +map2 [X|XS] [Y|YS] F [Z|ZS] :- F X Y Z, map2 XS YS F ZS. + +pred map2-filter i:list A, i:list B, i:(A -> B -> C -> prop), o:list C. +map2-filter [] [_|_] _ _ :- fatal-error "map2-filter on lists of different length". +map2-filter [_|_] [] _ _ :- fatal-error "map2-filter on lists of different length". +map2-filter [] [] _ []. +map2-filter [X|XS] [Y|YS] F [Z|ZS] :- F X Y Z, !, map2-filter XS YS F ZS. +map2-filter [_|XS] [_|YS] F ZS :- map2-filter XS YS F ZS. + +pred map-ok i:list A, i:(A -> B -> diagnostic -> prop), o:list A, o:diagnostic. +map-ok [X|L] P [Y|YS] S :- P X Y S0, if (S0 = ok) (map-ok L P YS S) (S = S0). +map-ok [] _ [] ok. + +pred fold-map i:list A, i:B, i:(A -> B -> C -> B -> prop), o:list C, o:B. +fold-map [] A _ [] A. +fold-map [X|XS] A F [Y|YS] A2 :- F X A Y A1, fold-map XS A1 F YS A2. + +pred omap i:option A, i:(A -> B -> prop), o:option B. +omap none _ none. +omap (some X) F (some Y) :- F X Y. + +% [nth N L X] picks in X the N-th element of L (L must be of length > N) +pred nth i:int, i:list A, o:A. +nth 0 [X|_ ] R :- !, X = R. +nth N [_|XS] R :- N > 0, !, N1 is N - 1, nth N1 XS R. +nth N _ _ :- N < 0, !, fatal-error "nth got a negative index". +nth _ _ _ :- fatal-error "nth run out of list items". + +% [lookup L K V] sees L as a map from K to V +pred lookup i:list (pair A B), i:A, o:B. +lookup [pr X Y|_] X Y. +lookup [_|LS] X Y :- lookup LS X Y. + +% [lookup! L K V] sees L as a map from K to V, stops at the first binding +pred lookup! i:list (pair A B), i:A, o:B. +lookup! [pr X Y|_] X Y :- !. +lookup! [_|LS] X Y :- lookup! LS X Y. + +% [mem! L X] succeeds once if X occurs inside L +pred mem! i:list A, o:A. +mem! [X|_] X :- !. +mem! [_|L] X :- mem! L X. + +% [mem L X] succeeds every time if X occurs inside L +pred mem i:list A, o:A. +mem [X|_] X. +mem [_|L] X :- mem L X. + +pred exists i:list A, i:(A -> prop). +exists [X|_] P :- P X. +exists [_|L] P :- exists L P. + +pred exists2 i:list A, i:list B, i:(A -> B -> prop). +exists2 [] [_|_] _ :- fatal-error "exists2 on lists of different length". +exists2 [_|_] [] _ :- fatal-error "exists2 on lists of different length". +exists2 [X|_] [Y|_] P :- P X Y. +exists2 [_|L] [_|M] P :- exists2 L M P. + +pred forall i:list A, i:(A -> prop). +forall [] _. +forall [X|L] P :- P X, forall L P. + +pred forall-ok i:list A, i:(A -> diagnostic -> prop), o:diagnostic. +forall-ok [X|L] P S :- P X S0, if (S0 = ok) (forall-ok L P S) (S = S0). +forall-ok [] _ ok. + +pred forall2 i:list A, i:list B, i:(A -> B -> prop). +forall2 [] [_|_] _ :- fatal-error "forall2 on lists of different length". +forall2 [_|_] [] _ :- fatal-error "forall2 on lists of different length". +forall2 [X|XS] [Y|YS] P :- P X Y, forall2 XS YS P. +forall2 [] [] _. + +pred filter i:list A, i:(A -> prop), o:list A. +filter [] _ []. +filter [X|L] P R :- if (P X) (R = X :: L1) (R = L1), filter L P L1. + +pred zip i:list A, i:list B, o:list (pair A B). +zip [_|_] [] _ :- fatal-error "zip on lists of different length". +zip [] [_|_] _ :- fatal-error "zip on lists of different length". +zip [X|LX] [Y|LY] [pr X Y|LR] :- zip LX LY LR. +zip [] [] []. + +pred unzip i:list (pair A B), o:list A, o:list B. +unzip [] [] []. +unzip [pr X Y|L] [X|LX] [Y|LY] :- unzip L LX LY. + +pred flatten i:list (list A), o:list A. +flatten [X|LS] R :- flatten LS LS', append X LS' R. +flatten [] []. + +pred null i:list A. +null []. + +pred iota i:int, o:list int. +iota N L :- iota.aux 0 N L. +iota.aux X X [] :- !. +iota.aux N X [N|R] :- M is N + 1, iota.aux M X R. + +% [intersperse X L R] R is [L0, X, ..., X, LN] +:index(_ 1) +pred intersperse i:A, i:list A, o:list A. +intersperse _ [] []. +intersperse _ [X] [X] :- !. +intersperse Sep [X|XS] [X,Sep|YS] :- intersperse Sep XS YS. + +% -- Misc -- + +pred flip i:(A -> B -> prop), i:B, i:A. +flip P X Y :- P Y X. + +pred time i:prop, o:float. +time P T :- gettimeofday Before, P, gettimeofday After, T is After - Before. + +pred do! i:list prop. +do! []. +do! [P|PS] :- P, !, do! PS. + +:index(_ 1) +pred do-ok! o:diagnostic, i:list (diagnostic -> prop). +do-ok! ok []. +do-ok! S [P|PS] :- P S0, !, if (S0 = ok) (do-ok! S PS) (S = S0). + +pred lift-ok i:prop, i:string, o:diagnostic. +lift-ok P Msg R :- (P, R = ok; R = error Msg). + +pred spy-do! i:list prop. +spy-do! L :- map L (x\y\y = spy x) L1, do! L1. + +pred while-ok-do! i:diagnostic, i:list (diagnostic -> prop), o:diagnostic. +while-ok-do! (error _ as E) _ E. +while-ok-do! ok [] ok. +while-ok-do! ok [P|PS] R :- P C, !, while-ok-do! C PS R. + +pred any->string i:A, o:string. +any->string X Y :- term_to_string X Y. + +pred max i:A, i:A, o:A. +max N M N :- N >= M, !. +max _ M M. + +% [findall P L] L is the list [P1,P2,P3..] where each Pi is a solution to P. +pred findall i:prop, o:list prop. +findall P L :- findall_solutions P L. + +} + +% [std.string.concat Separator Items Result] concatenates Items +% interspersing Separator +external pred std.string.concat i:string, i:list string, o:string. + +% CAVEAT: the type parameter of std.string.map must be a closed term + +kind std.string.map type -> type. + +% [std.string.map.empty M] The empty map +external pred std.string.map.empty o:std.string.map A. + +% [std.string.map.mem S M] Checks if S is bound in M +external pred std.string.map.mem i:string, i:std.string.map A. + +% [std.string.map.add S V M M1] M1 is M where V is bound to S +external pred std.string.map.add i:string, i:A, i:std.string.map A, + o:std.string.map A. + +% [std.string.map.remove S M M1] M1 is M where S is unbound +external pred std.string.map.remove i:string, i:std.string.map A, + o:std.string.map A. + +% [std.string.map.find S M V] V is the binding of S in M +external pred std.string.map.find i:string, i:std.string.map A, o:A. + +% [std.string.map.bindings M L] L is M transformed into an associative list +external pred std.string.map.bindings i:std.string.map A, + o:list (pair string A). + +% CAVEAT: the type parameter of std.int.map must be a closed term + +kind std.int.map type -> type. + +% [std.int.map.empty M] The empty map +external pred std.int.map.empty o:std.int.map A. + +% [std.int.map.mem S M] Checks if S is bound in M +external pred std.int.map.mem i:int, i:std.int.map A. + +% [std.int.map.add S V M M1] M1 is M where V is bound to S +external pred std.int.map.add i:int, i:A, i:std.int.map A, + o:std.int.map A. + +% [std.int.map.remove S M M1] M1 is M where S is unbound +external pred std.int.map.remove i:int, i:std.int.map A, o:std.int.map A. + +% [std.int.map.find S M V] V is the binding of S in M +external pred std.int.map.find i:int, i:std.int.map A, o:A. + +% [std.int.map.bindings M L] L is M transformed into an associative list +external pred std.int.map.bindings i:std.int.map A, o:list (pair int A). + +% CAVEAT: the type parameter of std.loc.map must be a closed term + +kind std.loc.map type -> type. + +% [std.loc.map.empty M] The empty map +external pred std.loc.map.empty o:std.loc.map A. + +% [std.loc.map.mem S M] Checks if S is bound in M +external pred std.loc.map.mem i:loc, i:std.loc.map A. + +% [std.loc.map.add S V M M1] M1 is M where V is bound to S +external pred std.loc.map.add i:loc, i:A, i:std.loc.map A, + o:std.loc.map A. + +% [std.loc.map.remove S M M1] M1 is M where S is unbound +external pred std.loc.map.remove i:loc, i:std.loc.map A, o:std.loc.map A. + +% [std.loc.map.find S M V] V is the binding of S in M +external pred std.loc.map.find i:loc, i:std.loc.map A, o:A. + +% [std.loc.map.bindings M L] L is M transformed into an associative list +external pred std.loc.map.bindings i:std.loc.map A, o:list (pair loc A). + +kind std.string.set type. + +% [std.string.set.empty A] The empty set +external pred std.string.set.empty o:std.string.set. + +% [std.string.set.mem Elem A] Checks if Elem is in a +external pred std.string.set.mem i:string, i:std.string.set. + +% [std.string.set.add Elem A B] B is A union {Elem} +external pred std.string.set.add i:string, i:std.string.set, + o:std.string.set. + +% [std.string.set.remove Elem A B] B is A \ {Elem} +external pred std.string.set.remove i:string, i:std.string.set, + o:std.string.set. + +% [std.string.set.union A B X] X is A union B +external pred std.string.set.union i:std.string.set, i:std.string.set, + o:std.string.set. + +% [std.string.set.inter A B X] X is A intersection B +external pred std.string.set.inter i:std.string.set, i:std.string.set, + o:std.string.set. + +% [std.string.set.diff A B X] X is A \ B +external pred std.string.set.diff i:std.string.set, i:std.string.set, + o:std.string.set. + +% [std.string.set.equal A B] tests A and B for equality +external pred std.string.set.equal i:std.string.set, i:std.string.set. + +% [std.string.set.subset A B] tests if A is a subset of B +external pred std.string.set.subset i:std.string.set, i:std.string.set. + +% [std.string.set.elements M L] L is M transformed into list +external pred std.string.set.elements i:std.string.set, o:list string. + +% [std.string.set.cardinal M N] N is the number of elements of M +external pred std.string.set.cardinal i:std.string.set, o:int. + +kind std.int.set type. + +% [std.int.set.empty A] The empty set +external pred std.int.set.empty o:std.int.set. + +% [std.int.set.mem Elem A] Checks if Elem is in a +external pred std.int.set.mem i:int, i:std.int.set. + +% [std.int.set.add Elem A B] B is A union {Elem} +external pred std.int.set.add i:int, i:std.int.set, o:std.int.set. + +% [std.int.set.remove Elem A B] B is A \ {Elem} +external pred std.int.set.remove i:int, i:std.int.set, o:std.int.set. + +% [std.int.set.union A B X] X is A union B +external pred std.int.set.union i:std.int.set, i:std.int.set, + o:std.int.set. + +% [std.int.set.inter A B X] X is A intersection B +external pred std.int.set.inter i:std.int.set, i:std.int.set, + o:std.int.set. + +% [std.int.set.diff A B X] X is A \ B +external pred std.int.set.diff i:std.int.set, i:std.int.set, + o:std.int.set. + +% [std.int.set.equal A B] tests A and B for equality +external pred std.int.set.equal i:std.int.set, i:std.int.set. + +% [std.int.set.subset A B] tests if A is a subset of B +external pred std.int.set.subset i:std.int.set, i:std.int.set. + +% [std.int.set.elements M L] L is M transformed into list +external pred std.int.set.elements i:std.int.set, o:list int. + +% [std.int.set.cardinal M N] N is the number of elements of M +external pred std.int.set.cardinal i:std.int.set, o:int. + +kind std.loc.set type. + +% [std.loc.set.empty A] The empty set +external pred std.loc.set.empty o:std.loc.set. + +% [std.loc.set.mem Elem A] Checks if Elem is in a +external pred std.loc.set.mem i:loc, i:std.loc.set. + +% [std.loc.set.add Elem A B] B is A union {Elem} +external pred std.loc.set.add i:loc, i:std.loc.set, o:std.loc.set. + +% [std.loc.set.remove Elem A B] B is A \ {Elem} +external pred std.loc.set.remove i:loc, i:std.loc.set, o:std.loc.set. + +% [std.loc.set.union A B X] X is A union B +external pred std.loc.set.union i:std.loc.set, i:std.loc.set, + o:std.loc.set. + +% [std.loc.set.inter A B X] X is A intersection B +external pred std.loc.set.inter i:std.loc.set, i:std.loc.set, + o:std.loc.set. + +% [std.loc.set.diff A B X] X is A \ B +external pred std.loc.set.diff i:std.loc.set, i:std.loc.set, + o:std.loc.set. + +% [std.loc.set.equal A B] tests A and B for equality +external pred std.loc.set.equal i:std.loc.set, i:std.loc.set. + +% [std.loc.set.subset A B] tests if A is a subset of B +external pred std.loc.set.subset i:std.loc.set, i:std.loc.set. + +% [std.loc.set.elements M L] L is M transformed into list +external pred std.loc.set.elements i:std.loc.set, o:list loc. + +% [std.loc.set.cardinal M N] N is the number of elements of M +external pred std.loc.set.cardinal i:std.loc.set, o:int. + +#line 0 "builtin_map.elpi" +kind std.map type -> type -> type. +type std.map std.map.private.map K V -> (K -> K -> cmp -> prop) -> std.map K V. + +namespace std.map { + +% [make Eq Ltn M] builds an empty map M where keys are compared using Eq and Ltn +pred make i:(K -> K -> cmp -> prop), o:std.map K V. +make Cmp (std.map private.empty Cmp). + +% [find K M V] looks in M for the value V associated to K +pred find i:K, i:std.map K V, o:V. +find K (std.map M Cmp) V :- private.find M Cmp K V. + +% [add K V M M1] M1 is M where K is bound to V +pred add i:K, i:V, i:std.map K V, o:std.map K V. +add K V (std.map M Cmp) (std.map M1 Cmp) :- private.add M Cmp K V M1. + +% [remove K M M1] M1 is M where K is unbound +pred remove i:K, i:std.map K V, o:std.map K V. +remove K (std.map M Cmp) (std.map M1 Cmp) :- private.remove M Cmp K M1. + +% [bindings M L] L is the key-value pairs in increasing order +pred bindings i:std.map K V, o:list (pair K V). +bindings (std.map M _) L :- private.bindings M [] L. + +namespace private { + +% Taken from OCaml's map.ml +kind map type -> type -> type. +type empty map K V. +type node map K V -> K -> V -> map K V -> int -> map K V. + +pred height i:map K V, o:int. +height empty 0. +height (node _ _ _ _ H) H. + +pred create i:map K V, i:K, i:V, i:map K V, o:map K V. +create L K V R (node L K V R H) :- H is {std.max {height L} {height R}} + 1. + +pred bal i:map K V, i:K, i:V, i:map K V, o:map K V. +bal L K V R T :- + height L HL, + height R HR, + HL2 is HL + 2, + HR2 is HR + 2, + bal.aux HL HR HL2 HR2 L K V R T. + +bal.aux HL _ _ HR2 (node LL LV LD LR _) X D R T :- + HL > HR2, {height LL} >= {height LR}, !, + create LL LV LD {create LR X D R} T. +bal.aux HL _ _ HR2 (node LL LV LD (node LRL LRV LRD LRR _) _) X D R T :- + HL > HR2, !, + create {create LL LV LD LRL} LRV LRD {create LRR X D R} T. +bal.aux _ HR HL2 _ L X D (node RL RV RD RR _) T :- + HR > HL2, {height RR} >= {height RL}, !, + create {create L X D RL} RV RD RR T. +bal.aux _ HR HL2 _ L X D (node (node RLL RLV RLD RLR _) RV RD RR _) T :- + HR > HL2, !, + create {create L X D RLL} RLV RLD {create RLR RV RD RR} T. +bal.aux _ _ _ _ L K V R T :- create L K V R T. + +pred add i:map K V, i:(K -> K -> cmp -> prop), i:K, i:V, o:map K V. +add empty _ K V T :- create empty K V empty T. +add (node _ X _ _ _ as M) Cmp X1 XD M1 :- Cmp X1 X E, add.aux E M Cmp X1 XD M1. +add.aux eq (node L _ _ R H) _ X XD T :- T = node L X XD R H. +add.aux lt (node L V D R _) Cmp X XD T :- bal {add L Cmp X XD} V D R T. +add.aux gt (node L V D R _) Cmp X XD T :- bal L V D {add R Cmp X XD} T. + +pred find i:map K V, i:(K -> K -> cmp -> prop), i:K, o:V. +find (node L K1 V1 R _) Cmp K V :- Cmp K K1 E, find.aux E Cmp L R V1 K V. +find.aux eq _ _ _ V _ V. +find.aux lt Cmp L _ _ K V :- find L Cmp K V. +find.aux gt Cmp _ R _ K V :- find R Cmp K V. + +pred remove-min-binding i:map K V, o:map K V. +remove-min-binding (node empty _ _ R _) R :- !. +remove-min-binding (node L V D R _) X :- bal {remove-min-binding L} V D R X. + +pred min-binding i:map K V, o:K, o:V. +min-binding (node empty V D _ _) V D :- !. +min-binding (node L _ _ _ _) V D :- min-binding L V D. + +pred merge i:map K V, i:map K V, o:map K V. +merge empty X X :- !. +merge X empty X :- !. +merge M1 M2 R :- + min-binding M2 X D, + bal M1 X D {remove-min-binding M2} R. + +pred remove i:map K V, i:(K -> K -> cmp -> prop), i:K, o:map K V. +remove empty _ _ empty :- !. +remove (node L V D R _) Cmp X M :- Cmp X V E, remove.aux E Cmp L R V D X M. +remove.aux eq _ L R _ _ _ M :- merge L R M. +remove.aux lt Cmp L R V D X M :- bal {remove L Cmp X} V D R M. +remove.aux gt Cmp L R V D X M :- bal L V D {remove R Cmp X} M. + +pred bindings i:map K V, i:list (pair K V), o:list (pair K V). +bindings empty X X. +bindings (node L V D R _) X X1 :- + bindings L [pr V D|{bindings R X}] X1. + + +} % std.map.private +} % std.map + + +#line 0 "builtin_set.elpi" +kind std.set type -> type. +type std.set std.set.private.set E -> (E -> E -> cmp -> prop) -> std.set E. + +namespace std.set { + +% [make Eq Ltn M] builds an empty set M where keys are compared using Eq and Ltn +pred make i:(E -> E -> cmp -> prop), o:std.set E. +make Cmp (std.set private.empty Cmp). + +% [mem E M] looks if E is in M +pred mem i:E, i:std.set E. +mem E (std.set M Cmp):- private.mem M Cmp E. + +% [add E M M1] M1 is M + {E} +pred add i:E, i:std.set E, o:std.set E. +add E (std.set M Cmp) (std.set M1 Cmp) :- private.add M Cmp E M1. + +% [remove E M M1] M1 is M - {E} +pred remove i:E, i:std.set E, o:std.set E. +remove E (std.set M Cmp) (std.set M1 Cmp) :- private.remove M Cmp E M1. + +% [cardinal S N] N is the number of elements of S +pred cardinal i:std.set E, o:int. +cardinal (std.set M _) N :- private.cardinal M N. + +pred elements i:std.set E, o:list E. +elements (std.set M _) L :- private.elements M [] L. + +namespace private { + +% Taken from OCaml's set.ml +kind set type -> type. +type empty set E. +type node set E -> E -> set E -> int -> set E. + +pred height i:set E, o:int. +height empty 0. +height (node _ _ _ H) H. + +pred create i:set E, i:E, i:set E, o:set E. +create L E R (node L E R H) :- H is {std.max {height L} {height R}} + 1. + +pred bal i:set E, i:E, i:set E, o:set E. +bal L E R T :- + height L HL, + height R HR, + HL2 is HL + 2, + HR2 is HR + 2, + bal.aux HL HR HL2 HR2 L E R T. + +bal.aux HL _ _ HR2 (node LL LV LR _) X R T :- + HL > HR2, {height LL} >= {height LR}, !, + create LL LV {create LR X R} T. +bal.aux HL _ _ HR2 (node LL LV (node LRL LRV LRR _) _) X R T :- + HL > HR2, !, + create {create LL LV LRL} LRV {create LRR X R} T. +bal.aux _ HR HL2 _ L X (node RL RV RR _) T :- + HR > HL2, {height RR} >= {height RL}, !, + create {create L X RL} RV RR T. +bal.aux _ HR HL2 _ L X (node (node RLL RLV RLR _) RV RR _) T :- + HR > HL2, !, + create {create L X RLL} RLV {create RLR RV RR} T. +bal.aux _ _ _ _ L E R T :- create L E R T. + +pred add i:set E, i:(E -> E -> cmp -> prop), i:E, o:set E. +add empty _ E T :- create empty E empty T. +add (node L X R H) Cmp X1 S :- Cmp X1 X E, add.aux E Cmp L R X X1 H S. +add.aux eq _ L R X _ H (node L X R H). +add.aux lt Cmp L R E X _ T :- bal {add L Cmp X} E R T. +add.aux gt Cmp L R E X _ T :- bal L E {add R Cmp X} T. + +pred mem i:set E, i:(E -> E -> cmp -> prop), i:E. +mem (node L K R _) Cmp E :- Cmp E K O, mem.aux O Cmp L R E. +mem.aux eq _ _ _ _. +mem.aux lt Cmp L _ E :- mem L Cmp E. +mem.aux gt Cmp _ R E :- mem R Cmp E. + +pred remove-min-binding i:set E, o:set E. +remove-min-binding (node empty _ R _) R :- !. +remove-min-binding (node L E R _) X :- bal {remove-min-binding L} E R X. + +pred min-binding i:set E, o:E. +min-binding (node empty E _ _) E :- !. +min-binding (node L _ _ _) E :- min-binding L E. + +pred merge i:set E, i:set E, o:set E. +merge empty X X :- !. +merge X empty X :- !. +merge M1 M2 R :- + min-binding M2 X, + bal M1 X {remove-min-binding M2} R. + +pred remove i:set E, i:(E -> E -> cmp -> prop), i:E, o:set E. +remove empty _ _ empty. +remove (node L E R _) Cmp X M :- Cmp X E O, remove.aux O Cmp L R E X M. +remove.aux eq _ L R _ _ M :- merge L R M. +remove.aux lt Cmp L R E X M :- bal {remove L Cmp X} E R M. +remove.aux gt Cmp L R E X M :- bal L E {remove R Cmp X} M. + +pred cardinal i:set E, o:int. +cardinal empty 0. +cardinal (node L _ R _) N :- N is {cardinal L} + 1 + {cardinal R}. + +pred elements i:set E, i:list E, o:list E. +elements empty X X. +elements (node L E R _) Acc X :- + elements L [E|{elements R Acc}] X. + +} % std.set.private +} % std.set + + +% == Elpi runtime builtins ===================================== + +% [trace.counter Name Value] reads the Value of a trace point Name +external pred trace.counter i:string, o:int. + +% [gc.get MinorHeapSize MajorHeapIncrement SpaceOverhead Verbose MaxOverhead +% StackLimit AllocationPolicy WindowSize] Reads the current settings of the +% garbage collector. See also OCaml's Gc.control type documentation. +external pred gc.get o:int, o:int, o:int, o:int, o:int, o:int, o:int, + o:int. + +% [gc.set MinorHeapSize MajorHeapIncrement SpaceOverhead Verbose MaxOverhead +% StackLimit AllocationPolicy WindowSize] Writes the current settings of the +% garbage collector. Any parameter left unspecificed (eg _) is not changed. +% See also OCaml's Gc.control type documentation. +external pred gc.set i:int, i:int, i:int, i:int, i:int, i:int, i:int, + i:int. + +% [gc.minor] See OCaml's Gc.minor documentation. +external pred gc.minor . + +% [gc.major] See OCaml's Gc.major documentation. +external pred gc.major . + +% [gc.full] See OCaml's Gc.full_major documentation. +external pred gc.full . + +% [gc.compact] See OCaml's Gc.compact documentation. +external pred gc.compact . + +% [gc.stat MinorWords PromotedWords MajorWords MinorCollections +% MajorCollections HeapWords HeapChunks LiveWords LiveBlocks FreeWords +% FreeBlocks LargestFree Fragments Compactions TopHeapWords StackSize] See +% OCaml's Gc.stat documentation. +external pred gc.stat o:float, o:float, o:float, o:int, o:int, o:int, + o:int, o:int, o:int, o:int, o:int, o:int, o:int, + o:int, o:int, o:int. + +% [gc.quick-stat MinorWords PromotedWords MajorWords MinorCollections +% MajorCollections HeapWords HeapChunks Compactions TopHeapWords StackSize] +% See OCaml's Gc.quick_stat documentation. +external pred gc.quick-stat o:float, o:float, o:float, o:int, o:int, + o:int, o:int, o:int, o:int, o:int. + + + + diff --git a/out/lambdapi.pkg b/out/lambdapi.pkg new file mode 100644 index 000000000..64a65dab8 --- /dev/null +++ b/out/lambdapi.pkg @@ -0,0 +1,2 @@ +package_name = coq +root_path = coq diff --git a/src/cli/lambdapi.ml b/src/cli/lambdapi.ml index 0b05e9fcb..2338b1501 100644 --- a/src/cli/lambdapi.ml +++ b/src/cli/lambdapi.ml @@ -23,6 +23,7 @@ let check_cmd : Config.t -> int option -> string list -> unit = let run _ = let open Timed in Config.init cfg; + Elpi_handle.init (); (* We save time to run each file in the same environment. *) let time = Time.save () in let handle file = diff --git a/src/common/error.ml b/src/common/error.ml index 975284cc7..4eebc33d2 100644 --- a/src/common/error.ml +++ b/src/common/error.ml @@ -73,4 +73,6 @@ let handle_exceptions : (unit -> unit) -> unit = fun f -> try f () with | Fatal(None, msg) -> exit_with "%s" msg | Fatal(Some(p), msg) -> exit_with "[%a] %s" Pos.pp p msg + (* | e -> exit_with "Uncaught [%s]." (Printexc.to_string e) + *) diff --git a/src/core/ctxt.ml b/src/core/ctxt.ml index abca46037..0ac9b4059 100644 --- a/src/core/ctxt.ml +++ b/src/core/ctxt.ml @@ -4,6 +4,17 @@ open Term open Lplib open Timed +(** [unbind ctx a def b] returns a triple [(x,t,new_ctx)] such that [(x,t)] is + an unbinding of [b] (in the sense of [Bindlib.unbind]) and [new_ctx] is an + extension of context [ctx] with the assumption that [x] has type [a] (only + if [x] occurs in [t]). If [def] is of the form [Some(u)], the context also + registers the term [u] as the definition of variable [x]. *) +let unbind : 'a actxt -> 'a -> term option -> tbinder -> + tvar * term * 'a actxt = + fun ctx a def b -> + let (x, t) = Bindlib.unbind b in + (x, t, if Bindlib.binder_occur b then (x, a, def) :: ctx else ctx) + (** [type_of x ctx] returns the type of [x] in the context [ctx] when it appears in it, and @raise [Not_found] otherwise. *) diff --git a/src/core/dune b/src/core/dune index d0f25c063..4bbaa79de 100644 --- a/src/core/dune +++ b/src/core/dune @@ -10,4 +10,4 @@ (public_name lambdapi.core) (synopsis "LambdaPi interactive theorem prover [core]") (modules :standard) - (libraries lambdapi.common lambdapi.lplib pratter bindlib why3)) + (libraries lambdapi.common lambdapi.lplib pratter bindlib why3 elpi)) diff --git a/src/core/elpi_lambdapi.ml b/src/core/elpi_lambdapi.ml new file mode 100644 index 000000000..457d9dd74 --- /dev/null +++ b/src/core/elpi_lambdapi.ml @@ -0,0 +1,259 @@ +open Elpi.API + +module Elpi_AUX = struct + let array_map_fold f st a = + let len = Array.length a in + let st = ref st in + let b = Array.make len RawData.mkNil in + for i = 0 to len-1 do + let st', x = f !st a.(i) in + st := st'; + b.(i) <- x + done; + !st, b + + let list_map_fold f s l = + let f st x = let st, x = f st x in st, x, [] in + let s, l, _ = Utils.map_acc f s l in + s, l + + let loc_of_pos = function + | None -> Ast.Loc.initial "(elpi)" + | Some x -> + let { Common.Pos.fname; start_line; start_col; _ } = x in + { + Ast.Loc.source_name = + (match fname with None -> "(.)" | Some x -> x); + source_start = 0; + source_stop = 0; + line = start_line; + line_starts_at = start_col; + } + +end + +(** Terms.sym is exposed to Elpi as an opaque data type (no syntax like int or + string). APIs are provided to manipulate symbols, eg get their type *) +let sym : Term.sym Conversion.t = OpaqueData.declare { + OpaqueData.name = "symbol"; + doc = "A symbol"; + pp = Print.sym; + compare = Term.Sym.compare; + hash = Hashtbl.hash; + hconsed = false; + constants = []; +} + +(** Waiting for a ppx to do all the work for us, we code by hand the + conversion of Terms.term *) + +(* Allocate Elpi symbols for the term constructors (type and kind are Elpi + keywords, hence typ and kin) *) +let typec = RawData.Constants.declare_global_symbol "typ" +let kindc = RawData.Constants.declare_global_symbol "kin" +let symbc = RawData.Constants.declare_global_symbol "symb" +let prodc = RawData.Constants.declare_global_symbol "prod" +let abstc = RawData.Constants.declare_global_symbol "abst" +let applc = RawData.Constants.declare_global_symbol "appl" + +(* A two way map linking Elpi's unification variable and Terms.meta. + An instance of this map is part of the Elpi state (threaded by many + APIs) *) +module M = struct + type t = Term.meta + let compare m1 m2 = Stdlib.compare m1.Term.meta_key m2.Term.meta_key + let pp = Print.meta + let show m = Format.asprintf "%a" pp m +end +module MM = FlexibleData.Map(M) +let metamap : MM.t State.component = MM.uvmap + +let pb = State.declare ~name:"elpi:problem" + ~pp:(fun fmt (p : Term.problem) -> + Format.fprintf fmt "@["; + Term.MetaSet.iter (fun m -> + Format.fprintf fmt "%d@ " m.Term.meta_key) Timed.(! p).Term.metas; + Format.fprintf fmt "@]"; + ) ~init:Term.new_problem ~start:(fun x -> x) + + +(* Terms.term -> Data.term, we use Ctxt.ctxt to carry a link between + Bindlib's var to Elpi's De Bruijn levels *) +let embed_term : ?pats:(int * string) list -> ?ctx:RawData.constant Term.actxt -> Term.term Conversion.embedding = fun ?(pats=[]) ?(ctx=[]) ~depth st t -> + let open RawData in + let open Term in + let gls = ref [] in + let call f ~depth s x = + let s, x, g = f ~depth s x in gls := g @ !gls; s, x in + let rec aux ~depth ctx st t = + match Term.unfold t with + | Vari v -> + let d = Ctxt.type_of v ctx in + st, mkBound d + | Type -> st, mkGlobal typec + | Kind -> st, mkGlobal kindc + | Symb s -> + let st, s = call sym.Conversion.embed ~depth st s in + st, mkApp symbc s [] + | Prod (src, tgt) -> + let st, src = aux ~depth ctx st src in + let _,tgt,ctx = Ctxt.unbind ctx depth None tgt in + let st, tgt = aux ~depth:(depth+1) ctx st tgt in + st, mkApp prodc src [mkLam tgt] + | Abst (ty, body) -> + let st, ty = aux ~depth ctx st ty in + let _,body,ctx = Ctxt.unbind ctx depth None body in + let st, body = aux ~depth:(depth+1) ctx st body in + st, mkApp prodc ty [mkLam body] + | Appl (hd, arg) -> + let st, hd = aux ~depth ctx st hd in + let st, arg = aux ~depth ctx st arg in + st, mkApp applc hd [arg] + | Meta (meta,args) -> + let st, flex = + try st, MM.elpi meta (State.get metamap st) + with Not_found -> + let st, flex = FlexibleData.Elpi.make st in + State.update metamap st (MM.add flex meta), flex in + let st, args = Elpi_AUX.array_map_fold (aux ~depth ctx) st args in + st, mkUnifVar flex ~args:(Array.to_list args) st + | Plac _ -> + let args = List.map (fun (_,x,_) -> mkBound x) ctx in + let st, flex = FlexibleData.Elpi.make st in + let st, meta = State.update_return pb st (fun pb -> + let m1 = LibMeta.fresh pb mk_Type 0 in + let m2 = LibMeta.fresh pb (mk_Meta (m1,[||])) + (List.length args) in (* empty context is surely wrong *) + pb, m2) in + let st = State.update metamap st (MM.add flex meta) in + st, mkUnifVar flex ~args st + | Patt(Some i,_,_) -> begin + try + let x = List.assoc i pats in + let st, flex = FlexibleData.Elpi.make ~name:x st in + st, mkUnifVar flex ~args:[] st + with Not_found -> + let pats = List.map (fun (i,n) -> Printf.sprintf "%d :-> %s; " i n) pats in + Common.Error.fatal_no_pos "embed_term: unnamed pattern %d in map: %s" i (String.concat "" pats); + end + | Patt _ -> Common.Error.fatal_no_pos "embed_term: Patt not implemented" + | TEnv _ -> Common.Error.fatal_no_pos "embed_term: TEnv not implemented" + | Wild -> Common.Error.fatal_no_pos "embed_term: Wild not implemented" + | TRef _ -> Common.Error.fatal_no_pos "embed_term: TRef not implemented" + | LLet _ -> Common.Error.fatal_no_pos "embed_term: LLet not implemented" + in + let st, t = aux ~depth ctx st t in + st, t, List.rev !gls + +module IntMap = Map.Make(struct type t = int let compare = compare end) + +(* Data.term -> Terms.term. We use and IntMap to link Elpi's De Bruijn + levels to Bindlib's var *) +let readback_term_box : Term.term Bindlib.box Conversion.readback = +fun ~depth st t -> + let open RawData in + let open Term in + let gls = ref [] in + let call f ~depth s x = + let s, x, g = f ~depth s x in gls := g @ !gls; s, x in + let rec aux ~depth ctx st t = + match look ~depth t with + | Const c when c == typec -> st, _Type + | Const c when c == kindc -> st, _Kind + | Const c when c >= 0 -> + begin try + let v = IntMap.find c ctx in + st, _Vari v + with Not_found -> Utils.type_error "readback_term: free variable" end + | App(c,s,[]) when c == symbc -> + let st, s = call sym.Conversion.readback ~depth st s in + st, _Symb s + | App(c,ty,[bo]) when c == prodc -> + let st, ty = aux ~depth ctx st ty in + let st, bo = aux_lam ~depth ctx st bo in + st, _Prod ty bo + | App(c,ty,[bo]) when c == abstc -> + let st, ty = aux ~depth ctx st ty in + let st, bo = aux_lam ~depth ctx st bo in + st, _Abst ty bo + | App(c,hd,[arg]) when c == applc -> + let st, hd = aux ~depth ctx st hd in + let st, arg = aux ~depth ctx st arg in + st, _Appl hd arg + | UnifVar(flex, args) -> + let st, meta = + try st, MM.host flex (State.get metamap st) + with Not_found -> + let st, m2 = State.update_return pb st (fun pb -> + let m1 = LibMeta.fresh pb mk_Type 0 in + let m2 = LibMeta.fresh pb (mk_Meta (m1,[||])) + (List.length args) in (* empty context is surely wrong *) + pb, m2) in + State.update metamap st (MM.add flex m2), m2 + in + let st, args = Elpi_AUX.list_map_fold (aux ~depth ctx) st args in + st, _Meta meta (Array.of_list args) + | _ -> Utils.type_error "readback_term" + and aux_lam ~depth ctx st t = + match look ~depth t with + | Lam bo -> + let v = Bindlib.new_var of_tvar "x" in + let ctx = IntMap.add depth v ctx in + let st, bo = aux ~depth:(depth+1) ctx st bo in + st, Bindlib.bind_var v bo + | _ -> Utils.type_error "readback_term" + in + let st, t = aux ~depth IntMap.empty st t in + st, t, List.rev !gls + +let readback_term ~depth st t = + let st, t, gls = readback_term_box ~depth st t in + st, Bindlib.unbox t, gls + +(** Terms.term has a HOAS *) +let term : Term.term Conversion.t = { + Conversion.ty = Conversion.TyName "term"; + pp = Print.term; + pp_doc = (fun fmt () -> Format.fprintf fmt {| +kind term type. +type typ term. +type kin term. +type symb symbol -> term. +type appl term -> term -> term. +type abst term -> (term -> term) -> term. +type prod term -> (term -> term) -> term. + |}); + readback = readback_term; + embed = embed_term ?ctx:None ?pats:None; +} + +(** Assignments to Elpi's unification variables are a spine of lambdas + followed by an actual term. We read them back as a Bindlib.mbinder *) +let readback_mbinder st t = + let open RawData in + let rec aux ~depth nvars t = + match look ~depth t with + | Lam bo -> aux ~depth:(depth+1) (nvars+1) bo + | _ -> + let open Bindlib in + let vs = Array.init nvars (fun i -> + new_var Term.of_tvar (Printf.sprintf "x%d" i)) in + let st, t, _ = readback_term_box ~depth st t in + st, unbox (bind_mvar vs t) + in + aux ~depth:0 0 t +let readback_assignments st = + let mmap = State.get metamap st in + MM.fold (fun meta _flex body st -> + match body with + | None -> st + | Some t -> + let open Timed in + match ! (meta.Term.meta_value) with + | Some _ -> assert false + | None -> + let st, t = readback_mbinder st t in + meta.Term.meta_value := Some t; + st + ) mmap st + diff --git a/src/core/sig_state.ml b/src/core/sig_state.ml index 4d1e8cca4..ea8412c8a 100644 --- a/src/core/sig_state.ml +++ b/src/core/sig_state.ml @@ -41,16 +41,19 @@ let dummy : sig_state = builtins = StrMap.empty; notations = SymMap.empty; open_paths = Path.Set.empty } -(** [add_symbol ss expo prop mstrat opaq id typ impl def] generates a new +(** [add_symbol ss expo prop mstrat opaq id typ impl tc def] generates a new signature state from [ss] by creating a new symbol with expo [e], property [p], strategy [st], name [x], type [a], implicit arguments [impl] and optional definition [def]. This new symbol is returned too. *) let add_symbol : sig_state -> expo -> prop -> match_strat - -> bool -> strloc -> term -> bool list -> term option -> sig_state * sym = - fun ss expo prop mstrat opaq id typ impl def -> + -> bool -> strloc -> term -> bool list -> bool -> bool -> term option -> + sig_state * sym = + fun ss expo prop mstrat opaq id typ impl tc tci def -> let sym = Sign.add_symbol ss.signature expo prop mstrat opaq id (cleanup typ) impl in + if tc then Sign.add_tc ss.signature sym; + if tci then Sign.add_tc_inst ss.signature sym; begin match def with | Some t -> sym.sym_def := Some (cleanup t) diff --git a/src/core/sign.ml b/src/core/sign.ml index a40db549d..0c474724f 100644 --- a/src/core/sign.ml +++ b/src/core/sign.ml @@ -36,6 +36,8 @@ type t = ; sign_notations: float notation SymMap.t ref (** Maps symbols to their notation if they have some. *) ; sign_ind : ind_data SymMap.t ref + ; sign_tc : SymSet.t ref + ; sign_tc_inst : SymSet.t ref ; sign_cp_pos : cp_pos list SymMap.t ref (** Maps a symbol to the critical pair positions it is heading in the rules. *) } @@ -51,6 +53,7 @@ let dummy : unit -> t = fun () -> { sign_symbols = ref StrMap.empty; sign_path = [] ; sign_deps = ref Path.Map.empty; sign_builtins = ref StrMap.empty ; sign_notations = ref SymMap.empty; sign_ind = ref SymMap.empty + ; sign_tc = ref SymSet.empty; sign_tc_inst = ref SymSet.empty ; sign_cp_pos = ref SymMap.empty } (** [find sign name] finds the symbol named [name] in [sign] if it exists, and @@ -265,6 +268,8 @@ let read : string -> t = fun fname -> unsafe_reset sign.sign_notations; unsafe_reset sign.sign_ind; unsafe_reset sign.sign_cp_pos; + unsafe_reset sign.sign_tc; + unsafe_reset sign.sign_tc_inst; let shallow_reset_sym s = unsafe_reset s.sym_type; unsafe_reset s.sym_def; @@ -301,6 +306,8 @@ let read : string -> t = fun fname -> StrMap.iter (fun _ s -> reset_sym s) !(sign.sign_symbols); StrMap.iter (fun _ s -> shallow_reset_sym s) !(sign.sign_builtins); SymMap.iter (fun s _ -> shallow_reset_sym s) !(sign.sign_notations); + SymSet.iter (fun s -> reset_sym s) !(sign.sign_tc); + SymSet.iter (fun s -> reset_sym s) !(sign.sign_tc_inst); let f _ sm = StrMap.iter (fun _ rs -> List.iter reset_rule rs) sm in Path.Map.iter f !(sign.sign_deps); let reset_ind i = @@ -413,3 +420,12 @@ let rec dependencies : t -> (Path.t * t) list = fun sign -> | d::deps -> minimize ((List.filter not_here d) :: acc) deps in List.concat (minimize [] deps) + +let add_tc : t -> sym -> unit = + fun sign sym -> + sign.sign_tc := SymSet.add sym !(sign.sign_tc) + +let add_tc_inst : t -> sym -> unit = + fun sign sym -> + sign.sign_tc_inst := SymSet.add sym !(sign.sign_tc_inst); + diff --git a/src/core/term.ml b/src/core/term.ml index 0fa4f9f86..3981c3159 100644 --- a/src/core/term.ml +++ b/src/core/term.ml @@ -276,7 +276,8 @@ end first). Note that it cannot be implemented by a map as the order is important. *) type 'a actxt = (tvar * 'a * term option) list -type ctxt = term actx +type ctxt = term actxt + (** Typing context with lifted terms. Used to optimise type checking and avoid lifting terms several times. Definitions are not included because these contexts are used to create meta variables types, which do not use [let] @@ -350,7 +351,7 @@ type problem = problem_aux ref let new_problem : unit -> problem = fun () -> ref {to_solve = []; unsolved = []; recompute = false; metas = MetaSet.empty} -(** [create_sym path expo prop opaq name typ impl] creates a new symbol with +(** [create_sym path expo prop opaq name typ impl tc] creates a new symbol with path [path], exposition [expo], property [prop], opacity [opaq], matching strategy [mstrat], name [name.elt], type [typ], implicit arguments [impl], position [name.pos], no definition and no rules. *) diff --git a/src/core/term.mli b/src/core/term.mli index 4f528cb30..6cdcac610 100644 --- a/src/core/term.mli +++ b/src/core/term.mli @@ -257,7 +257,7 @@ end list [xn:An;..;x1:A1] in reverse order (last added variable comes first). *) type 'a actxt = (tvar * 'a * term option) list -type ctxt = term actx +type ctxt = term actxt (** Typing context with lifted terms. Used to optimise type checking and avoid lifting terms several times. Definitions are not included because these @@ -294,10 +294,11 @@ module Sym : Map.OrderedType with type t = sym module SymSet : Set.S with type elt = sym module SymMap : Map.S with type key = sym -(** [create_sym path expo prop opaq name typ impl] creates a new symbol with +(** [create_sym path expo prop opaq name typ impl tc] creates a new symbol with position [pos], path [path], exposition [expo], property [prop], opacity [opaq], matching strategy [mstrat], name [name.elt], type [typ], implicit - arguments [impl], position [name.pos], no definition and no rules. *) + arguments [impl], position [name.pos], typeclass [tc] no definition and no + rules. *) val create_sym : Path.t -> expo -> prop -> match_strat -> bool -> Pos.strloc -> term -> bool list -> sym diff --git a/src/core/unif_rule.ml b/src/core/unif_rule.ml index eb2b73f89..1a9972201 100644 --- a/src/core/unif_rule.ml +++ b/src/core/unif_rule.ml @@ -10,13 +10,15 @@ open Term (** Symbol "≡". *) let equiv : sym = let id = Pos.none "≡" in - let s = Sign.add_symbol Ghost.sign Public Defin Eager false id mk_Kind [] in + let s = + Sign.add_symbol Ghost.sign Public Defin Eager false id mk_Kind [] in Sign.add_notation Ghost.sign s (Infix(Pratter.Neither, 2.0)); s (** Symbol ";". *) let cons : sym = let id = Pos.none ";" in - let s = Sign.add_symbol Ghost.sign Public Const Eager true id mk_Kind [] in + let s = + Sign.add_symbol Ghost.sign Public Const Eager true id mk_Kind [] in Sign.add_notation Ghost.sign s (Infix(Pratter.Right, 1.0)); s (** [unpack eqs] transforms a term of the form diff --git a/src/export/coq.ml b/src/export/coq.ml index c7b86cdf8..f944afb75 100644 --- a/src/export/coq.ml +++ b/src/export/coq.ml @@ -55,6 +55,8 @@ let modifier : p_modifier pp = fun ppf {elt; _} -> | P_mstrat(s) -> Print.match_strat ppf s | P_prop(p) -> Print.prop ppf p | P_opaq -> out ppf "opaque " + | P_typeclass -> assert false (* TODO *) + | P_typeclass_instance -> assert false (* TODO *) (* ends with a space if the list is not empty *) let modifiers : p_modifier list pp = List.pp modifier "" @@ -271,6 +273,7 @@ let command : p_command pp = fun ppf { elt; _ } -> (Option.pp (unit "@," |+ proof)) p_sym_prf end | P_unif_rule _ -> () + | P_elpi _ -> () end let ast : ast pp = fun ppf -> Stream.iter (command ppf) diff --git a/src/handle/command.ml b/src/handle/command.ml index 07924450b..6f4b3c052 100644 --- a/src/handle/command.ml +++ b/src/handle/command.ml @@ -80,17 +80,19 @@ let handle_require_as : compiler -> sig_state -> p_path -> p_ident -> (** [handle_modifiers ms] verifies that the modifiers in [ms] are compatible. If so, they are returned as a tuple. Otherwise, it fails. *) -let handle_modifiers : p_modifier list -> prop * expo * match_strat = +let handle_modifiers : p_modifier list -> prop * expo * match_strat * bool * bool = fun ms -> - let rec get_modifiers ((props, expos, strats) as acc) = function + let rec get_modifiers ((props, expos, strats,tc,tci) as acc) = function | [] -> acc - | {elt=P_prop _;_} as p::ms -> get_modifiers (p::props, expos, strats) ms - | {elt=P_expo _;_} as e::ms -> get_modifiers (props, e::expos, strats) ms + | {elt=P_typeclass;_}::ms -> get_modifiers (props, expos, strats,true, tci) ms + | {elt=P_typeclass_instance;_}::ms -> get_modifiers (props, expos, strats,tc, true) ms + | {elt=P_prop _;_} as p::ms -> get_modifiers (p::props, expos, strats,tc,tci) ms + | {elt=P_expo _;_} as e::ms -> get_modifiers (props, e::expos, strats,tc,tci) ms | {elt=P_mstrat _;_} as s::ms -> - get_modifiers (props, expos, s::strats) ms + get_modifiers (props, expos, s::strats,tc,tci) ms | {elt=P_opaq;_}::ms -> get_modifiers acc ms in - let props, expos, strats = get_modifiers ([],[],[]) ms in + let props, expos, strats, tc, tci = get_modifiers ([],[],[],false,false) ms in let prop = match props with | [{elt=P_prop (Assoc b);_};{elt=P_prop Commu;_}] @@ -119,7 +121,7 @@ let handle_modifiers : p_modifier list -> prop * expo * match_strat = | [] -> Eager | _ -> assert false in - (prop, expo, strat) + (prop, expo, strat, tc, tci) (** [check_rule ss syms r] checks rule [r] and returns the head symbol of the lhs and the rule itself. *) @@ -163,7 +165,8 @@ let handle_inductive_symbol : sig_state -> expo -> prop -> match_strat end; (* Actually add the symbol to the signature and the state. *) Console.out 2 (Color.red "symbol %a : %a") uid name term typ; - let r = Sig_state.add_symbol ss expo prop mstrat false id typ impl None in + let r = + Sig_state.add_symbol ss expo prop mstrat false id typ impl false false None in sig_state := fst r; r (** Representation of a yet unchecked proof. The structure is initialized when @@ -195,6 +198,8 @@ let get_proof_data : compiler -> sig_state -> p_command -> cmd_output = Console.out 3 (Color.cya "%a") Pos.pp pos; Console.out 4 "%a" Pretty.command cmd; match elt with + | P_elpi(file,pred,arg) -> + (ss, None, (Elpi_handle.run ss file pred arg; None)) | P_query(q) -> (ss, None, Query.handle ss None q) | P_require(b,ps) -> (List.fold_left (handle_require compile b) ss ps, None, None) @@ -286,7 +291,11 @@ let get_proof_data : compiler -> sig_state -> p_command -> cmd_output = | P_inductive(ms, params, p_ind_list) -> (* Check modifiers. *) - let (prop, expo, mstrat) = handle_modifiers ms in + let (prop, expo, mstrat,tc,tci) = handle_modifiers ms in + if tc then + fatal pos "Property typeclass cannot be used on inductive types."; + if tci then + fatal pos "Property instance cannot be used on inductive types."; if prop <> Defin then fatal pos "Property modifiers cannot be used on inductive types."; if mstrat <> Eager then @@ -354,7 +363,8 @@ let get_proof_data : compiler -> sig_state -> p_command -> cmd_output = let pos = after (end_pos pos) in let id = Pos.make pos rec_name in let r = - Sig_state.add_symbol ss expo Defin Eager false id rec_typ [] None + Sig_state.add_symbol + ss expo Defin Eager false id rec_typ [] false false None in sig_state := fst r; r in (ss, rec_sym::rec_sym_list) @@ -392,7 +402,7 @@ let get_proof_data : compiler -> sig_state -> p_command -> cmd_output = | _ -> () end; (* Verify modifiers. *) - let prop, expo, mstrat = handle_modifiers p_sym_mod in + let prop, expo, mstrat, tc, tci = handle_modifiers p_sym_mod in let opaq = List.exists Syntax.is_opaq p_sym_mod in let pdata_prv = expo = Privat || (p_sym_def && opaq) in (match p_sym_def, opaq, prop, mstrat with @@ -495,7 +505,7 @@ let get_proof_data : compiler -> sig_state -> p_command -> cmd_output = wrn pe.pos "Proof admitted."; let t = Option.map (fun t -> t.elt) t in fst (Sig_state.add_symbol - ss expo prop mstrat opaq p_sym_nam a impl t) + ss expo prop mstrat opaq p_sym_nam a impl tc tci t) | P_proof_end -> (* Check that the proof is indeed finished. *) if not (finished ps) then @@ -506,7 +516,7 @@ let get_proof_data : compiler -> sig_state -> p_command -> cmd_output = (* Add the symbol in the signature. *) Console.out 2 (Color.red "symbol %a : %a") uid id term a; fst (Sig_state.add_symbol - ss expo prop mstrat opaq p_sym_nam a impl d) + ss expo prop mstrat opaq p_sym_nam a impl tc tci d) in (* Create the proof state. *) let pdata_state = @@ -554,8 +564,10 @@ let get_proof_data : compiler -> sig_state -> p_command -> cmd_output = | Fatal(Some(Some(_)),_) as e -> raise e | Fatal(None ,m) -> fatal pos "Error on command.@.%s" m | Fatal(Some(None) ,m) -> fatal pos "Error on command.@.%s" m + (* | e -> fatal pos "Uncaught exception: %s." (Printexc.to_string e) +*) (** [handle compile_mod ss cmd] retrieves proof data from [cmd] (with {!val:get_proof_data}) and handles proofs using functions from diff --git a/src/handle/dune b/src/handle/dune index 05be56087..8703f3f57 100644 --- a/src/handle/dune +++ b/src/handle/dune @@ -2,4 +2,4 @@ (name handle) (public_name lambdapi.handle) (modules :standard) - (libraries lambdapi.core lambdapi.parsing lambdapi.tool)) + (libraries lambdapi.core lambdapi.parsing lambdapi.tool elpi)) diff --git a/src/handle/elpi_handle.ml b/src/handle/elpi_handle.ml new file mode 100644 index 000000000..e177a29d2 --- /dev/null +++ b/src/handle/elpi_handle.ml @@ -0,0 +1,269 @@ +open Elpi.API +open Core +open Elpi_lambdapi + +let ss : Sig_state.t State.component = + State.declare ~name:"elpi:ss" + ~pp:(fun _fmt _ -> ()) ~init:(fun () -> Sig_state.dummy) ~start:(fun x -> x) + +let goalc = RawData.Constants.declare_global_symbol "goal" +let ofc = RawData.Constants.declare_global_symbol "of" +let nablac = RawData.Constants.declare_global_symbol "nabla" +let sealc = RawData.Constants.declare_global_symbol "seal" + +let embed_goal : Term.meta Conversion.embedding = fun ~depth st m -> + let open Term in + let ty = + let open Timed in + !(m.meta_type) in + let open RawData in + let open Utils in + let rec aux ~depth st (c,ctx,i,args) ty = + match unfold ty with + | Prod (dom,b) -> + let st, dom, gls = embed_term ~ctx:c ~depth st dom in + let x,b,c = Ctxt.unbind c depth None b in + let st, g, gls1 = + aux ~depth:(depth+1) st + (c,(depth+1,mkApp ofc (mkBound depth) [dom])::ctx,i,x::args) b in + st, mkApp nablac (mkLam g) [], gls @ gls1 + | _ -> + let ctx = List.map (fun (from,t) -> move ~from ~to_:depth t) ctx in + let st, ty, gls = embed_term ~ctx:c ~depth st ty in + let m = mk_Meta (i,List.rev args |> List.map Term.of_tvar |> Array.of_list) in + let st, i, gls1 = embed_term ~ctx:c ~depth st m in + st, mkApp sealc (mkApp goalc (list_to_lp_list ctx) [ty; i]) [], gls @ gls1 + in + aux ~depth st ([],[],m,[]) ty + +let goal : Term.meta Conversion.t = { + Conversion.embed = embed_goal; + readback = (fun ~depth:_ _ _ -> assert false); + pp_doc = (fun fmt _ -> Format.fprintf fmt "TODO"); + pp = (fun fmt _ -> Format.fprintf fmt "TODO"); + ty = Conversion.TyName "sealed-goal" +} + +(** APIs (data types and predicates) exposed to Elpi *) +let lambdapi_builtin_declarations : BuiltIn.declaration list = + let open BuiltIn in + let open BuiltInPredicate in + let open BuiltInData in + let open BuiltInPredicate.Notation in +[ + LPDoc "---- Lambdapi datatypes ----"; + + MLData sym; + MLData term; + LPCode {| +kind sealed-goal type. +type nabla (term -> sealed-goal) -> sealed-goal. +type seal goal -> sealed-goal. + +kind goal type. +type goal list prop -> term -> term -> goal. + +type of term -> term -> prop. + +pred msolve o:list sealed-goal. + +|}; + + LPDoc "---- Builtin predicates ----"; + + MLCode(Pred("lp.sig", + In(sym,"S", + Out(term,"T", + Easy "Gives the type of a symbol")), + (fun s _ ~depth:_ -> !: (Timed.(!) s.Term.sym_type))), + DocAbove); + + MLCode(Pred("lp.tc-instances", + Out(list sym,"L", + Read (ContextualConversion.unit_ctx, "Gives the list of type class instances")), + (fun _ ~depth:_ _ _ state -> + let s = (State.get ss state).Sig_state.signature in + !: ((Timed.(!) s.Sign.sign_tc_inst) |> Term.SymSet.elements))), + DocAbove); + + MLCode(Pred("lp.tc?", + In(sym,"S", + Read (ContextualConversion.unit_ctx, "Tells if S is a type class")), + (fun sym ~depth:_ _ _ state -> + let s = (State.get ss state).Sig_state.signature in + if ((Timed.(!) s.Sign.sign_tc) |> Term.SymSet.mem sym) then () + else raise No_clause)), + DocAbove); + + MLCode(Pred("lp.term->string", + In(term,"T", + Out(string,"S", + Easy "Pretty prints a term T to the string S")), + (fun t _ ~depth:_ -> !: (Format.asprintf "%a" Print.term t))), + DocAbove); + + LPDoc "---- Elpi predicates ----"; + +] @ Elpi.Builtin.std_declarations + +let lambdapi_builtins = + BuiltIn.declare ~file_name:"lambdap.elpi" lambdapi_builtin_declarations + +let document () = + BuiltIn.document_file ~header:"% automatically generated" lambdapi_builtins + +(** The runtime of Elpi (we need only one I guess) *) +let elpi = ref None + +let init () = + let e = Setup.init + ~builtins:[lambdapi_builtins] + ~legacy_parser:false + ~file_resolver:(Parse.std_resolver ~paths:[] ()) () in + elpi := Some e; + document () + +(** Given an Elpi file, a predicate name and a Terms.term argument we + run Elpi and print the term before/after the execution *) +let run : Sig_state.t -> string -> string -> Parsing.Syntax.p_term -> unit = +fun ss file predicate arg -> + let pos = arg.Common.Pos.pos in + let loc = Elpi_AUX.loc_of_pos pos in + let arg = Parsing.Scope.scope_term false ss Env.empty arg in + let elpi = match !elpi with None -> assert false | Some x -> x in + + let ast = Parse.program ~elpi ~files:[file] in + let prog = + let flags = Elpi.API.Compile.default_flags in + ast |> + Elpi.API.Compile.unit ~flags ~elpi |> + (fun u -> Elpi.API.Compile.assemble ~elpi ~flags [u]) in + + let query = + let open Elpi.API.Query in + compile prog loc + (Query { predicate; arguments = (D(term,arg,Q(term,"it",N))) }) in + + if not (Elpi.API.Compile.static_check + ~checker:(Elpi.Builtin.default_checker ()) query) then + Common.Error.fatal pos "elpi: type error in %s" file; + + Common.Console.out 1 "\nelpi: before: %a\n" Print.term arg; + match Execute.once (Elpi.API.Compile.optimize query) with + | Execute.Success { + Data.state; pp_ctx; constraints; output = (arg1,()); _ + } -> + let _ = readback_assignments state in + Common.Console.out 1 "\nelpi: after: %a\n" Print.term arg1; + Common.Console.out 1 "elpi: constraints:@ @[%a@]\n" + Pp.(constraints pp_ctx) constraints + | Failure -> Common.Error.fatal_no_pos "elpi: failure" + | NoMoreSteps -> assert false + +let extend (cctx) v ?def ty = (v, ty, def) :: cctx + +let is_tc_instance : Sig_state.t -> Term.ctxt -> Term.meta -> bool = + fun ss c m -> + let open Timed in + let open Term in + let is_tc symb = + SymSet.mem symb !(ss.Sig_state.signature.Sign.sign_tc) in + let rec aux c t = + match get_args (unfold t) with + | Symb s, _ -> is_tc s + | Prod(dom, b), [] -> + let (x, b) = Bindlib.unbind b in + let c = extend c x dom in + aux c b + | _ -> false + in + aux c !(m.meta_type) + +let tc_metas_of_term : Sig_state.t -> Term.ctxt -> Term.term -> Term.meta list = + fun ss c t -> + let open Term in + let acc = ref [] in + let rec aux c t = + match unfold t with + | Meta(m,_) when is_tc_instance ss c m && not (List.memq m !acc) -> + acc := m :: !acc + | Abst (dom, b) | Prod(dom, b) -> + aux c dom; + let (x, b) = Bindlib.unbind b in + let c = extend c x dom in + aux c b + | LLet (dom, t, b) -> + aux c dom; + aux c t; + let (x, b) = Bindlib.unbind b in + let c = extend c x dom in + aux c b + | Appl(t,u) -> aux c t; aux c u + | Plac _ -> assert false (* term was inferred before *) + | _ -> () + in + aux c t; + !acc + + +let scope_ref : (Parsing.Syntax.p_term -> Term.term * (int * string) list) ref = ref (fun _ -> assert false) + +(* we set the state, Elpi.API.Qery lacks this function *) +let hack s c = { c with Conversion.embed = + fun ~depth state x -> + let state = State.set ss state s in + c.Conversion.embed ~depth state x +} + +let solve_tc : ?scope:(Parsing.Syntax.p_term -> Term.term * (int * string) list) -> Sig_state.t -> Common.Pos.popt -> Term.problem -> Term.ctxt -> + Term.term * Term.term -> Term.term * Term.term = + fun ?scope ss pos _pb ctxt (t,ty) -> + let tc = tc_metas_of_term ss ctxt t in + if tc <> [] then begin + let elpi = match !elpi with None -> assert false | Some x -> x in + Option.iter (fun f -> scope_ref := f) scope; + + Common.Console.out 1 "BEFORE TC RESOLUTION:@ %a : %a@\n" Print.term t Print.term ty; + + let file = "tcsolver.elpi" in + let predicate = "msolve" in + let pos = Elpi_AUX.loc_of_pos pos in + + let ast = Parse.program ~elpi ~files:[file] in + let prog = + let flags = Elpi.API.Compile.default_flags in + ast |> + Elpi.API.Compile.unit ~flags ~elpi |> + (fun u -> Elpi.API.Compile.assemble ~elpi ~flags [u]) in + + let query = + let open Elpi.API.Query in + let open Elpi.API.BuiltInData in + compile prog pos + (Query { predicate; arguments = (D(hack ss (list goal),tc,N)) }) in + + if not (Elpi.API.Compile.static_check + ~checker:(Elpi.Builtin.default_checker ()) query) then + Common.Error.fatal_no_pos "elpi: type error in %s" file; + + match Execute.once (Elpi.API.Compile.optimize query) with + | Execute.Success { Data.state; output = (); _ } -> + let _ = readback_assignments state in + () + | Failure -> Common.Error.fatal_no_pos "elpi: failure" + | NoMoreSteps -> assert false + end; + t, ty + +let lpq : Quotation.quotation = fun ~depth st _loc text -> + let open Parsing in + let ast = Parser.Lp.parse_string "xxx" ("type " ^ text ^ ";") in + match ast |> Stream.next |> fun x -> x.Common.Pos.elt with + | Syntax.P_query { Common.Pos.elt = Syntax.P_query_infer(t,_); _ } -> + (*Printf.eprintf "Q %s\n" text;*) + let t, pats = !scope_ref t in + let st, t, _ = embed_term ~pats ~depth st t in + st, t + | _ -> assert false + +let () = Quotation.set_default_quotation lpq diff --git a/src/handle/query.ml b/src/handle/query.ml index e12b4c5d0..5c94e11de 100644 --- a/src/handle/query.ml +++ b/src/handle/query.ml @@ -7,15 +7,20 @@ open Proof open Lplib open Base open Timed -let infer : Pos.popt -> problem -> ctxt -> term -> term * term = - fun pos p ctx t -> +let infer : ?scope:(Parsing.Syntax.p_term -> Term.term * (int * string) list) -> Sig_state.t -> Pos.popt -> problem -> ctxt -> term -> term * term = + fun ?scope ss pos p ctx t -> match Infer.infer_noexn p ctx t with | None -> fatal pos "%a is not typable." term t | Some (t, a) -> if Unif.solve_noexn p then begin - if !p.unsolved = [] then (t, a) - else + if !p.unsolved = [] then begin + let t, a = Elpi_handle.solve_tc ?scope ss pos p ctx (t, a) in + match Infer.infer_noexn p ctx t with + | None -> fatal pos "%a is not typable AFTER TC RESOLUTION!!!!" term t + | _ -> Common.Console.out 1 "TC RESOLUTION OK!@ "; + t, a + end else (List.iter (wrn pos "Cannot solve %a." constr) !p.unsolved; fatal pos "Failed to infer the type of %a." term t) end @@ -179,9 +184,9 @@ let handle : Sig_state.t -> proof_state option -> p_query -> result = Console.out 2 "assertion: it is %b that %a" (not must_fail) constr (ctxt, t, u); (* Check that [t] is typable. *) - let (t, a) = infer pt.pos p ctxt t in + let (t, a) = infer ss pt.pos p ctxt t in (* Check that [u] is typable. *) - let (u, b) = infer pu.pos p ctxt u in + let (u, b) = infer ss pu.pos p ctxt u in (* Check that [t] and [u] have the same type. *) p := {!p with to_solve = (ctxt,a,b)::!p.to_solve}; if Unif.solve_noexn p then @@ -200,8 +205,12 @@ let handle : Sig_state.t -> proof_state option -> p_query -> result = None | P_query_infer(pt, cfg) -> let t = scope pt in - return term (Eval.eval cfg ctxt (snd (infer pt.pos p ctxt t))) + let scope = Scope.scope_term_w_pats ~typ:false ~mok true ss env in + let t, ty = infer ss ~scope pt.pos p ctxt t in + let t = Eval.eval cfg ctxt t in + let ty = Eval.eval cfg ctxt ty in + return typing (ctxt,t,ty) | P_query_normalize(pt, cfg) -> let t = scope pt in - let t, _ = infer pt.pos p ctxt t in + let t, _ = infer ss pt.pos p ctxt t in return term (Eval.eval cfg ctxt t) diff --git a/src/handle/query.mli b/src/handle/query.mli index 83eadb07f..23db36608 100644 --- a/src/handle/query.mli +++ b/src/handle/query.mli @@ -10,7 +10,7 @@ open Proof refinement of [t]. Note that [p] gets modified. Context [c] must well sorted. @raise Fatal if [t] cannot be typed. *) -val infer : popt -> problem -> ctxt -> term -> term * term +val infer : ?scope:(Parsing.Syntax.p_term -> Core.Term.term * (int * string) list) -> Sig_state.t -> popt -> problem -> ctxt -> term -> term * term (** [check pos p c t a] checks that the term [t] has type [a] in context [c] and under the constraints of [p], and returns [t] refined. Context [c] diff --git a/src/handle/rewrite.ml b/src/handle/rewrite.ml index ffd8f8c02..4ae6aa60c 100644 --- a/src/handle/rewrite.ml +++ b/src/handle/rewrite.ml @@ -321,7 +321,7 @@ let rewrite : Sig_state.t -> problem -> popt -> goal_typ -> bool -> (* Infer the type of [t] (the argument given to the tactic). *) let g_ctxt = Env.to_ctxt g_env in - let (t, t_type) = Query.infer pos p g_ctxt t in + let (t, t_type) = Query.infer ss pos p g_ctxt t in (* Check that [t_type ≡ Π x1:a1, ..., Π xn:an, P (eq a l r)]. *) let (a, l, r), vars = get_eq_data cfg pos t_type in diff --git a/src/handle/tactic.ml b/src/handle/tactic.ml index 3fdd7bea1..c0dbf3745 100644 --- a/src/handle/tactic.ml +++ b/src/handle/tactic.ml @@ -41,7 +41,7 @@ let add_axiom : Sig_state.t -> popt -> meta -> sym = (* We ignore the new ss returned by Sig_state.add_symbol: axioms do not need to be in scope. *) snd (Sig_state.add_symbol ss - Public Defin Eager true id !(m.meta_type) [] None) + Public Defin Eager true id !(m.meta_type) [] false false None) in (* Create the value which will be substituted for the metavariable. This value is [sym x0 ... xn] where [xi] are variables that will be diff --git a/src/parsing/lpLexer.ml b/src/parsing/lpLexer.ml index ff575c8c2..715f26573 100644 --- a/src/parsing/lpLexer.ml +++ b/src/parsing/lpLexer.ml @@ -45,6 +45,7 @@ type token = | COMPUTE | CONSTANT | DEBUG + | ELPI | END | FAIL | FLAG @@ -55,6 +56,7 @@ type token = | INDUCTIVE | INFIX | INJECTIVE + | INSTANCE | LET | NOTATION | OPAQUE @@ -80,6 +82,7 @@ type token = | SYMMETRY | TYPE_QUERY | TYPE_TERM + | TYPECLASS | UNIF_RULE | VERBOSE | WHY3 @@ -208,6 +211,7 @@ let rec token lb = | "compute" -> COMPUTE | "constant" -> CONSTANT | "debug" -> DEBUG + | "elpi" -> ELPI | "end" -> END | "fail" -> FAIL | "flag" -> FLAG @@ -218,6 +222,7 @@ let rec token lb = | "inductive" -> INDUCTIVE | "infix" -> INFIX | "injective" -> INJECTIVE + | "instance" -> INSTANCE | "left" -> SIDE(Pratter.Left) | "let" -> LET | "notation" -> NOTATION @@ -246,6 +251,7 @@ let rec token lb = | "symbol" -> SYMBOL | "symmetry" -> SYMMETRY | "type" -> TYPE_QUERY + | "typeclass" -> TYPECLASS | "TYPE" -> TYPE_TERM | "unif_rule" -> UNIF_RULE | "verbose" -> VERBOSE diff --git a/src/parsing/lpParser.mly b/src/parsing/lpParser.mly index 8c24556b8..42fd53310 100644 --- a/src/parsing/lpParser.mly +++ b/src/parsing/lpParser.mly @@ -38,6 +38,7 @@ %token COMPUTE %token CONSTANT %token DEBUG +%token ELPI %token END %token FAIL %token FLAG @@ -48,6 +49,7 @@ %token INDUCTIVE %token INFIX %token INJECTIVE +%token INSTANCE %token LET %token NOTATION %token OPAQUE @@ -73,6 +75,7 @@ %token SYMMETRY %token TYPE_QUERY %token TYPE_TERM +%token TYPECLASS %token UNIF_RULE %token VERBOSE %token WHY3 @@ -162,6 +165,8 @@ command: | NOTATION i=qid_or_nat n=notation SEMICOLON { make_pos $loc (P_notation(i,n)) } | q=query SEMICOLON { make_pos $sloc (P_query(q)) } + | ELPI f=STRINGLIT p=STRINGLIT a=term SEMICOLON + { make_pos $sloc (P_elpi(f,p,a)) } | EOF { raise End_of_file } query: @@ -202,6 +207,8 @@ modifier: | PRIVATE { make_pos $sloc (P_expo Term.Privat) } | PROTECTED { make_pos $sloc (P_expo Term.Protec) } | SEQUENTIAL { make_pos $sloc (P_mstrat Term.Sequen) } + | TYPECLASS { make_pos $sloc P_typeclass } + | INSTANCE { make_pos $sloc P_typeclass_instance } uid: s=UID { make_pos $sloc s} diff --git a/src/parsing/scope.ml b/src/parsing/scope.ml index 964e18d1d..525d05e0a 100644 --- a/src/parsing/scope.ml +++ b/src/parsing/scope.ml @@ -483,12 +483,28 @@ let scope = a type (defaults to [false]). No {b new} metavariables may appear in [t], but metavariables in the image of [mok] may be used. The function [mok] defaults to the function constant to [None] *) -let scope_term : ?typ:bool -> ?mok:(int -> meta option) -> +let scope_term: ?typ:bool -> ?mok:(int -> meta option) -> bool -> sig_state -> env -> p_term -> term = fun ?(typ=false) ?(mok=fun _ -> None) m_term_prv ss env t -> let md = M_Term {m_term_meta_of_key=mok; m_term_prv} in Bindlib.unbox (scope ~typ 0 md ss env t) +let scope_term_w_pats: ?typ:bool -> ?mok:(int -> meta option) -> + bool -> sig_state -> env -> p_term -> term * (int * string) list = + fun ?(typ=false) ?(mok=fun _ -> None) m_term_prv ss env t -> + let names = Hashtbl.create 1 in + let md = M_LHS { + m_lhs_prv = false; + m_lhs_indices = Hashtbl.create 1; + m_lhs_arities = Hashtbl.create 1; + m_lhs_names = names; + m_lhs_size = 0; + m_lhs_in_env = [] + } in + let t = Bindlib.unbox (scope ~typ 0 md ss env t) in + t, List.of_seq (Hashtbl.to_seq names) + + (** [patt_vars t] returns a couple [(pvs,nl)]. The first compoment [pvs] is an association list giving the arity of all the “pattern variables” appearing in the parser-level term [t]. The second component [nl] contains the names diff --git a/src/parsing/scope.mli b/src/parsing/scope.mli index 91d2cb6c8..5478c442f 100644 --- a/src/parsing/scope.mli +++ b/src/parsing/scope.mli @@ -18,6 +18,11 @@ val scope_term : ?mok:(int -> meta option) -> bool -> sig_state -> env -> p_term -> term +val scope_term_w_pats : + ?typ:bool (* default: false *) -> + ?mok:(int -> meta option) -> + bool -> sig_state -> env -> p_term -> term * (int * string) list + (** Representation of a rewriting rule prior to SR-checking. *) type pre_rule = { pr_sym : sym diff --git a/src/parsing/syntax.ml b/src/parsing/syntax.ml index 7780c76c4..869118c25 100644 --- a/src/parsing/syntax.ml +++ b/src/parsing/syntax.ml @@ -234,6 +234,8 @@ type p_modifier_aux = | P_expo of Term.expo (** visibility of symbol outside its modules *) | P_prop of Term.prop (** symbol properties: constant, definable, ... *) | P_opaq (** opacity *) + | P_typeclass + | P_typeclass_instance type p_modifier = p_modifier_aux loc @@ -266,6 +268,7 @@ type p_command_aux = | P_unif_rule of p_rule | P_coercion of p_rule | P_query of p_query + | P_elpi of string * string * p_term (** Parser-level representation of a single (located) command. *) type p_command = p_command_aux loc diff --git a/tcsolver.elpi b/tcsolver.elpi new file mode 100644 index 000000000..1c2319244 --- /dev/null +++ b/tcsolver.elpi @@ -0,0 +1,34 @@ +msolve L :- + compile_all Cl, + Cl => std.forall L (open tc). + +pred open i:(term -> term -> prop), i:sealed-goal. +open P (nabla F) :- pi x\ open P (F x). +open P (seal (goal Ctx Ty Wit)) :- + print "TC PROBLEM:\n" Ctx "|-\n" Wit ":" Ty "\n", + Ctx => P Ty Wit, + print "TC INSTANCE:\n" Ctx "|-\n" Wit ":" Ty "\n". + +:index(_ 4) +pred tc o:term, o:term. + +pred compile i:term, i:list prop, i:term, o:prop. +compile (prod (appl T U) F) TODO H (pi x\ C x) :- + T = symb S, lp.tc? S, + !, + pi x\ + compile (F x) [tc (appl T U) x |TODO] (appl H x) (C x). + +compile (prod _ F) TODO H (pi x\ C x) :- + !, + pi x\ + compile (F x) TODO (appl H x) (C x). + +compile End TODO HD (tc End HD :- TODO). + +pred compile_all o:list prop. +compile_all Props :- + lp.tc-instances Candidates, + std.map Candidates + (N\claus\ sigma T\ lp.sig N T, compile T [] (symb N) claus) + Props. diff --git a/tests/OK/elpitest.elpi b/tests/OK/elpitest.elpi new file mode 100644 index 000000000..748347590 --- /dev/null +++ b/tests/OK/elpitest.elpi @@ -0,0 +1,29 @@ +main T T1 :- + print "before type inference" T, + of T _, + print "\nafter type inference" T, + T1 = T. + + +pred of i:term, o:term. + +% API to access the type of a symbol +of (symb S) T :- lp.sig S T. + +% silly rules +of (prod A B) T :- + of A typ, + pi x\ of x A => of (B x) T. + +of (appl H A) Ta :- + of H (prod S T), + of A S, + Ta = T A. + +% suspension of typing on holes (type constraint) +of U T :- var U, declare_constraint (of U T) [U]. + +% uniqueness of typing +constraint of { + rule (of (uvar X _) T) \ (of (uvar X _) S) <=> (S = T). +} diff --git a/tests/OK/elpitest.lp b/tests/OK/elpitest.lp new file mode 100644 index 000000000..48eaedb35 --- /dev/null +++ b/tests/OK/elpitest.lp @@ -0,0 +1,44 @@ +/* dune exec lambdapi check tests/OK/elpitest.lp */ +require open tests.OK.logic; +require open tests.OK.bool; +require open tests.OK.nat; + +flag "print_implicits" on; // default is off +flag "print_contexts" on; // default is off +flag "print_domains" on; // default is off +flag "print_meta_types" on; // default is off + +constant typeclass symbol eqdec : U → TYPE; +constant symbol mk_eqdec [i] : + Π (f : T i → T i → B), + // (Π (x y : T i), P ({|and|} (imp (x = y) (f x y = true))(imp (f x y = true) (x = y)))) → + eqdec i; + +symbol == [i:U] [j:eqdec i] : T i → T i → B; +notation == infix 1; + +/* extra stdlib */ +symbol eq_bool : B → B → B ≔ λ b1 b2, bool_neg (bool_xor b1 b2); +symbol Pair : U → U → TYPE; +symbol mk_Pair [i] [j] : T i → T j → Pair i j; +symbol pair : U → U → U; +rule T (pair $i $j) ↪ Pair $i $j; + +instance symbol bool_eqdec : eqdec bool ≔ mk_eqdec eq_bool /* p */; +instance symbol nat_eqdec : eqdec nat ≔ mk_eqdec eq_nat /* p */; +instance symbol pair_eqdec : Π i j, eqdec i → eqdec j → eqdec (pair i j); + /* ≔ mk_eqdec eq_nat ; */ + +type λ b , true == b; +type λ n , 1 == n; +type λ n : T nat, λ b : T bool, mk_Pair 1 true == mk_Pair n b; + +rule @== _ (@mk_eqdec _ $f /* _ */) ↪ $f; + +compute true == true; +compute true == false; +compute 1 == 0; + +// assert ⊢ eqop _ true false : B; +// elpi "tests/OK/elpitest.elpi" "main" +// (Π x y , P (x = y) → P (y = x)); diff --git a/tests/rewriting.ml b/tests/rewriting.ml index 4a945c469..ec8609bed 100644 --- a/tests/rewriting.ml +++ b/tests/rewriting.ml @@ -15,7 +15,7 @@ let scope_term ss = Scope.scope_term true ss [] let add_sym ss id = Sig_state.add_symbol ss Public Defin Eager true (Pos.none id) Term.mk_Kind [] - None + false false None (* Define a signature state and some symbols. *) From c58aef2acac63426a4bca4cbcf0d4cea7689ffa5 Mon Sep 17 00:00:00 2001 From: Enrico Tassi Date: Fri, 27 Jan 2023 15:34:28 +0100 Subject: [PATCH 04/25] xxx --- out/Coq__Init__Logic.lp | 30148 +++++++++++++++++++++++++++++++++++++ out/Makefile | 4 - out/compile.sh | 9 + out/{ => coq}/cupicef.lp | 0 out/lambdapi.pkg | 2 - 5 files changed, 30157 insertions(+), 6 deletions(-) create mode 100644 out/Coq__Init__Logic.lp delete mode 100644 out/Makefile create mode 100755 out/compile.sh rename out/{ => coq}/cupicef.lp (100%) delete mode 100644 out/lambdapi.pkg diff --git a/out/Coq__Init__Logic.lp b/out/Coq__Init__Logic.lp new file mode 100644 index 000000000..18cd3f66e --- /dev/null +++ b/out/Coq__Init__Logic.lp @@ -0,0 +1,30148 @@ +flag "eta_equality" on; +require coq.cupicef as cupicef; +symbol True : cupicef.Univ cupicef.prop; +symbol I : cupicef.Term cupicef.prop True; +rule cupicef.code_guarded? _ I ↪ cupicef.guarded; +symbol match____True + : Π (s : cupicef.Sort), + Π (P : cupicef.Term cupicef.prop True → cupicef.Univ s), + Π (case__I : cupicef.Term s (P I)), + Π (x : cupicef.Term cupicef.prop True), cupicef.Term s (P x); +rule match____True $s $P $case__I I ↪ $case__I; +rule match____True $s + (λ x, cupicef.uncode $_.[] (cupicef.code (cupicef.cU $s'.[]) $P.[x])) ↪ +match____True $s' (λ x, $P.[x]); +symbol True__rect + : Π (P : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (f : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) P), + Π (t : cupicef.Term cupicef.prop True), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) P + ≔ λ (P : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (f : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) P), + λ (t : cupicef.Term cupicef.prop True), + match____True (cupicef.{|type|} (cupicef.s cupicef.z)) + (λ (__ : cupicef.Term cupicef.prop True), P) f t; +symbol True__ind + : Π (P : cupicef.Univ cupicef.prop), + Π (f : cupicef.Term cupicef.prop P), + Π (t : cupicef.Term cupicef.prop True), cupicef.Term cupicef.prop P + ≔ λ (P : cupicef.Univ cupicef.prop), + λ (f : cupicef.Term cupicef.prop P), + λ (t : cupicef.Term cupicef.prop True), + match____True cupicef.prop + (λ (__ : cupicef.Term cupicef.prop True), P) f t; +symbol True__rec + : Π (P : cupicef.Univ cupicef.set), + Π (f : cupicef.Term cupicef.set P), + Π (t : cupicef.Term cupicef.prop True), cupicef.Term cupicef.set P + ≔ λ (P : cupicef.Univ cupicef.set), + True__rect + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) cupicef.I) + cupicef.I P); +symbol False : cupicef.Univ cupicef.prop; +symbol match____False + : Π (s : cupicef.Sort), + Π (P : cupicef.Term cupicef.prop False → cupicef.Univ s), + Π (x : cupicef.Term cupicef.prop False), cupicef.Term s (P x); +rule match____False $s + (λ x, cupicef.uncode $_.[] (cupicef.code (cupicef.cU $s'.[]) $P.[x])) ↪ +match____False $s' (λ x, $P.[x]); +symbol False__rect + : Π (P : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (f : cupicef.Term cupicef.prop False), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) P + ≔ λ (P : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (f : cupicef.Term cupicef.prop False), + match____False (cupicef.{|type|} (cupicef.s cupicef.z)) + (λ (__ : cupicef.Term cupicef.prop False), P) f; +symbol False__ind + : Π (P : cupicef.Univ cupicef.prop), + Π (f : cupicef.Term cupicef.prop False), cupicef.Term cupicef.prop P + ≔ λ (P : cupicef.Univ cupicef.prop), + λ (f : cupicef.Term cupicef.prop False), + match____False cupicef.prop + (λ (__ : cupicef.Term cupicef.prop False), P) f; +symbol False__rec + : Π (P : cupicef.Univ cupicef.set), + Π (f : cupicef.Term cupicef.prop False), cupicef.Term cupicef.set P + ≔ λ (P : cupicef.Univ cupicef.set), + False__rect + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) cupicef.I) + cupicef.I P); +symbol not : Π (A : cupicef.Univ cupicef.prop), cupicef.Univ cupicef.prop + ≔ λ (A : cupicef.Univ cupicef.prop), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), False); +symbol and + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), cupicef.Univ cupicef.prop; +symbol conj + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (__ : cupicef.Term cupicef.prop A), + Π (__0 : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (and A B); +rule cupicef.code_guarded? _ (conj _ _ _ _) ↪ cupicef.guarded; +symbol match____and + : Π (s : cupicef.Sort), + Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (P : cupicef.Term cupicef.prop (and A B) → cupicef.Univ s), + Π + ( + case__conj : Π (__ : cupicef.Term cupicef.prop A), + Π (__0 : cupicef.Term cupicef.prop B), + cupicef.Term s (P (conj A B __ __0)) + ), + Π (x : cupicef.Term cupicef.prop (and A B)), + cupicef.Term s (P x); +rule match____and $s $A $B $P $case__conj (conj $A $B $__ $__0) ↪ $case__conj + $__ $__0; +rule match____and $s $A $B + (λ x, cupicef.uncode $_.[] (cupicef.code (cupicef.cU $s'.[]) $P.[x])) ↪ +match____and $s' $A $B (λ x, $P.[x]); +symbol and__rect + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (P : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + f : Π (__ : cupicef.Term cupicef.prop A), + Π (__0 : cupicef.Term cupicef.prop B), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) P + ), + Π (a : cupicef.Term cupicef.prop (and A B)), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) P + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (P : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + f : Π (__ : cupicef.Term cupicef.prop A), + Π (__0 : cupicef.Term cupicef.prop B), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) P + ), + λ (a : cupicef.Term cupicef.prop (and A B)), + match____and (cupicef.{|type|} (cupicef.s cupicef.z)) A B + (λ (__ : cupicef.Term cupicef.prop (and A B)), P) f a; +symbol and__ind + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (P : cupicef.Univ cupicef.prop), + Π + ( + f : Π (__ : cupicef.Term cupicef.prop A), + Π (__0 : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop P + ), + Π (a : cupicef.Term cupicef.prop (and A B)), + cupicef.Term cupicef.prop P + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (P : cupicef.Univ cupicef.prop), + λ + ( + f : Π (__ : cupicef.Term cupicef.prop A), + Π (__0 : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop P + ), + λ (a : cupicef.Term cupicef.prop (and A B)), + match____and cupicef.prop A B + (λ (__ : cupicef.Term cupicef.prop (and A B)), P) f a; +symbol and__rec + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (P : cupicef.Univ cupicef.set), + Π + ( + f : Π (__ : cupicef.Term cupicef.prop A), + Π (__0 : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.set P + ), + Π (a : cupicef.Term cupicef.prop (and A B)), + cupicef.Term cupicef.set P + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (P : cupicef.Univ cupicef.set), + and__rect A B + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.univ cupicef.set + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I P); +opaque symbol proj1 + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (__ : cupicef.Term cupicef.prop (and A B)), + cupicef.Term cupicef.prop A + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (H : cupicef.Term cupicef.prop (and A B)), + match____and cupicef.prop A B + (λ (__ : cupicef.Term cupicef.prop (and A B)), A) + (λ (H0 : cupicef.Term cupicef.prop A), + λ (H1 : cupicef.Term cupicef.prop B), H0) + H; +opaque symbol proj2 + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (__ : cupicef.Term cupicef.prop (and A B)), + cupicef.Term cupicef.prop B + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (H : cupicef.Term cupicef.prop (and A B)), + match____and cupicef.prop A B + (λ (__ : cupicef.Term cupicef.prop (and A B)), B) + (λ (H0 : cupicef.Term cupicef.prop A), + λ (H1 : cupicef.Term cupicef.prop B), H1) + H; +symbol or + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), cupicef.Univ cupicef.prop; +symbol or__introl + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (__ : cupicef.Term cupicef.prop A), + cupicef.Term cupicef.prop (or A B); +symbol or__intror + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (or A B); +rule cupicef.code_guarded? _ (or__introl _ _ _) ↪ cupicef.guarded; +rule cupicef.code_guarded? _ (or__intror _ _ _) ↪ cupicef.guarded; +symbol match____or + : Π (s : cupicef.Sort), + Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (P : cupicef.Term cupicef.prop (or A B) → cupicef.Univ s), + Π + ( + case__or__introl : Π (__ : cupicef.Term cupicef.prop A), + cupicef.Term s (P (or__introl A B __)) + ), + Π + ( + case__or__intror : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term s (P (or__intror A B __)) + ), + Π (x : cupicef.Term cupicef.prop (or A B)), + cupicef.Term s (P x); +rule match____or $s $A $B $P $case__or__introl $case__or__intror + (or__introl $A $B $__) ↪ $case__or__introl $__; +rule match____or $s $A $B $P $case__or__introl $case__or__intror + (or__intror $A $B $__) ↪ $case__or__intror $__; +rule match____or $s $A $B + (λ x, cupicef.uncode $_.[] (cupicef.code (cupicef.cU $s'.[]) $P.[x])) ↪ +match____or $s' $A $B (λ x, $P.[x]); +symbol or__ind + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (P : cupicef.Univ cupicef.prop), + Π + ( + f : Π (__ : cupicef.Term cupicef.prop A), + cupicef.Term cupicef.prop P + ), + Π + ( + f0 : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop P + ), + Π (o : cupicef.Term cupicef.prop (or A B)), + cupicef.Term cupicef.prop P + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (P : cupicef.Univ cupicef.prop), + λ + ( + f : Π (__ : cupicef.Term cupicef.prop A), + cupicef.Term cupicef.prop P + ), + λ + ( + f0 : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop P + ), + λ (o : cupicef.Term cupicef.prop (or A B)), + match____or cupicef.prop A B + (λ (__ : cupicef.Term cupicef.prop (or A B)), P) f f0 o; +symbol iff + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), cupicef.Univ cupicef.prop + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), A)); +opaque symbol iff__refl + : Π (A : cupicef.Univ cupicef.prop), cupicef.Term cupicef.prop (iff A A) + ≔ λ (A : cupicef.Univ cupicef.prop), + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), A)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), A)) + (λ (H : cupicef.Term cupicef.prop A), H) + (λ (H : cupicef.Term cupicef.prop A), H); +opaque symbol iff__trans + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + Π (__ : cupicef.Term cupicef.prop (iff A B)), + Π (__0 : cupicef.Term cupicef.prop (iff B C)), + cupicef.Term cupicef.prop (iff A C) + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + λ (H : cupicef.Term cupicef.prop (iff A B)), + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), A)) + (λ + ( + __ : cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), A))) + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (iff B C) + (λ (__0 : cupicef.Term cupicef.prop (iff B C)), iff A C)) + (λ + ( + H1 : Π (__ : cupicef.Term cupicef.prop A), + cupicef.Term cupicef.prop B + ), + λ + ( + H2 : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop A + ), + λ (H0 : cupicef.Term cupicef.prop (iff B C)), + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + B (λ (__ : cupicef.Term cupicef.prop B), C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + C (λ (__ : cupicef.Term cupicef.prop C), B)) + (λ + ( + __ : cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I C + (λ (__ : cupicef.Term cupicef.prop C), B))) + ), iff A C) + (λ + ( + H3 : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop C + ), + λ + ( + H4 : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop B + ), + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I C + (λ (__ : cupicef.Term cupicef.prop C), A)) + (λ (H5 : cupicef.Term cupicef.prop A), H3 (H1 H5)) + (λ (H5 : cupicef.Term cupicef.prop C), + H2 (H1 (H2 (H4 H5))))) + H0) + H; +opaque symbol iff__sym + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (__ : cupicef.Term cupicef.prop (iff A B)), + cupicef.Term cupicef.prop (iff B A) + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (H : cupicef.Term cupicef.prop (iff A B)), + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), A)) + (λ + ( + __ : cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + A (λ (__ : cupicef.Term cupicef.prop A), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + B (λ (__ : cupicef.Term cupicef.prop B), A))) + ), iff B A) + (λ + ( + H1 : Π (__ : cupicef.Term cupicef.prop A), + cupicef.Term cupicef.prop B + ), + λ + ( + H2 : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop A + ), + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), A)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), B)) + H2 H1) + H; +opaque symbol and__iff__compat__l + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + Π (__ : cupicef.Term cupicef.prop (iff B C)), + cupicef.Term cupicef.prop (iff (and A B) (and A C)) + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + λ (H : cupicef.Term cupicef.prop (iff B C)), + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I C + (λ (__ : cupicef.Term cupicef.prop C), B)) + (λ + ( + __ : cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I C + (λ (__ : cupicef.Term cupicef.prop C), B))) + ), iff (and A B) (and A C)) + (λ + ( + Hl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop C + ), + λ + ( + Hr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop B + ), + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (and A B) + (λ (__ : cupicef.Term cupicef.prop (and A B)), and A C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (and A C) + (λ (__ : cupicef.Term cupicef.prop (and A C)), and A B)) + (λ (H0 : cupicef.Term cupicef.prop (and A B)), + match____and cupicef.prop A B + (λ (__ : cupicef.Term cupicef.prop (and A B)), and A C) + (λ (H1 : cupicef.Term cupicef.prop A), + λ (H2 : cupicef.Term cupicef.prop B), + conj A C H1 (Hl H2)) + H0) + (λ (H0 : cupicef.Term cupicef.prop (and A C)), + match____and cupicef.prop A C + (λ (__ : cupicef.Term cupicef.prop (and A C)), and A B) + (λ (H1 : cupicef.Term cupicef.prop A), + λ (H2 : cupicef.Term cupicef.prop C), + conj A B H1 (Hr H2)) + H0)) + H; +opaque symbol and__iff__compat__r + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + Π (__ : cupicef.Term cupicef.prop (iff B C)), + cupicef.Term cupicef.prop (iff (and B A) (and C A)) + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + λ (H : cupicef.Term cupicef.prop (iff B C)), + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I C + (λ (__ : cupicef.Term cupicef.prop C), B)) + (λ + ( + __ : cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I C + (λ (__ : cupicef.Term cupicef.prop C), B))) + ), iff (and B A) (and C A)) + (λ + ( + Hl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop C + ), + λ + ( + Hr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop B + ), + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (and B A) + (λ (__ : cupicef.Term cupicef.prop (and B A)), and C A)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (and C A) + (λ (__ : cupicef.Term cupicef.prop (and C A)), and B A)) + (λ (H0 : cupicef.Term cupicef.prop (and B A)), + match____and cupicef.prop B A + (λ (__ : cupicef.Term cupicef.prop (and B A)), and C A) + (λ (H1 : cupicef.Term cupicef.prop B), + λ (H2 : cupicef.Term cupicef.prop A), + conj C A (Hl H1) H2) + H0) + (λ (H0 : cupicef.Term cupicef.prop (and C A)), + match____and cupicef.prop C A + (λ (__ : cupicef.Term cupicef.prop (and C A)), and B A) + (λ (H1 : cupicef.Term cupicef.prop C), + λ (H2 : cupicef.Term cupicef.prop A), + conj B A (Hr H1) H2) + H0)) + H; +opaque symbol or__iff__compat__l + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + Π (__ : cupicef.Term cupicef.prop (iff B C)), + cupicef.Term cupicef.prop (iff (or A B) (or A C)) + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + λ (H : cupicef.Term cupicef.prop (iff B C)), + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I C + (λ (__ : cupicef.Term cupicef.prop C), B)) + (λ + ( + __ : cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I C + (λ (__ : cupicef.Term cupicef.prop C), B))) + ), iff (or A B) (or A C)) + (λ + ( + Hl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop C + ), + λ + ( + Hr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop B + ), + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (or A B) + (λ (__ : cupicef.Term cupicef.prop (or A B)), or A C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (or A C) + (λ (__ : cupicef.Term cupicef.prop (or A C)), or A B)) + (λ (H0 : cupicef.Term cupicef.prop (or A B)), + match____or cupicef.prop A B + (λ (__ : cupicef.Term cupicef.prop (or A B)), or A C) + (λ (H1 : cupicef.Term cupicef.prop A), + or__introl A C H1) + (λ (H1 : cupicef.Term cupicef.prop B), + or__intror A C (Hl H1)) + H0) + (λ (H0 : cupicef.Term cupicef.prop (or A C)), + match____or cupicef.prop A C + (λ (__ : cupicef.Term cupicef.prop (or A C)), or A B) + (λ (H1 : cupicef.Term cupicef.prop A), + or__introl A B H1) + (λ (H1 : cupicef.Term cupicef.prop C), + or__intror A B (Hr H1)) + H0)) + H; +opaque symbol or__iff__compat__r + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + Π (__ : cupicef.Term cupicef.prop (iff B C)), + cupicef.Term cupicef.prop (iff (or B A) (or C A)) + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + λ (H : cupicef.Term cupicef.prop (iff B C)), + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I C + (λ (__ : cupicef.Term cupicef.prop C), B)) + (λ + ( + __ : cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I C + (λ (__ : cupicef.Term cupicef.prop C), B))) + ), iff (or B A) (or C A)) + (λ + ( + Hl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop C + ), + λ + ( + Hr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop B + ), + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (or B A) + (λ (__ : cupicef.Term cupicef.prop (or B A)), or C A)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (or C A) + (λ (__ : cupicef.Term cupicef.prop (or C A)), or B A)) + (λ (H0 : cupicef.Term cupicef.prop (or B A)), + match____or cupicef.prop B A + (λ (__ : cupicef.Term cupicef.prop (or B A)), or C A) + (λ (H1 : cupicef.Term cupicef.prop B), + or__introl C A (Hl H1)) + (λ (H1 : cupicef.Term cupicef.prop A), + or__intror C A H1) + H0) + (λ (H0 : cupicef.Term cupicef.prop (or C A)), + match____or cupicef.prop C A + (λ (__ : cupicef.Term cupicef.prop (or C A)), or B A) + (λ (H1 : cupicef.Term cupicef.prop C), + or__introl B A (Hr H1)) + (λ (H1 : cupicef.Term cupicef.prop A), + or__intror B A H1) + H0)) + H; +opaque symbol imp__iff__compat__l + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + Π (__ : cupicef.Term cupicef.prop (iff B C)), + cupicef.Term cupicef.prop + (iff + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__0 : cupicef.Term cupicef.prop A), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__0 : cupicef.Term cupicef.prop A), C))) + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + λ (H : cupicef.Term cupicef.prop (iff B C)), + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I C + (λ (__ : cupicef.Term cupicef.prop C), B)) + (λ + ( + __ : cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I C + (λ (__ : cupicef.Term cupicef.prop C), B))) + ), + iff + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__0 : cupicef.Term cupicef.prop A), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__0 : cupicef.Term cupicef.prop A), C))) + (λ + ( + Hl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop C + ), + λ + ( + Hr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop B + ), + conj + (cupicef.prod (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), B)) + (λ + ( + __ : Π (__ : cupicef.Term cupicef.prop A), + cupicef.Term cupicef.prop B + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + A (λ (__0 : cupicef.Term cupicef.prop A), C))) + (cupicef.prod (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), C)) + (λ + ( + __ : Π (__ : cupicef.Term cupicef.prop A), + cupicef.Term cupicef.prop C + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + A (λ (__0 : cupicef.Term cupicef.prop A), B))) + (λ + ( + H0 : Π (__ : cupicef.Term cupicef.prop A), + cupicef.Term cupicef.prop B + ), λ (H1 : cupicef.Term cupicef.prop A), Hl (H0 H1)) + (λ + ( + H0 : Π (__ : cupicef.Term cupicef.prop A), + cupicef.Term cupicef.prop C + ), λ (H1 : cupicef.Term cupicef.prop A), Hr (H0 H1))) + H; +opaque symbol imp__iff__compat__r + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + Π (__ : cupicef.Term cupicef.prop (iff B C)), + cupicef.Term cupicef.prop + (iff + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__0 : cupicef.Term cupicef.prop B), A)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I C + (λ (__0 : cupicef.Term cupicef.prop C), A))) + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + λ (H : cupicef.Term cupicef.prop (iff B C)), + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I C + (λ (__ : cupicef.Term cupicef.prop C), B)) + (λ + ( + __ : cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I C + (λ (__ : cupicef.Term cupicef.prop C), B))) + ), + iff + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__0 : cupicef.Term cupicef.prop B), A)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I C + (λ (__0 : cupicef.Term cupicef.prop C), A))) + (λ + ( + Hl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop C + ), + λ + ( + Hr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop B + ), + conj + (cupicef.prod (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), A)) + (λ + ( + __ : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + C (λ (__0 : cupicef.Term cupicef.prop C), A))) + (cupicef.prod (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I C + (λ (__ : cupicef.Term cupicef.prop C), A)) + (λ + ( + __ : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + B (λ (__0 : cupicef.Term cupicef.prop B), A))) + (λ + ( + H0 : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop A + ), λ (H1 : cupicef.Term cupicef.prop C), H0 (Hr H1)) + (λ + ( + H0 : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop A + ), λ (H1 : cupicef.Term cupicef.prop B), H0 (Hl H1))) + H; +opaque symbol not__iff__compat + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (__ : cupicef.Term cupicef.prop (iff A B)), + cupicef.Term cupicef.prop (iff (not A) (not B)) + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (H : cupicef.Term cupicef.prop (iff A B)), + imp__iff__compat__r False A B H; +opaque symbol neg__false + : Π (A : cupicef.Univ cupicef.prop), + cupicef.Term cupicef.prop (iff (not A) (iff A False)) + ≔ λ (A : cupicef.Univ cupicef.prop), + cupicef.cast cupicef.prop cupicef.prop + (and + (cupicef.prod (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.{|rule|} (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), False)) + (λ + ( + __ : Π (__ : cupicef.Term cupicef.prop A), + cupicef.Term cupicef.prop False + ), iff A False)) + (cupicef.prod cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I (iff A False) + (λ (__ : cupicef.Term cupicef.prop (iff A False)), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__0 : cupicef.Term cupicef.prop A), False)))) + (iff (not A) (iff A False)) cupicef.I + (conj + (cupicef.prod (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.{|rule|} (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), False)) + (λ + ( + __ : Π (__ : cupicef.Term cupicef.prop A), + cupicef.Term cupicef.prop False + ), iff A False)) + (cupicef.prod cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I (iff A False) + (λ (__ : cupicef.Term cupicef.prop (iff A False)), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__0 : cupicef.Term cupicef.prop A), False))) + (λ + ( + H : Π (__ : cupicef.Term cupicef.prop A), + cupicef.Term cupicef.prop False + ), + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), False)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I False + (λ (__ : cupicef.Term cupicef.prop False), A)) + H (λ (H1 : cupicef.Term cupicef.prop False), False__ind A H1)) + (λ (H0 : cupicef.Term cupicef.prop (iff A False)), + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), False)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I False + (λ (__ : cupicef.Term cupicef.prop False), A)) + (λ + ( + __ : cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + A (λ (__ : cupicef.Term cupicef.prop A), False)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + False (λ (__ : cupicef.Term cupicef.prop False), A))) + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__0 : cupicef.Term cupicef.prop A), False)) + (λ + ( + H : Π (__ : cupicef.Term cupicef.prop A), + cupicef.Term cupicef.prop False + ), + λ + ( + __tmp : Π (__ : cupicef.Term cupicef.prop False), + cupicef.Term cupicef.prop A + ), H) + H0)); +symbol let__H0 + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + Π + ( + Hl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop A + ), + Π + ( + Hr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop A + ), + Π (H : cupicef.Term cupicef.prop (iff (and A B) (and A C))), + Π + ( + HypL : Π (__ : cupicef.Term cupicef.prop (and A B)), + cupicef.Term cupicef.prop (and A C) + ), + Π + ( + HypR : Π (__ : cupicef.Term cupicef.prop (and A C)), + cupicef.Term cupicef.prop (and A B) + ), + Π (H0 : cupicef.Term cupicef.prop B), + Π (__ : cupicef.Term cupicef.prop (and A B)), + cupicef.Term cupicef.prop C + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + λ + ( + Hl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop A + ), + λ + ( + Hr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop A + ), + λ (H : cupicef.Term cupicef.prop (iff (and A B) (and A C))), + λ + ( + HypL : Π (__ : cupicef.Term cupicef.prop (and A B)), + cupicef.Term cupicef.prop (and A C) + ), + λ + ( + HypR : Π (__ : cupicef.Term cupicef.prop (and A C)), + cupicef.Term cupicef.prop (and A B) + ), + λ (H0 : cupicef.Term cupicef.prop B), + λ (__ : cupicef.Term cupicef.prop (and A B)), + match____and cupicef.prop A C + (λ (__0 : cupicef.Term cupicef.prop (and A C)), C) + (λ (__0 : cupicef.Term cupicef.prop A), + λ (__1 : cupicef.Term cupicef.prop C), __1) + (HypL __); +symbol let__H1 + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + Π + ( + Hl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop A + ), + Π + ( + Hr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop A + ), + Π (H : cupicef.Term cupicef.prop (iff (and A B) (and A C))), + Π + ( + HypL : Π (__ : cupicef.Term cupicef.prop (and A B)), + cupicef.Term cupicef.prop (and A C) + ), + Π + ( + HypR : Π (__ : cupicef.Term cupicef.prop (and A C)), + cupicef.Term cupicef.prop (and A B) + ), + Π (H0 : cupicef.Term cupicef.prop C), + Π (__ : cupicef.Term cupicef.prop (and A C)), + cupicef.Term cupicef.prop B + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + λ + ( + Hl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop A + ), + λ + ( + Hr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop A + ), + λ (H : cupicef.Term cupicef.prop (iff (and A B) (and A C))), + λ + ( + HypL : Π (__ : cupicef.Term cupicef.prop (and A B)), + cupicef.Term cupicef.prop (and A C) + ), + λ + ( + HypR : Π (__ : cupicef.Term cupicef.prop (and A C)), + cupicef.Term cupicef.prop (and A B) + ), + λ (H0 : cupicef.Term cupicef.prop C), + λ (__ : cupicef.Term cupicef.prop (and A C)), + match____and cupicef.prop A B + (λ (__0 : cupicef.Term cupicef.prop (and A B)), B) + (λ (__0 : cupicef.Term cupicef.prop A), + λ (__1 : cupicef.Term cupicef.prop B), __1) + (HypR __); +opaque symbol and__cancel__l + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + Π + ( + __ : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop A + ), + Π + ( + __0 : Π (__0 : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop A + ), + cupicef.Term cupicef.prop + (iff (iff (and A B) (and A C)) (iff B C)) + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + λ + ( + Hl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop A + ), + λ + ( + Hr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop A + ), + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (iff (and A B) (and A C)) + (λ + (__ : cupicef.Term cupicef.prop (iff (and A B) (and A C)) + ), iff B C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (iff B C) + (λ (__ : cupicef.Term cupicef.prop (iff B C)), + iff (and A B) (and A C))) + (λ + (H : cupicef.Term cupicef.prop (iff (and A B) (and A C))), + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (and A B) + (λ (__ : cupicef.Term cupicef.prop (and A B)), and A C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (and A C) + (λ (__ : cupicef.Term cupicef.prop (and A C)), and A B)) + (λ + ( + __ : cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I (and A B) + (λ (__ : cupicef.Term cupicef.prop (and A B)), + and A C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I (and A C) + (λ (__ : cupicef.Term cupicef.prop (and A C)), + and A B))) + ), iff B C) + (λ + ( + HypL : Π (__ : cupicef.Term cupicef.prop (and A B)), + cupicef.Term cupicef.prop (and A C) + ), + λ + ( + HypR : Π (__ : cupicef.Term cupicef.prop (and A C)), + cupicef.Term cupicef.prop (and A B) + ), + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I C + (λ (__ : cupicef.Term cupicef.prop C), B)) + (λ (H0 : cupicef.Term cupicef.prop B), + let__H0 A B C Hl Hr H HypL HypR H0 + (conj A B (Hl H0) H0)) + (λ (H0 : cupicef.Term cupicef.prop C), + let__H1 A B C Hl Hr H HypL HypR H0 + (conj A C (Hr H0) H0))) + H) + (and__iff__compat__l A B C); +symbol let__H2 + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + Π + ( + Hl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop A + ), + Π + ( + Hr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop A + ), + Π (H : cupicef.Term cupicef.prop (iff (and B A) (and C A))), + Π + ( + HypL : Π (__ : cupicef.Term cupicef.prop (and B A)), + cupicef.Term cupicef.prop (and C A) + ), + Π + ( + HypR : Π (__ : cupicef.Term cupicef.prop (and C A)), + cupicef.Term cupicef.prop (and B A) + ), + Π (H0 : cupicef.Term cupicef.prop B), + Π (__ : cupicef.Term cupicef.prop (and B A)), + cupicef.Term cupicef.prop C + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + λ + ( + Hl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop A + ), + λ + ( + Hr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop A + ), + λ (H : cupicef.Term cupicef.prop (iff (and B A) (and C A))), + λ + ( + HypL : Π (__ : cupicef.Term cupicef.prop (and B A)), + cupicef.Term cupicef.prop (and C A) + ), + λ + ( + HypR : Π (__ : cupicef.Term cupicef.prop (and C A)), + cupicef.Term cupicef.prop (and B A) + ), + λ (H0 : cupicef.Term cupicef.prop B), + λ (__ : cupicef.Term cupicef.prop (and B A)), + match____and cupicef.prop C A + (λ (__0 : cupicef.Term cupicef.prop (and C A)), C) + (λ (__0 : cupicef.Term cupicef.prop C), + λ (__1 : cupicef.Term cupicef.prop A), __0) + (HypL __); +symbol let__H3 + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + Π + ( + Hl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop A + ), + Π + ( + Hr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop A + ), + Π (H : cupicef.Term cupicef.prop (iff (and B A) (and C A))), + Π + ( + HypL : Π (__ : cupicef.Term cupicef.prop (and B A)), + cupicef.Term cupicef.prop (and C A) + ), + Π + ( + HypR : Π (__ : cupicef.Term cupicef.prop (and C A)), + cupicef.Term cupicef.prop (and B A) + ), + Π (H0 : cupicef.Term cupicef.prop C), + Π (__ : cupicef.Term cupicef.prop (and C A)), + cupicef.Term cupicef.prop B + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + λ + ( + Hl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop A + ), + λ + ( + Hr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop A + ), + λ (H : cupicef.Term cupicef.prop (iff (and B A) (and C A))), + λ + ( + HypL : Π (__ : cupicef.Term cupicef.prop (and B A)), + cupicef.Term cupicef.prop (and C A) + ), + λ + ( + HypR : Π (__ : cupicef.Term cupicef.prop (and C A)), + cupicef.Term cupicef.prop (and B A) + ), + λ (H0 : cupicef.Term cupicef.prop C), + λ (__ : cupicef.Term cupicef.prop (and C A)), + match____and cupicef.prop B A + (λ (__0 : cupicef.Term cupicef.prop (and B A)), B) + (λ (__0 : cupicef.Term cupicef.prop B), + λ (__1 : cupicef.Term cupicef.prop A), __0) + (HypR __); +opaque symbol and__cancel__r + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + Π + ( + __ : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop A + ), + Π + ( + __0 : Π (__0 : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop A + ), + cupicef.Term cupicef.prop + (iff (iff (and B A) (and C A)) (iff B C)) + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + λ + ( + Hl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop A + ), + λ + ( + Hr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop A + ), + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (iff (and B A) (and C A)) + (λ + (__ : cupicef.Term cupicef.prop (iff (and B A) (and C A)) + ), iff B C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (iff B C) + (λ (__ : cupicef.Term cupicef.prop (iff B C)), + iff (and B A) (and C A))) + (λ + (H : cupicef.Term cupicef.prop (iff (and B A) (and C A))), + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (and B A) + (λ (__ : cupicef.Term cupicef.prop (and B A)), and C A)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (and C A) + (λ (__ : cupicef.Term cupicef.prop (and C A)), and B A)) + (λ + ( + __ : cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I (and B A) + (λ (__ : cupicef.Term cupicef.prop (and B A)), + and C A)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I (and C A) + (λ (__ : cupicef.Term cupicef.prop (and C A)), + and B A))) + ), iff B C) + (λ + ( + HypL : Π (__ : cupicef.Term cupicef.prop (and B A)), + cupicef.Term cupicef.prop (and C A) + ), + λ + ( + HypR : Π (__ : cupicef.Term cupicef.prop (and C A)), + cupicef.Term cupicef.prop (and B A) + ), + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I C + (λ (__ : cupicef.Term cupicef.prop C), B)) + (λ (H0 : cupicef.Term cupicef.prop B), + let__H2 A B C Hl Hr H HypL HypR H0 + (conj B A H0 (Hl H0))) + (λ (H0 : cupicef.Term cupicef.prop C), + let__H3 A B C Hl Hr H HypL HypR H0 + (conj C A H0 (Hr H0)))) + H) + (and__iff__compat__r A B C); +opaque symbol and__comm + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + cupicef.Term cupicef.prop (iff (and A B) (and B A)) + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I (and A B) + (λ (__ : cupicef.Term cupicef.prop (and A B)), and B A)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I (and B A) + (λ (__ : cupicef.Term cupicef.prop (and B A)), and A B)) + (λ (H : cupicef.Term cupicef.prop (and A B)), + match____and cupicef.prop A B + (λ (__ : cupicef.Term cupicef.prop (and A B)), and B A) + (λ (H0 : cupicef.Term cupicef.prop A), + λ (H1 : cupicef.Term cupicef.prop B), conj B A H1 H0) + H) + (λ (H : cupicef.Term cupicef.prop (and B A)), + match____and cupicef.prop B A + (λ (__ : cupicef.Term cupicef.prop (and B A)), and A B) + (λ (H0 : cupicef.Term cupicef.prop B), + λ (H1 : cupicef.Term cupicef.prop A), conj A B H1 H0) + H); +opaque symbol and__assoc + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + cupicef.Term cupicef.prop (iff (and (and A B) C) (and A (and B C))) + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (and (and A B) C) + (λ (__ : cupicef.Term cupicef.prop (and (and A B) C)), + and A (and B C))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (and A (and B C)) + (λ (__ : cupicef.Term cupicef.prop (and A (and B C))), + and (and A B) C)) + (λ (H : cupicef.Term cupicef.prop (and (and A B) C)), + match____and cupicef.prop (and A B) C + (λ (__ : cupicef.Term cupicef.prop (and (and A B) C)), + and A (and B C)) + (λ (H0 : cupicef.Term cupicef.prop (and A B)), + match____and cupicef.prop A B + (λ (__ : cupicef.Term cupicef.prop (and A B)), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I C + (λ (__0 : cupicef.Term cupicef.prop C), and A (and B C))) + (λ (H1 : cupicef.Term cupicef.prop A), + λ (H2 : cupicef.Term cupicef.prop B), + λ (H3 : cupicef.Term cupicef.prop C), + conj A (and B C) H1 (conj B C H2 H3)) + H0) + H) + (λ (H : cupicef.Term cupicef.prop (and A (and B C))), + match____and cupicef.prop A (and B C) + (λ (__ : cupicef.Term cupicef.prop (and A (and B C))), + and (and A B) C) + (λ (H0 : cupicef.Term cupicef.prop A), + λ (H1 : cupicef.Term cupicef.prop (and B C)), + match____and cupicef.prop B C + (λ (__ : cupicef.Term cupicef.prop (and B C)), + and (and A B) C) + (λ (H2 : cupicef.Term cupicef.prop B), + λ (H3 : cupicef.Term cupicef.prop C), + conj (and A B) C (conj A B H0 H2) H3) + H1) + H); +symbol let__o + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + Π + ( + Fl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (not A) + ), + Π + ( + Fr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop (not A) + ), + Π (H : cupicef.Term cupicef.prop (iff (or A B) (or A C))), + Π + ( + Hl : Π (__ : cupicef.Term cupicef.prop (or A B)), + cupicef.Term cupicef.prop (or A C) + ), + Π + ( + Hr : Π (__ : cupicef.Term cupicef.prop (or A C)), + cupicef.Term cupicef.prop (or A B) + ), + Π (H0 : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (or A C) + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + λ + ( + Fl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (not A) + ), + λ + ( + Fr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop (not A) + ), + λ (H : cupicef.Term cupicef.prop (iff (or A B) (or A C))), + λ + ( + Hl : Π (__ : cupicef.Term cupicef.prop (or A B)), + cupicef.Term cupicef.prop (or A C) + ), + λ + ( + Hr : Π (__ : cupicef.Term cupicef.prop (or A C)), + cupicef.Term cupicef.prop (or A B) + ), + λ (H0 : cupicef.Term cupicef.prop B), + Hl (or__intror A B H0); +symbol let__n + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + Π + ( + Fl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (not A) + ), + Π + ( + Fr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop (not A) + ), + Π (H : cupicef.Term cupicef.prop (iff (or A B) (or A C))), + Π + ( + Hl : Π (__ : cupicef.Term cupicef.prop (or A B)), + cupicef.Term cupicef.prop (or A C) + ), + Π + ( + Hr : Π (__ : cupicef.Term cupicef.prop (or A C)), + cupicef.Term cupicef.prop (or A B) + ), + Π (H0 : cupicef.Term cupicef.prop B), + Π (H1 : cupicef.Term cupicef.prop A), + cupicef.Term cupicef.prop False + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + λ + ( + Fl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (not A) + ), + λ + ( + Fr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop (not A) + ), + λ (H : cupicef.Term cupicef.prop (iff (or A B) (or A C))), + λ + ( + Hl : Π (__ : cupicef.Term cupicef.prop (or A B)), + cupicef.Term cupicef.prop (or A C) + ), + λ + ( + Hr : Π (__ : cupicef.Term cupicef.prop (or A C)), + cupicef.Term cupicef.prop (or A B) + ), + λ (H0 : cupicef.Term cupicef.prop B), + λ (H1 : cupicef.Term cupicef.prop A), Fl H0 H1; +symbol let__o0 + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + Π + ( + Fl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (not A) + ), + Π + ( + Fr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop (not A) + ), + Π (H : cupicef.Term cupicef.prop (iff (or A B) (or A C))), + Π + ( + Hl : Π (__ : cupicef.Term cupicef.prop (or A B)), + cupicef.Term cupicef.prop (or A C) + ), + Π + ( + Hr : Π (__ : cupicef.Term cupicef.prop (or A C)), + cupicef.Term cupicef.prop (or A B) + ), + Π (H0 : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop (or A B) + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + λ + ( + Fl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (not A) + ), + λ + ( + Fr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop (not A) + ), + λ (H : cupicef.Term cupicef.prop (iff (or A B) (or A C))), + λ + ( + Hl : Π (__ : cupicef.Term cupicef.prop (or A B)), + cupicef.Term cupicef.prop (or A C) + ), + λ + ( + Hr : Π (__ : cupicef.Term cupicef.prop (or A C)), + cupicef.Term cupicef.prop (or A B) + ), + λ (H0 : cupicef.Term cupicef.prop C), + Hr (or__intror A C H0); +symbol let__n0 + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + Π + ( + Fl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (not A) + ), + Π + ( + Fr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop (not A) + ), + Π (H : cupicef.Term cupicef.prop (iff (or A B) (or A C))), + Π + ( + Hl : Π (__ : cupicef.Term cupicef.prop (or A B)), + cupicef.Term cupicef.prop (or A C) + ), + Π + ( + Hr : Π (__ : cupicef.Term cupicef.prop (or A C)), + cupicef.Term cupicef.prop (or A B) + ), + Π (H0 : cupicef.Term cupicef.prop C), + Π (H1 : cupicef.Term cupicef.prop A), + cupicef.Term cupicef.prop False + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + λ + ( + Fl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (not A) + ), + λ + ( + Fr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop (not A) + ), + λ (H : cupicef.Term cupicef.prop (iff (or A B) (or A C))), + λ + ( + Hl : Π (__ : cupicef.Term cupicef.prop (or A B)), + cupicef.Term cupicef.prop (or A C) + ), + λ + ( + Hr : Π (__ : cupicef.Term cupicef.prop (or A C)), + cupicef.Term cupicef.prop (or A B) + ), + λ (H0 : cupicef.Term cupicef.prop C), + λ (H1 : cupicef.Term cupicef.prop A), Fr H0 H1; +opaque symbol or__cancel__l + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + Π + ( + __ : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (not A) + ), + Π + ( + __0 : Π (__0 : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop (not A) + ), + cupicef.Term cupicef.prop + (iff (iff (or A B) (or A C)) (iff B C)) + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + λ + ( + Fl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (not A) + ), + λ + ( + Fr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop (not A) + ), + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (iff (or A B) (or A C)) + (λ + (__ : cupicef.Term cupicef.prop (iff (or A B) (or A C))), + iff B C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (iff B C) + (λ (__ : cupicef.Term cupicef.prop (iff B C)), + iff (or A B) (or A C))) + (λ (H : cupicef.Term cupicef.prop (iff (or A B) (or A C))), + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (or A B) + (λ (__ : cupicef.Term cupicef.prop (or A B)), or A C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (or A C) + (λ (__ : cupicef.Term cupicef.prop (or A C)), or A B)) + (λ + ( + __ : cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I (or A B) + (λ (__ : cupicef.Term cupicef.prop (or A B)), + or A C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I (or A C) + (λ (__ : cupicef.Term cupicef.prop (or A C)), + or A B))) + ), iff B C) + (λ + ( + Hl : Π (__ : cupicef.Term cupicef.prop (or A B)), + cupicef.Term cupicef.prop (or A C) + ), + λ + ( + Hr : Π (__ : cupicef.Term cupicef.prop (or A C)), + cupicef.Term cupicef.prop (or A B) + ), + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I C + (λ (__ : cupicef.Term cupicef.prop C), B)) + (λ (H0 : cupicef.Term cupicef.prop B), + match____or cupicef.prop A C + (λ (__ : cupicef.Term cupicef.prop (or A C)), C) + (λ (H1 : cupicef.Term cupicef.prop A), + match____False cupicef.prop + (λ (__ : cupicef.Term cupicef.prop False), C) + (let__n A B C Fl Fr H Hl Hr H0 H1)) + (λ (H1 : cupicef.Term cupicef.prop C), H1) + (let__o A B C Fl Fr H Hl Hr H0)) + (λ (H0 : cupicef.Term cupicef.prop C), + match____or cupicef.prop A B + (λ (__ : cupicef.Term cupicef.prop (or A B)), B) + (λ (H1 : cupicef.Term cupicef.prop A), + match____False cupicef.prop + (λ (__ : cupicef.Term cupicef.prop False), B) + (let__n0 A B C Fl Fr H Hl Hr H0 H1)) + (λ (H1 : cupicef.Term cupicef.prop B), H1) + (let__o0 A B C Fl Fr H Hl Hr H0))) + H) + (or__iff__compat__l A B C); +symbol let__o1 + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + Π + ( + Fl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (not A) + ), + Π + ( + Fr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop (not A) + ), + Π (H : cupicef.Term cupicef.prop (iff (or B A) (or C A))), + Π + ( + Hl : Π (__ : cupicef.Term cupicef.prop (or B A)), + cupicef.Term cupicef.prop (or C A) + ), + Π + ( + Hr : Π (__ : cupicef.Term cupicef.prop (or C A)), + cupicef.Term cupicef.prop (or B A) + ), + Π (H0 : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (or C A) + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + λ + ( + Fl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (not A) + ), + λ + ( + Fr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop (not A) + ), + λ (H : cupicef.Term cupicef.prop (iff (or B A) (or C A))), + λ + ( + Hl : Π (__ : cupicef.Term cupicef.prop (or B A)), + cupicef.Term cupicef.prop (or C A) + ), + λ + ( + Hr : Π (__ : cupicef.Term cupicef.prop (or C A)), + cupicef.Term cupicef.prop (or B A) + ), + λ (H0 : cupicef.Term cupicef.prop B), + Hl (or__introl B A H0); +symbol let__n1 + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + Π + ( + Fl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (not A) + ), + Π + ( + Fr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop (not A) + ), + Π (H : cupicef.Term cupicef.prop (iff (or B A) (or C A))), + Π + ( + Hl : Π (__ : cupicef.Term cupicef.prop (or B A)), + cupicef.Term cupicef.prop (or C A) + ), + Π + ( + Hr : Π (__ : cupicef.Term cupicef.prop (or C A)), + cupicef.Term cupicef.prop (or B A) + ), + Π (H0 : cupicef.Term cupicef.prop B), + Π (H1 : cupicef.Term cupicef.prop A), + cupicef.Term cupicef.prop False + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + λ + ( + Fl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (not A) + ), + λ + ( + Fr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop (not A) + ), + λ (H : cupicef.Term cupicef.prop (iff (or B A) (or C A))), + λ + ( + Hl : Π (__ : cupicef.Term cupicef.prop (or B A)), + cupicef.Term cupicef.prop (or C A) + ), + λ + ( + Hr : Π (__ : cupicef.Term cupicef.prop (or C A)), + cupicef.Term cupicef.prop (or B A) + ), + λ (H0 : cupicef.Term cupicef.prop B), + λ (H1 : cupicef.Term cupicef.prop A), Fl H0 H1; +symbol let__o2 + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + Π + ( + Fl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (not A) + ), + Π + ( + Fr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop (not A) + ), + Π (H : cupicef.Term cupicef.prop (iff (or B A) (or C A))), + Π + ( + Hl : Π (__ : cupicef.Term cupicef.prop (or B A)), + cupicef.Term cupicef.prop (or C A) + ), + Π + ( + Hr : Π (__ : cupicef.Term cupicef.prop (or C A)), + cupicef.Term cupicef.prop (or B A) + ), + Π (H0 : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop (or B A) + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + λ + ( + Fl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (not A) + ), + λ + ( + Fr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop (not A) + ), + λ (H : cupicef.Term cupicef.prop (iff (or B A) (or C A))), + λ + ( + Hl : Π (__ : cupicef.Term cupicef.prop (or B A)), + cupicef.Term cupicef.prop (or C A) + ), + λ + ( + Hr : Π (__ : cupicef.Term cupicef.prop (or C A)), + cupicef.Term cupicef.prop (or B A) + ), + λ (H0 : cupicef.Term cupicef.prop C), + Hr (or__introl C A H0); +symbol let__n2 + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + Π + ( + Fl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (not A) + ), + Π + ( + Fr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop (not A) + ), + Π (H : cupicef.Term cupicef.prop (iff (or B A) (or C A))), + Π + ( + Hl : Π (__ : cupicef.Term cupicef.prop (or B A)), + cupicef.Term cupicef.prop (or C A) + ), + Π + ( + Hr : Π (__ : cupicef.Term cupicef.prop (or C A)), + cupicef.Term cupicef.prop (or B A) + ), + Π (H0 : cupicef.Term cupicef.prop C), + Π (H1 : cupicef.Term cupicef.prop A), + cupicef.Term cupicef.prop False + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + λ + ( + Fl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (not A) + ), + λ + ( + Fr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop (not A) + ), + λ (H : cupicef.Term cupicef.prop (iff (or B A) (or C A))), + λ + ( + Hl : Π (__ : cupicef.Term cupicef.prop (or B A)), + cupicef.Term cupicef.prop (or C A) + ), + λ + ( + Hr : Π (__ : cupicef.Term cupicef.prop (or C A)), + cupicef.Term cupicef.prop (or B A) + ), + λ (H0 : cupicef.Term cupicef.prop C), + λ (H1 : cupicef.Term cupicef.prop A), Fr H0 H1; +opaque symbol or__cancel__r + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + Π + ( + __ : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (not A) + ), + Π + ( + __0 : Π (__0 : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop (not A) + ), + cupicef.Term cupicef.prop + (iff (iff (or B A) (or C A)) (iff B C)) + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + λ + ( + Fl : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (not A) + ), + λ + ( + Fr : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop (not A) + ), + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (iff (or B A) (or C A)) + (λ + (__ : cupicef.Term cupicef.prop (iff (or B A) (or C A))), + iff B C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (iff B C) + (λ (__ : cupicef.Term cupicef.prop (iff B C)), + iff (or B A) (or C A))) + (λ (H : cupicef.Term cupicef.prop (iff (or B A) (or C A))), + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (or B A) + (λ (__ : cupicef.Term cupicef.prop (or B A)), or C A)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (or C A) + (λ (__ : cupicef.Term cupicef.prop (or C A)), or B A)) + (λ + ( + __ : cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I (or B A) + (λ (__ : cupicef.Term cupicef.prop (or B A)), + or C A)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I (or C A) + (λ (__ : cupicef.Term cupicef.prop (or C A)), + or B A))) + ), iff B C) + (λ + ( + Hl : Π (__ : cupicef.Term cupicef.prop (or B A)), + cupicef.Term cupicef.prop (or C A) + ), + λ + ( + Hr : Π (__ : cupicef.Term cupicef.prop (or C A)), + cupicef.Term cupicef.prop (or B A) + ), + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I C + (λ (__ : cupicef.Term cupicef.prop C), B)) + (λ (H0 : cupicef.Term cupicef.prop B), + match____or cupicef.prop C A + (λ (__ : cupicef.Term cupicef.prop (or C A)), C) + (λ (H1 : cupicef.Term cupicef.prop C), H1) + (λ (H1 : cupicef.Term cupicef.prop A), + match____False cupicef.prop + (λ (__ : cupicef.Term cupicef.prop False), C) + (let__n1 A B C Fl Fr H Hl Hr H0 H1)) + (let__o1 A B C Fl Fr H Hl Hr H0)) + (λ (H0 : cupicef.Term cupicef.prop C), + match____or cupicef.prop B A + (λ (__ : cupicef.Term cupicef.prop (or B A)), B) + (λ (H1 : cupicef.Term cupicef.prop B), H1) + (λ (H1 : cupicef.Term cupicef.prop A), + match____False cupicef.prop + (λ (__ : cupicef.Term cupicef.prop False), B) + (let__n2 A B C Fl Fr H Hl Hr H0 H1)) + (let__o2 A B C Fl Fr H Hl Hr H0))) + H) + (or__iff__compat__r A B C); +opaque symbol or__comm + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + cupicef.Term cupicef.prop (iff (or A B) (or B A)) + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I (or A B) + (λ (__ : cupicef.Term cupicef.prop (or A B)), or B A)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I (or B A) + (λ (__ : cupicef.Term cupicef.prop (or B A)), or A B)) + (λ (H : cupicef.Term cupicef.prop (or A B)), + match____or cupicef.prop A B + (λ (__ : cupicef.Term cupicef.prop (or A B)), or B A) + (λ (H0 : cupicef.Term cupicef.prop A), or__intror B A H0) + (λ (H0 : cupicef.Term cupicef.prop B), or__introl B A H0) H) + (λ (H : cupicef.Term cupicef.prop (or B A)), + match____or cupicef.prop B A + (λ (__ : cupicef.Term cupicef.prop (or B A)), or A B) + (λ (H0 : cupicef.Term cupicef.prop B), or__intror A B H0) + (λ (H0 : cupicef.Term cupicef.prop A), or__introl A B H0) H); +opaque symbol or__assoc + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + cupicef.Term cupicef.prop (iff (or (or A B) C) (or A (or B C))) + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (or (or A B) C) + (λ (__ : cupicef.Term cupicef.prop (or (or A B) C)), + or A (or B C))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (or A (or B C)) + (λ (__ : cupicef.Term cupicef.prop (or A (or B C))), + or (or A B) C)) + (λ (H : cupicef.Term cupicef.prop (or (or A B) C)), + match____or cupicef.prop (or A B) C + (λ (__ : cupicef.Term cupicef.prop (or (or A B) C)), + or A (or B C)) + (λ (H0 : cupicef.Term cupicef.prop (or A B)), + match____or cupicef.prop A B + (λ (__ : cupicef.Term cupicef.prop (or A B)), + or A (or B C)) + (λ (H1 : cupicef.Term cupicef.prop A), + or__introl A (or B C) H1) + (λ (H1 : cupicef.Term cupicef.prop B), + or__intror A (or B C) (or__introl B C H1)) + H0) + (λ (H0 : cupicef.Term cupicef.prop C), + or__intror A (or B C) (or__intror B C H0)) + H) + (λ (H : cupicef.Term cupicef.prop (or A (or B C))), + match____or cupicef.prop A (or B C) + (λ (__ : cupicef.Term cupicef.prop (or A (or B C))), + or (or A B) C) + (λ (H0 : cupicef.Term cupicef.prop A), + or__introl (or A B) C (or__introl A B H0)) + (λ (H0 : cupicef.Term cupicef.prop (or B C)), + match____or cupicef.prop B C + (λ (__ : cupicef.Term cupicef.prop (or B C)), + or (or A B) C) + (λ (H1 : cupicef.Term cupicef.prop B), + or__introl (or A B) C (or__intror A B H1)) + (λ (H1 : cupicef.Term cupicef.prop C), + or__intror (or A B) C H1) + H0) + H); +opaque symbol iff__and + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (__ : cupicef.Term cupicef.prop (iff A B)), + cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__0 : cupicef.Term cupicef.prop A), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__0 : cupicef.Term cupicef.prop B), A))) + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (H : cupicef.Term cupicef.prop (iff A B)), + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), A)) + (λ + ( + __ : cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + A (λ (__ : cupicef.Term cupicef.prop A), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + B (λ (__ : cupicef.Term cupicef.prop B), A))) + ), + and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__0 : cupicef.Term cupicef.prop A), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__0 : cupicef.Term cupicef.prop B), A))) + (λ + ( + H0 : Π (__ : cupicef.Term cupicef.prop A), + cupicef.Term cupicef.prop B + ), + λ + ( + H1 : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop A + ), + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), A)) + H0 H1) + H; +opaque symbol iff__to__and + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + cupicef.Term cupicef.prop + (iff (iff A B) + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), A)))) + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I (iff A B) + (λ (__ : cupicef.Term cupicef.prop (iff A B)), + and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__0 : cupicef.Term cupicef.prop A), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__0 : cupicef.Term cupicef.prop B), A)))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), A))) + (λ + ( + __ : cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), A))) + ), iff A B)) + (λ (H : cupicef.Term cupicef.prop (iff A B)), + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), A)) + (λ + ( + __ : cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + A (λ (__ : cupicef.Term cupicef.prop A), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + B (λ (__ : cupicef.Term cupicef.prop B), A))) + ), + and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__0 : cupicef.Term cupicef.prop A), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__0 : cupicef.Term cupicef.prop B), A))) + (λ + ( + Hl : Π (__ : cupicef.Term cupicef.prop A), + cupicef.Term cupicef.prop B + ), + λ + ( + Hr : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop A + ), + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), A)) + (λ (H0 : cupicef.Term cupicef.prop A), Hl H0) + (λ (H0 : cupicef.Term cupicef.prop B), Hr H0)) + H) + (λ + ( + H : cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), A))) + ), + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), A)) + (λ + ( + __ : cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + A (λ (__ : cupicef.Term cupicef.prop A), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + B (λ (__ : cupicef.Term cupicef.prop B), A))) + ), iff A B) + (λ + ( + Hl : Π (__ : cupicef.Term cupicef.prop A), + cupicef.Term cupicef.prop B + ), + λ + ( + Hr : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop A + ), + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), A)) + (λ (H0 : cupicef.Term cupicef.prop A), Hl H0) + (λ (H0 : cupicef.Term cupicef.prop B), Hr H0)) + H); +symbol IF__then__else + : Π (P : cupicef.Univ cupicef.prop), + Π (Q : cupicef.Univ cupicef.prop), + Π (R : cupicef.Univ cupicef.prop), cupicef.Univ cupicef.prop + ≔ λ (P : cupicef.Univ cupicef.prop), + λ (Q : cupicef.Univ cupicef.prop), + λ (R : cupicef.Univ cupicef.prop), or (and P Q) (and (not P) R); +symbol ex + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ cupicef.prop + ), cupicef.Univ cupicef.prop; +symbol ex__intro + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ cupicef.prop + ), + Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (__ : cupicef.Term cupicef.prop (P x)), + cupicef.Term cupicef.prop (ex A P); +rule cupicef.code_guarded? _ (ex__intro _ _ _ _) ↪ cupicef.guarded; +symbol match____ex + : Π (s : cupicef.Sort), + Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + Π (P0 : cupicef.Term cupicef.prop (ex A P) → cupicef.Univ s), + Π + ( + case__ex__intro : Π + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + Π (__ : cupicef.Term cupicef.prop (P x0)), + cupicef.Term s (P0 (ex__intro A P x0 __)) + ), + Π (x : cupicef.Term cupicef.prop (ex A P)), + cupicef.Term s (P0 x); +rule match____ex $s $A $P $P0 $case__ex__intro (ex__intro $A $P $x0 $__) ↪ +$case__ex__intro $x0 $__; +rule match____ex $s $A $P + (λ x, cupicef.uncode $_.[] (cupicef.code (cupicef.cU $s'.[]) $P0.[x])) ↪ +match____ex $s' $A $P (λ x, $P0.[x]); +symbol ex__ind + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ cupicef.prop + ), + Π (P0 : cupicef.Univ cupicef.prop), + Π + ( + f : Π + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π (__ : cupicef.Term cupicef.prop (P x)), + cupicef.Term cupicef.prop P0 + ), + Π (e : cupicef.Term cupicef.prop (ex A P)), + cupicef.Term cupicef.prop P0 + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + λ (P0 : cupicef.Univ cupicef.prop), + λ + ( + f : Π + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + Π (__ : cupicef.Term cupicef.prop (P x)), + cupicef.Term cupicef.prop P0 + ), + λ (e : cupicef.Term cupicef.prop (ex A P)), + match____ex cupicef.prop A P + (λ (__ : cupicef.Term cupicef.prop (ex A P)), P0) f e; +symbol ex2 + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ cupicef.prop + ), + Π + ( + Q : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), cupicef.Univ cupicef.prop; +symbol ex__intro2 + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ cupicef.prop + ), + Π + ( + Q : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (__ : cupicef.Term cupicef.prop (P x)), + Π (__0 : cupicef.Term cupicef.prop (Q x)), + cupicef.Term cupicef.prop (ex2 A P Q); +rule cupicef.code_guarded? _ (ex__intro2 _ _ _ _ _ _) ↪ cupicef.guarded; +symbol match____ex2 + : Π (s : cupicef.Sort), + Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + Π + ( + Q : Π + ( + __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), cupicef.Univ cupicef.prop + ), + Π + (P0 : cupicef.Term cupicef.prop (ex2 A P Q) → cupicef.Univ s), + Π + ( + case__ex__intro2 : Π + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + Π + (__ : cupicef.Term cupicef.prop (P x0) + ), + Π + ( + __0 : cupicef.Term cupicef.prop + (Q x0) + ), + cupicef.Term s + (P0 (ex__intro2 A P Q x0 __ __0)) + ), + Π (x : cupicef.Term cupicef.prop (ex2 A P Q)), + cupicef.Term s (P0 x); +rule match____ex2 $s $A $P $Q $P0 $case__ex__intro2 + (ex__intro2 $A $P $Q $x0 $__ $__0) ↪ $case__ex__intro2 $x0 $__ $__0; +rule match____ex2 $s $A $P $Q + (λ x, cupicef.uncode $_.[] (cupicef.code (cupicef.cU $s'.[]) $P0.[x])) ↪ +match____ex2 $s' $A $P $Q (λ x, $P0.[x]); +symbol ex2__ind + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ cupicef.prop + ), + Π + ( + Q : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + Π (P0 : cupicef.Univ cupicef.prop), + Π + ( + f : Π + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + Π (__ : cupicef.Term cupicef.prop (P x)), + Π (__0 : cupicef.Term cupicef.prop (Q x)), + cupicef.Term cupicef.prop P0 + ), + Π (e : cupicef.Term cupicef.prop (ex2 A P Q)), + cupicef.Term cupicef.prop P0 + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + λ + ( + Q : Π + ( + __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), cupicef.Univ cupicef.prop + ), + λ (P0 : cupicef.Univ cupicef.prop), + λ + ( + f : Π + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π (__ : cupicef.Term cupicef.prop (P x)), + Π (__0 : cupicef.Term cupicef.prop (Q x)), + cupicef.Term cupicef.prop P0 + ), + λ (e : cupicef.Term cupicef.prop (ex2 A P Q)), + match____ex2 cupicef.prop A P Q + (λ (__ : cupicef.Term cupicef.prop (ex2 A P Q)), P0) f e; +symbol all + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ cupicef.prop + ), cupicef.Univ cupicef.prop + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.prop + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + P x); +opaque symbol inst + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ cupicef.prop + ), + Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + __ : cupicef.Term cupicef.prop + (all A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x0)) + ), cupicef.Term cupicef.prop (P x) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + cupicef.cast + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.prop)) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.prop) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.prop)) + cupicef.I A + (λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.prod + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.prop + (cupicef.{|rule|} + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I A + (λ + ( + x0 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), P x0)) + (λ + ( + H : Π + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Term cupicef.prop (P x0) + ), P x))) + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (all A + (λ + ( + x0 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), P x0)) + (λ + ( + __ : cupicef.Term cupicef.prop + (all A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x0)) + ), P x))) + cupicef.I + (λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + H : Π + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Term cupicef.prop (P x0) + ), H x); +opaque symbol gen + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ cupicef.prop + ), + Π (B : cupicef.Univ cupicef.prop), + Π + ( + f : Π + ( + y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (P y) + ), + Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (all A P) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + cupicef.cast + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.prop)))) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + (cupicef.{|rule|} cupicef.prop cupicef.prop))) + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.prop))) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.prop)))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (λ (B : cupicef.Univ cupicef.prop), + cupicef.prod + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.prop)) + (cupicef.{|rule|} + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.prop))) + cupicef.I + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), P y))) + (λ + ( + f : Π + ( + y : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (P y) + ), + cupicef.prod cupicef.prop + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.prop) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.prop)) + cupicef.I B + (λ (H : cupicef.Term cupicef.prop B), + cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x))))) + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + (cupicef.{|rule|} cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (λ (B : cupicef.Univ cupicef.prop), + cupicef.prod + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), P y))) + (λ + ( + f : Π + ( + y : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (P y) + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), all A P)))) + cupicef.I + (λ (B : cupicef.Univ cupicef.prop), + λ + ( + f : Π + ( + y : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop (P y) + ), + λ (H : cupicef.Term cupicef.prop B), + λ + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), f x H); +symbol eq + : Π + (A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))), + Π + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + A + ), + Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), cupicef.Univ cupicef.prop; +symbol eq__refl + : Π + (A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))), + Π + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + A + ), cupicef.Term cupicef.prop (eq A x x); +rule cupicef.code_guarded? _ (eq__refl _ _) ↪ cupicef.guarded; +symbol match____eq + : Π (s : cupicef.Sort), + Π + ( + A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + ), + Π + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + Π + ( + P : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), cupicef.Term cupicef.prop (eq A x __) → cupicef.Univ s + ), + Π (case__eq__refl : cupicef.Term s (P x (eq__refl A x))), + Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + Π (x0 : cupicef.Term cupicef.prop (eq A x __)), + cupicef.Term s (P __ x0); +rule match____eq $s $A $x $P $case__eq__refl $x (eq__refl $A $x) ↪ $case__eq__refl; +rule match____eq $s $A $x + (λ __, + λ x, + cupicef.uncode $_.[] (cupicef.code (cupicef.cU $s'.[]) $P.[__; x])) ↪ +match____eq $s' $A $x (λ __, λ x, $P.[__; x]); +symbol eq__rect + : Π + (A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))), + Π + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + A + ), + Π + ( + P : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) + ), + Π + (f : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) (P x)), + Π + ( + y : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + Π (e : cupicef.Term cupicef.prop (eq A x y)), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) (P y) + ≔ λ + ( + A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + ), + λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + λ + ( + P : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) + ), + λ + ( + f : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) (P x) + ), + λ + ( + y : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + λ (e : cupicef.Term cupicef.prop (eq A x y)), + match____eq (cupicef.{|type|} (cupicef.s cupicef.z)) A x + (λ + ( + y0 : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + A + ), λ (__ : cupicef.Term cupicef.prop (eq A x y0)), P y0) + f y e; +symbol eq__ind + : Π + (A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))), + Π + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + A + ), + Π + ( + P : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), cupicef.Univ cupicef.prop + ), + Π (f : cupicef.Term cupicef.prop (P x)), + Π + ( + y : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + Π (e : cupicef.Term cupicef.prop (eq A x y)), + cupicef.Term cupicef.prop (P y) + ≔ λ + ( + A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + ), + λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + λ + ( + P : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), cupicef.Univ cupicef.prop + ), + λ (f : cupicef.Term cupicef.prop (P x)), + λ + ( + y : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + λ (e : cupicef.Term cupicef.prop (eq A x y)), + match____eq cupicef.prop A x + (λ + ( + y0 : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + A + ), λ (__ : cupicef.Term cupicef.prop (eq A x y0)), P y0) + f y e; +symbol eq__rec + : Π + (A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))), + Π + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + A + ), + Π + ( + P : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), cupicef.Univ cupicef.set + ), + Π (f : cupicef.Term cupicef.set (P x)), + Π + ( + y : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + Π (e : cupicef.Term cupicef.prop (eq A x y)), + cupicef.Term cupicef.set (P y) + ≔ λ + ( + A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + ), + λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + λ + ( + P : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), cupicef.Univ cupicef.set + ), + eq__rect A x + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + cupicef.univ cupicef.set + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P); +symbol let__f + : Π (A : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + Π (h1 : cupicef.Term cupicef.prop A), + Π + ( + h2 : Π (__ : cupicef.Term cupicef.prop A), + cupicef.Term cupicef.prop False + ), cupicef.Term cupicef.prop False + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + λ (h1 : cupicef.Term cupicef.prop A), + λ + ( + h2 : Π (__ : cupicef.Term cupicef.prop A), + cupicef.Term cupicef.prop False + ), h2 h1; +opaque symbol absurd + : Π (A : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + Π (__ : cupicef.Term cupicef.prop A), + Π (__0 : cupicef.Term cupicef.prop (not A)), + cupicef.Term cupicef.prop C + ≔ cupicef.cast + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.prop)))) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop)))) + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.prop))) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.prop)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (λ (A : cupicef.Univ cupicef.prop), + cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.prop)) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (λ (C : cupicef.Univ cupicef.prop), + cupicef.prod cupicef.prop + (cupicef.{|rule|} (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.prop) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.prop)) + cupicef.I A + (λ (h1 : cupicef.Term cupicef.prop A), + cupicef.prod (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.prop + (cupicef.{|rule|} + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), False)) + (λ + ( + h2 : Π (__ : cupicef.Term cupicef.prop A), + cupicef.Term cupicef.prop False + ), C))))) + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop))) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop)))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (λ (A : cupicef.Univ cupicef.prop), + cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop))) + cupicef.I + (cupicef.univ cupicef.prop (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (λ (C : cupicef.Univ cupicef.prop), + cupicef.prod cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (not A) (λ (__0 : cupicef.Term cupicef.prop (not A)), C))))) + cupicef.I + (λ (A : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + λ (h1 : cupicef.Term cupicef.prop A), + λ + ( + h2 : Π (__ : cupicef.Term cupicef.prop A), + cupicef.Term cupicef.prop False + ), + match____False cupicef.prop + (λ (__ : cupicef.Term cupicef.prop False), C) + (let__f A C h1 h2)); +symbol eq__sym + : Π + (A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))), + Π + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + A + ), + Π + ( + y : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + Π (__ : cupicef.Term cupicef.prop (eq A x y)), + cupicef.Term cupicef.prop (eq A y x) + ≔ λ + ( + A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + ), + λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + λ + ( + y : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + λ (H : cupicef.Term cupicef.prop (eq A x y)), + match____eq cupicef.prop A x + (λ + ( + y0 : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + λ (__ : cupicef.Term cupicef.prop (eq A x y0)), eq A y0 x) + (eq__refl A x) y H; +symbol eq__trans + : Π + (A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))), + Π + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + A + ), + Π + ( + y : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + Π + ( + z : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + Π (__ : cupicef.Term cupicef.prop (eq A x y)), + Π (__0 : cupicef.Term cupicef.prop (eq A y z)), + cupicef.Term cupicef.prop (eq A x z) + ≔ λ + ( + A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + ), + λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + λ + ( + y : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + λ + ( + z : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + λ (H : cupicef.Term cupicef.prop (eq A x y)), + λ (H0 : cupicef.Term cupicef.prop (eq A y z)), + match____eq cupicef.prop A y + (λ + ( + y0 : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + A + ), + λ (__ : cupicef.Term cupicef.prop (eq A y y0)), + eq A x y0) + H z H0; +symbol f__equal + : Π + (A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))), + Π + ( + B : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + ), + Π + ( + f : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) B + ), + Π + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + Π + ( + y : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + Π (__ : cupicef.Term cupicef.prop (eq A x y)), + cupicef.Term cupicef.prop (eq B (f x) (f y)) + ≔ λ + ( + A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + ), + λ + ( + B : cupicef.Univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + ), + λ + ( + f : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) B + ), + λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + λ + ( + y : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + λ (H : cupicef.Term cupicef.prop (eq A x y)), + match____eq cupicef.prop A x + (λ + ( + y0 : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + A + ), + λ (__ : cupicef.Term cupicef.prop (eq A x y0)), + eq B (f x) (f y0)) + (eq__refl B (f x)) y H; +opaque symbol not__eq__sym + : Π + (A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))), + Π + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + A + ), + Π + ( + y : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + Π (__ : cupicef.Term cupicef.prop (not (eq A x y))), + cupicef.Term cupicef.prop (not (eq A y x)) + ≔ λ + ( + A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + ), + λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + λ + ( + y : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A + ), + cupicef.cast + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.prod cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I (not (eq A x y)) + (λ (h1 : cupicef.Term cupicef.prop (not (eq A x y))), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (eq A y x) + (λ (h2 : cupicef.Term cupicef.prop (eq A y x)), False))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (not (eq A x y)) + (λ (__ : cupicef.Term cupicef.prop (not (eq A x y))), + not (eq A y x))) + cupicef.I + (λ (h1 : cupicef.Term cupicef.prop (not (eq A x y))), + λ (h2 : cupicef.Term cupicef.prop (eq A y x)), + h1 + (match____eq cupicef.prop A y + (λ + ( + y0 : cupicef.Term + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + A + ), + λ (__ : cupicef.Term cupicef.prop (eq A y y0)), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (not (eq A y0 y)) + (λ + (h3 : cupicef.Term cupicef.prop (not (eq A y0 y))), + eq A y0 y)) + (λ (h3 : cupicef.Term cupicef.prop (not (eq A y y))), + eq__refl A y) + x h2 h1)); +symbol eq__ind__r + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + Π (__ : cupicef.Term cupicef.prop (P x)), + Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + __0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y x) + ), cupicef.Term cupicef.prop (P y) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + P : Π + ( + __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), cupicef.Univ cupicef.prop + ), + λ (H : cupicef.Term cupicef.prop (P x)), + λ + (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + H0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y x) + ), + eq__ind + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + y0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P y0) + H y + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y x H0); +symbol eq__rec__r + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.set + ), + Π (__ : cupicef.Term cupicef.set (P x)), + Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + __0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y x) + ), cupicef.Term cupicef.set (P y) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + P : Π + ( + __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), cupicef.Univ cupicef.set + ), + λ (H : cupicef.Term cupicef.set (P x)), + λ + (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + H0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y x) + ), + eq__rec + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + y0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P y0) + H y + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y x H0); +symbol eq__rect__r + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) + ), + Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) (P x) + ), + Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + __0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y x) + ), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) (P y) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + P : Π + ( + __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) + ), + λ + ( + H : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) (P x) + ), + λ + (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + H0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y x) + ), + eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + y0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P y0) + H y + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y x H0); +symbol rew__opp__r + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) + ), + Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + H : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + Π + ( + a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + (P y) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P y)) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P (eq__rect__r A y P a x H) y H) + a) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) + ), + λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + H : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + λ + ( + a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + (P y) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + y0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) + ), + cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I (P y0) + (λ + ( + a0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P y0) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P y0)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P (eq__rect__r A y0 P a0 x e) y0 e) + a0)) + (λ + ( + a0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) (P x) + ), + eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + a0) + y H a; +symbol rew__opp__l + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) + ), + Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + H : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + Π + ( + a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (eq__rect__r A y P + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P a y H) + x H) + a) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) + ), + λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + H : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + λ + ( + a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + y0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (eq__rect__r A y0 P + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P a y0 e) + x e) + a) + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + a) + y H; +opaque symbol f__equal2 + : Π (A1 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (A2 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + f : Π + ( + __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A1 + ), + Π + ( + __0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A2 + ), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) B + ), + Π + (x1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A1), + Π + ( + y1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A1 + ), + Π + ( + x2 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A2 + ), + Π + ( + y2 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A2 + ), + Π + ( + __ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 y1) + ), + Π + ( + __0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2) (f y1 y2)) + ≔ λ (A1 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (A2 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + f : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A1 + ), + Π + ( + __0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A2 + ), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) B + ), + λ + ( + x1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A1 + ), + λ + ( + y1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A1 + ), + λ + ( + x2 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A2 + ), + λ + ( + y2 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A2 + ), + λ + ( + H : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 y1) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 + (λ + ( + y : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A1 + ), + λ + ( + __ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 y) + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) + (λ + ( + __0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2) (f y y2))) + (λ + ( + H0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 + (λ + ( + y : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A2 + ), + λ + ( + __ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2) (f x1 y)) + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2)) + y2 H0) + y1 H; +opaque symbol f__equal3 + : Π (A1 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (A2 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (A3 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + f : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A1 + ), + Π + ( + __0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A2 + ), + Π + ( + __1 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A3 + ), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + B + ), + Π + ( + x1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A1 + ), + Π + ( + y1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A1 + ), + Π + ( + x2 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A2 + ), + Π + ( + y2 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A2 + ), + Π + ( + x3 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A3 + ), + Π + ( + y3 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A3 + ), + Π + ( + __ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 y1) + ), + Π + ( + __0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) + ), + Π + ( + __1 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3) (f y1 y2 y3)) + ≔ λ (A1 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (A2 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (A3 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + f : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A1 + ), + Π + ( + __0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A2 + ), + Π + ( + __1 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A3 + ), + cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) B + ), + λ + ( + x1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A1 + ), + λ + ( + y1 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A1 + ), + λ + ( + x2 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A2 + ), + λ + ( + y2 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A2 + ), + λ + ( + x3 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A3 + ), + λ + ( + y3 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A3 + ), + λ + ( + H : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 y1) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 + (λ + ( + y : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A1 + ), + λ + ( + __ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 y) + ), + cupicef.prod cupicef.prop + (cupicef.{|rule|} cupicef.prop + cupicef.prop) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop + cupicef.prop)) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) + (λ + ( + __0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop + cupicef.prop) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) + (λ + ( + __1 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3) (f y y2 y3)))) + (λ + ( + H0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 + (λ + ( + y : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A2 + ), + λ + ( + __ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y) + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop + cupicef.prop) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) + (λ + ( + __0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3) (f x1 y y3))) + (λ + ( + H1 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 + (λ + ( + y : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A3 + ), + λ + ( + __ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3) (f x1 x2 y)) + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3)) + y3 H1) + y2 H0) + y1 H; +opaque symbol f__equal4 + : Π (A1 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (A2 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (A3 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (A4 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + f : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A1 + ), + Π + ( + __0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A2 + ), + Π + ( + __1 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A3 + ), + Π + ( + __2 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A4 + ), + cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) B + ), + Π + ( + x1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A1 + ), + Π + ( + y1 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A1 + ), + Π + ( + x2 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A2 + ), + Π + ( + y2 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A2 + ), + Π + ( + x3 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A3 + ), + Π + ( + y3 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A3 + ), + Π + ( + x4 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A4 + ), + Π + ( + y4 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A4 + ), + Π + ( + __ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 y1) + ), + Π + ( + __0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) + ), + Π + ( + __1 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) + ), + Π + ( + __2 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3 x4) (f y1 y2 y3 y4)) + ≔ λ (A1 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (A2 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (A3 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (A4 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + f : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A1 + ), + Π + ( + __0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A2 + ), + Π + ( + __1 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A3 + ), + Π + ( + __2 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A4 + ), + cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) B + ), + λ + ( + x1 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A1 + ), + λ + ( + y1 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A1 + ), + λ + ( + x2 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A2 + ), + λ + ( + y2 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A2 + ), + λ + ( + x3 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A3 + ), + λ + ( + y3 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A3 + ), + λ + ( + x4 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A4 + ), + λ + ( + y4 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A4 + ), + λ + ( + H : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 y1) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 + (λ + ( + y : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A1 + ), + λ + ( + __ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 y) + ), + cupicef.prod cupicef.prop + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop + cupicef.prop)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop + cupicef.prop))) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) + (λ + ( + __0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) + ), + cupicef.prod cupicef.prop + (cupicef.{|rule|} cupicef.prop + cupicef.prop) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop + cupicef.prop)) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) + (λ + ( + __1 : cupicef.Term + cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) + ), + cupicef.prod cupicef.prop + cupicef.prop + (cupicef.{|rule|} cupicef.prop + cupicef.prop) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) + (λ + ( + __2 : cupicef.Term + cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3 x4) + (f y y2 y3 y4))))) + (λ + ( + H0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 + (λ + ( + y : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A2 + ), + λ + ( + __ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y) + ), + cupicef.prod cupicef.prop + (cupicef.{|rule|} cupicef.prop + cupicef.prop) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop + cupicef.prop)) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) + (λ + ( + __0 : cupicef.Term + cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) + ), + cupicef.prod cupicef.prop + cupicef.prop + (cupicef.{|rule|} cupicef.prop + cupicef.prop) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) + (λ + ( + __1 : cupicef.Term + cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3 x4) + (f x1 y y3 y4)))) + (λ + ( + H1 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 + (λ + ( + y : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A3 + ), + λ + ( + __ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y) + ), + cupicef.prod cupicef.prop + cupicef.prop + (cupicef.{|rule|} cupicef.prop + cupicef.prop) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) + (λ + ( + __0 : cupicef.Term + cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3 x4) + (f x1 x2 y y4))) + (λ + ( + H2 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 + (λ + ( + y : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A4 + ), + λ + ( + __ : cupicef.Term + cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3 x4) + (f x1 x2 x3 y)) + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3 x4)) + y4 H2) + y3 H1) + y2 H0) + y1 H; +opaque symbol f__equal5 + : Π (A1 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (A2 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (A3 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (A4 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (A5 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + f : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A1 + ), + Π + ( + __0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A2 + ), + Π + ( + __1 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A3 + ), + Π + ( + __2 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A4 + ), + Π + ( + __3 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A5 + ), + cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) B + ), + Π + ( + x1 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A1 + ), + Π + ( + y1 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A1 + ), + Π + ( + x2 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A2 + ), + Π + ( + y2 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A2 + ), + Π + ( + x3 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A3 + ), + Π + ( + y3 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A3 + ), + Π + ( + x4 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A4 + ), + Π + ( + y4 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A4 + ), + Π + ( + x5 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A5 + ), + Π + ( + y5 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A5 + ), + Π + ( + __ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 y1) + ), + Π + ( + __0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) + ), + Π + ( + __1 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) + ), + Π + ( + __2 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) + ), + Π + ( + __3 : cupicef.Term + cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A5) + x5 y5) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3 x4 x5) + (f y1 y2 y3 y4 y5)) + ≔ λ (A1 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (A2 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (A3 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (A4 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + (A5 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + f : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A1 + ), + Π + ( + __0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A2 + ), + Π + ( + __1 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A3 + ), + Π + ( + __2 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A4 + ), + Π + ( + __3 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A5 + ), + cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) B + ), + λ + ( + x1 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A1 + ), + λ + ( + y1 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A1 + ), + λ + ( + x2 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A2 + ), + λ + ( + y2 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A2 + ), + λ + ( + x3 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A3 + ), + λ + ( + y3 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A3 + ), + λ + ( + x4 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A4 + ), + λ + ( + y4 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A4 + ), + λ + ( + x5 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A5 + ), + λ + ( + y5 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A5 + ), + λ + ( + H : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 y1) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 + (λ + ( + y : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A1 + ), + λ + ( + __ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A1) + x1 y) + ), + cupicef.prod cupicef.prop + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} + cupicef.prop + (cupicef.{|rule|} + cupicef.prop cupicef.prop))) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} + cupicef.prop + (cupicef.{|rule|} + cupicef.prop + (cupicef.{|rule|} + cupicef.prop cupicef.prop)))) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) + (λ + ( + __0 : cupicef.Term + cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) + ), + cupicef.prod cupicef.prop + (cupicef.{|rule|} + cupicef.prop + (cupicef.{|rule|} + cupicef.prop cupicef.prop)) + (cupicef.{|rule|} + cupicef.prop + (cupicef.{|rule|} + cupicef.prop + (cupicef.{|rule|} + cupicef.prop + cupicef.prop))) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) + (λ + ( + __1 : cupicef.Term + cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) + ), + cupicef.prod + cupicef.prop + (cupicef.{|rule|} + cupicef.prop + cupicef.prop) + (cupicef.{|rule|} + cupicef.prop + (cupicef.{|rule|} + cupicef.prop + cupicef.prop)) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) + (λ + ( + __2 : cupicef.Term + cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + ( + cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) + ), + cupicef.prod + cupicef.prop + cupicef.prop + (cupicef.{|rule|} + cupicef.prop + cupicef.prop) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A5) + x5 y5) + (λ + ( + __3 : cupicef.Term + cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + ( + cupicef.{|type|} + ( + cupicef.s + cupicef.z))) + (cupicef.axiom + ( + cupicef.{|type|} + ( + cupicef.s + ( + cupicef.s + cupicef.z)))) + (cupicef.univ + ( + cupicef.{|type|} + ( + cupicef.s + cupicef.z)) + ( + cupicef.axiom + ( + cupicef.{|type|} + ( + cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + ( + cupicef.{|type|} + ( + cupicef.s + ( + cupicef.s + cupicef.z))) + ( + cupicef.axiom + ( + cupicef.{|type|} + ( + cupicef.s + ( + cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I + A5) + x5 y5) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3 x4 x5) + (f y y2 y3 y4 y5)))))) + (λ + ( + H0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y2) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 + (λ + ( + y : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A2 + ), + λ + ( + __ : cupicef.Term + cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A2) + x2 y) + ), + cupicef.prod cupicef.prop + (cupicef.{|rule|} + cupicef.prop + (cupicef.{|rule|} + cupicef.prop cupicef.prop)) + (cupicef.{|rule|} + cupicef.prop + (cupicef.{|rule|} + cupicef.prop + (cupicef.{|rule|} + cupicef.prop + cupicef.prop))) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) + (λ + ( + __0 : cupicef.Term + cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) + ), + cupicef.prod + cupicef.prop + (cupicef.{|rule|} + cupicef.prop + cupicef.prop) + (cupicef.{|rule|} + cupicef.prop + (cupicef.{|rule|} + cupicef.prop + cupicef.prop)) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) + (λ + ( + __1 : cupicef.Term + cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + ( + cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) + ), + cupicef.prod + cupicef.prop + cupicef.prop + (cupicef.{|rule|} + cupicef.prop + cupicef.prop) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A5) + x5 y5) + (λ + ( + __2 : cupicef.Term + cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + ( + cupicef.{|type|} + ( + cupicef.s + cupicef.z))) + (cupicef.axiom + ( + cupicef.{|type|} + ( + cupicef.s + ( + cupicef.s + cupicef.z)))) + (cupicef.univ + ( + cupicef.{|type|} + ( + cupicef.s + cupicef.z)) + ( + cupicef.axiom + ( + cupicef.{|type|} + ( + cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + ( + cupicef.{|type|} + ( + cupicef.s + ( + cupicef.s + cupicef.z))) + ( + cupicef.axiom + ( + cupicef.{|type|} + ( + cupicef.s + ( + cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I + A5) + x5 y5) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3 x4 x5) + (f x1 y y3 y4 y5))))) + (λ + ( + H1 : cupicef.Term + cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y3) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 + (λ + ( + y : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A3 + ), + λ + ( + __ : cupicef.Term + cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A3) + x3 y) + ), + cupicef.prod + cupicef.prop + (cupicef.{|rule|} + cupicef.prop + cupicef.prop) + (cupicef.{|rule|} + cupicef.prop + (cupicef.{|rule|} + cupicef.prop + cupicef.prop)) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) + (λ + ( + __0 : cupicef.Term + cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + ( + cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) + ), + cupicef.prod + cupicef.prop + cupicef.prop + (cupicef.{|rule|} + cupicef.prop + cupicef.prop) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A5) + x5 y5) + (λ + ( + __1 : cupicef.Term + cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + ( + cupicef.{|type|} + ( + cupicef.s + cupicef.z))) + (cupicef.axiom + ( + cupicef.{|type|} + ( + cupicef.s + ( + cupicef.s + cupicef.z)))) + (cupicef.univ + ( + cupicef.{|type|} + ( + cupicef.s + cupicef.z)) + ( + cupicef.axiom + ( + cupicef.{|type|} + ( + cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + ( + cupicef.{|type|} + ( + cupicef.s + ( + cupicef.s + cupicef.z))) + ( + cupicef.axiom + ( + cupicef.{|type|} + ( + cupicef.s + ( + cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I + A5) + x5 y5) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3 x4 x5) + (f x1 x2 y y4 y5)))) + (λ + ( + H2 : cupicef.Term + cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y4) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 + (λ + ( + y : cupicef.Term + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + A4 + ), + λ + ( + __ : cupicef.Term + cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A4) + x4 y) + ), + cupicef.prod + cupicef.prop + cupicef.prop + (cupicef.{|rule|} + cupicef.prop + cupicef.prop) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A5) + x5 y5) + (λ + ( + __0 : cupicef.Term + cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + ( + cupicef.{|type|} + ( + cupicef.s + cupicef.z))) + (cupicef.axiom + ( + cupicef.{|type|} + ( + cupicef.s + ( + cupicef.s + cupicef.z)))) + (cupicef.univ + ( + cupicef.{|type|} + ( + cupicef.s + cupicef.z)) + ( + cupicef.axiom + ( + cupicef.{|type|} + ( + cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + ( + cupicef.{|type|} + ( + cupicef.s + ( + cupicef.s + cupicef.z))) + ( + cupicef.axiom + ( + cupicef.{|type|} + ( + cupicef.s + ( + cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I + A5) + x5 y5) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3 x4 x5) + (f x1 x2 x3 y y5))) + (λ + ( + H3 : cupicef.Term + cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A5) + x5 y5) + ), + match____eq + cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A5) + x5 + (λ + ( + y : cupicef.Term + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + A5 + ), + λ + ( + __ : cupicef.Term + cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + ( + cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + ( + cupicef.s + ( + cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + ( + cupicef.s + cupicef.z)) + (cupicef.axiom + ( + cupicef.{|type|} + ( + cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + ( + cupicef.s + ( + cupicef.s + cupicef.z))) + (cupicef.axiom + ( + cupicef.{|type|} + ( + cupicef.s + ( + cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I + A5) + x5 y) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3 x4 x5) + (f x1 x2 x3 x4 y)) + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x1 x2 x3 x4 x5)) + y5 H3) + y4 H2) + y3 H1) + y2 H0) + y1 H; +symbol f__equal__compose + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (C : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (b : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + f : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) B + ), + Π + ( + g : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) B + ), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + C + ), + Π + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a b) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + (g (f a)) (g (f b)))) + (f__equal + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + g (f a) (f b) + (f__equal + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f a b e)) + (f__equal + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + (λ + ( + a0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), g (f a0)) + a b e)) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (C : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + (b : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + f : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + B + ), + λ + ( + g : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) B + ), + cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) C + ), + λ + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a b) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a + (λ + ( + y : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + e0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a y) + ), + eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + (g (f a)) (g (f y)))) + (f__equal + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + g (f a) (f y) + (f__equal + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f a y e0)) + (f__equal + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + (λ + ( + a0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), g (f a0)) + a y e0)) + (eq__refl + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + (g (f a)) (g (f a)))) + (f__equal + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I C) + (λ + ( + a0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), g (f a0)) + a a + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a))) + b e; +symbol eq__trans__refl__l + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x y + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x) + e) + e) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + y0 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + e0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) + ), + eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0)) + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x y0 + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x) + e0) + e0) + (eq__refl + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + y e; +symbol eq__trans__refl__r + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y y e + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y)) + e) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + y0 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + e0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) + ), + eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0)) + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0 y0 e0 + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0)) + e0) + (eq__refl + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + y e; +symbol eq__sym__involutive + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)) + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y x + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y e)) + e) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + y0 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + e0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) + ), + eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0)) + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0 x + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0 e0)) + e0) + (eq__refl + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + y e; +symbol eq__trans__sym__inv__l + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y y)) + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y x y + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y e) + e) + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y)) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + y0 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + e0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) + ), + eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0 y0)) + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0 x y0 + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0 e0) + e0) + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0)) + (eq__refl + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + y e; +symbol eq__trans__sym__inv__r + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y x e + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y e)) + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + y0 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + e0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) + ), + eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0 x e0 + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0 e0)) + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + (eq__refl + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + y e; +symbol eq__trans__assoc + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (z : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (t : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + Π + ( + e_27_ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y z) + ), + Π + ( + e_27__27_ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + z t) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x t)) + (eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y t e + (eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y z t e_27_ e_27__27_)) + (eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x z t + (eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y z e e_27_) + e_27__27_)) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ (z : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + (t : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + λ + ( + e_27_ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y z) + ), + λ + ( + e_27__27_ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + z t) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + z + (λ + ( + y0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + e0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + z y0) + ), + eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0)) + (eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y y0 e + (eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y z y0 e_27_ e0)) + (eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x z y0 + (eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y z e e_27_) + e0)) + (eq__refl + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x z)) + (eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x z z + (eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y z e e_27_) + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + z))) + t e_27__27_; +symbol let__e + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + f : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π + ( + Hf : Π + (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a (f a)) + ), + Π (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a (f a)) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + f : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + Hf : Π + ( + a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a (f a)) + ), + λ (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Hf a; +symbol let__a0 + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + f : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π + ( + Hf : Π + (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a (f a)) + ), + Π (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + f : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + Hf : Π + ( + a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a (f a)) + ), + λ (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + f a; +symbol let__e0 + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + f : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π + ( + Hf : Π + (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a (f a)) + ), + Π (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a (f a)) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + f : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + Hf : Π + ( + a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a (f a)) + ), + λ (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Hf a; +symbol let__a1 + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + f : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π + ( + Hf : Π + (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a (f a)) + ), + Π (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + f : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + Hf : Π + ( + a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a (f a)) + ), + λ (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + f a; +symbol eq__id__comm__l + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + f : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π + ( + Hf : Π + (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a (f a)) + ), + Π (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) (f (f a)))) + (f__equal + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + f a (f a) (Hf a)) + (Hf (f a))) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + f : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + Hf : Π + ( + a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a (f a)) + ), + λ (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.cast cupicef.prop cupicef.prop + ((λ + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) (f a)) + ), + eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) (f (f a)))) + (match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a + (λ + ( + y : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + __ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + 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(cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a y e0) + (Hf a)) + y e0) + (Hf y)) + (match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a + (λ + ( + y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + λ + ( + e0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a y) + ), + eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a y)) + (eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a a y + (eq__sym + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a a + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a)) + e0) + e0) + (eq__refl + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a a)) + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a)) + (f a) (let__e0 A f Hf a)) + (f a) (let__e A f Hf a)) + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a)) + (eq__trans__sym__inv__l A a (f a) (Hf a))); +symbol let__Hfsymf + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + f : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π + ( + Hf : Π + (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a) + ), + Π (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + (a0 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + a0 (f a0)) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + f : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + Hf : Π + ( + a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a) + ), + λ (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + (a0 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a0) a0 (Hf a0); +symbol let__e1 + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + f : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π + ( + Hf : Π + (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a) + ), + Π (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f a)) (f (f (f a))))) + (f__equal + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + f (f a) (f (f a)) (let__Hfsymf A f Hf a (f a))) + (let__Hfsymf A f Hf a (f (f a)))) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + f : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + Hf : Π + ( + a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a) + ), + λ (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + eq__id__comm__l A f (let__Hfsymf A f Hf a) (f a); +symbol let__e2 + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + f : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π + ( + Hf : Π + (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a) + ), + Π (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f a)) (f (f (f a)))) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + f : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + Hf : Π + ( + a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a) + ), + λ (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + let__Hfsymf A f Hf a (f (f a)); +symbol let__e3 + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + f : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π + ( + Hf : Π + (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a) + ), + Π (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) (f (f a)))) + (f__equal + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + f a (f a) (let__Hfsymf A f Hf a a)) + (let__Hfsymf A f Hf a (f a))) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + f : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + Hf : Π + ( + a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a) + ), + λ (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + eq__id__comm__l A f (let__Hfsymf A f Hf a) a; +symbol let__e4 + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + f : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π + ( + Hf : Π + (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a) + ), + Π (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) (f (f a))) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + f : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + Hf : Π + ( + a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a) + ), + λ (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + let__Hfsymf A f Hf a (f a); +symbol let__e5 + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + f : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π + ( + Hf : Π + (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a) + ), + Π (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + f : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + Hf : Π + ( + a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a) + ), + λ (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Hf a; +symbol eq__id__comm__r + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + f : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π + ( + Hf : Π + (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a) + ), + Π (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f a)) (f a))) + (f__equal + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + f (f a) a (Hf a)) + (Hf (f a))) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + f : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + Hf : Π + ( + a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) a) + ), + λ (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.cast cupicef.prop cupicef.prop + ((λ + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f a)) (f (f a))) + ), + eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f a)) (f a))) + (match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) + (λ + ( + y : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + __ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) y) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f a)) (f y)) + e a (Hf a)) + (Hf (f a))) + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f a)))) + (eq + (cupicef.cast 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cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f a)) (f (f a))) + ), + eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + 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(cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) y) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f a)) (f y)) + e a (Hf a)) + (Hf (f a))) + (cupicef.cast cupicef.prop cupicef.prop + (eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f a)) (f a))) + (match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) + (λ + ( + y : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + __ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) y) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f a)) (f y)) + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f a)) (f (f (f a))) (f (f a)) + (let__Hfsymf A f Hf a (f (f a))) (Hf (f (f a)))) + a (Hf a)) + (Hf (f a))) + (eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + 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cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) + (λ + ( + y : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + __ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) y) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + 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+ (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f a)) (f (f (f a)))) + ), + λ + ( + __ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + 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(cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + f (f a) (f (f a)) (let__Hfsymf A f Hf a (f a))) + y) + ), + eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f a)) (f a))) + (match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) + (λ + ( + y0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + __0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) y0) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f a)) (f y0)) + (eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f a)) (f (f (f a))) (f (f a)) y (Hf (f (f a)))) + a (Hf a)) + (Hf (f a))) + (match____eq cupicef.prop + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) (f (f a)))) + (f__equal + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + f a (f a) (let__Hfsymf A f Hf a a)) + (λ + ( + y : 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(cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) + (λ + ( + y : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + __ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f a) y) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + 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+ (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + f a (f a) (let__Hfsymf A f Hf a a))) + (Hf (f (f a)))) + a (Hf a)) + (Hf (f a))) + cupicef.I + (match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} 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+ (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f (f a))) (f (f a)))) + e1 (Hf (f (f a)))) + (eq__refl + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f (f a))) (f (f a)))) + (Hf (f (f a)))) + (eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f (f a))) (f (f (f a))) (f (f a)) + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f (f a)))) + (Hf (f (f a)))) + (eq__trans__refl__l A (f (f (f a))) (f (f a)) + (Hf (f (f a)))))) + a (let__e5 A f Hf a))) + (let__Hfsymf A f Hf a (f a)) (let__e3 A f Hf a)) + (let__Hfsymf A f Hf a (f (f a))) (let__e1 A f Hf a))) + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (f (f a))) + (eq__trans__sym__inv__l A (f (f (f a))) (f (f a)) + (Hf (f (f a))))); +opaque symbol eq__refl__map__distr + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + f : Π + ( + __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) B + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f x))) + (f__equal + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f x x + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x))) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + f : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) B + ), + eq__refl + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f x))) + (eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x)); +symbol eq__trans__map__distr + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (z : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + f : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) B + ), + Π + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + Π + ( + e_27_ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y z) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f z))) + (f__equal + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f x z + (eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y z e e_27_)) + (eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y) (f z) + (f__equal + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f x y e) + (f__equal + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f y z e_27_))) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + (z : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + f : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + B + ), + λ + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + λ + ( + e_27_ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y z) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y + (λ + ( + y0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + e0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y y0) + ), + eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y0))) + (f__equal + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f x y0 + (eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y y0 e e0)) + (eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y) (f y0) + (f__equal + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f x y e) + (f__equal + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f y y0 e0))) + (eq__refl + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y))) + (eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y) (f y) + (f__equal + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f x y e) + (f__equal + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f y y + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y)))) + z e_27_; +symbol eq__sym__map__distr + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + f : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) B + ), + Π + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f y) (f x))) + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y) + (f__equal + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f x y e)) + (f__equal + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f y x + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y e))) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + f : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) B + ), + λ + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + y0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + e0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) + ), + eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f y0) (f x))) + (eq__sym + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f y0) + (f__equal + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f x y0 e0)) + (f__equal + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f y0 x + (eq__sym + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0 e0))) + (eq__refl + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + (f x) (f x))) + (f__equal + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I B) + f x x + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)))) + y e; +symbol eq__trans__sym__distr + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (z : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + Π + ( + e_27_ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y z) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + z x)) + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x z + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y z e e_27_)) + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + z y x + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y z e_27_) + (eq__sym + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y e))) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ (z : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + y0 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + λ + ( + e0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0 z) + (λ + ( + e_27_ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0 z) + ), + eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + z x)) + (eq__sym + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x z + (eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0 z e0 e_27_)) + (eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + z y0 x + (eq__sym + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0 z e_27_) + (eq__sym + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0 e0)))) + (λ + ( + e_27_ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x z) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + y0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + e0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) + ), + eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0 x)) + (eq__sym + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0 + (eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x y0 + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x) + e0)) + (eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0 x x + (eq__sym + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0 e0) + (eq__sym + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)))) + (eq__refl + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x)) + (eq__trans + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x x + (eq__sym + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + (eq__sym + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)))) + z e_27_) + y e; +opaque symbol eq__trans__rew__distr + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) + ), + Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (z : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + Π + ( + e_27_ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y z) + ), + Π + ( + k : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P z)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P k z + (eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y z e e_27_)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y P + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P k y e) + z e_27_)) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) + ), + λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + (z : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + y0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + e0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) + ), + cupicef.prod cupicef.prop + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.prop) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.prop)) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0 z) + (λ + ( + e_27_ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0 z) + ), + cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I (P x) + (λ + ( + k : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P z)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P k z + (eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0 z e0 e_27_)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + y0 P + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P k y0 e0) + z e_27_)))) + (λ + ( + e_27_ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x z) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + y0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + e0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) + ), + cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.prop) + cupicef.I (P x) + (λ + ( + k : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P y0)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P k y0 + (eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x y0 + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x) + e0)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P k x + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + y0 e0))) + (λ + ( + k : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) ( + P x) + ), + eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P k x + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + x + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x))) + z e_27_) + y e; +opaque symbol rew__const + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (P : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + Π + (k : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) P), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I P) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P) + k y e) + k) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (P : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + y0 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + λ + ( + e0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) + ), + cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.prop + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.prop) + cupicef.I P + (λ + ( + k : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) P + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I P) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P) + k y0 e0) + k)) + (λ + ( + k : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + P + ), + eq__refl + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I P) + k) + y e; +symbol subrelation + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + R : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π + ( + __0 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + B + ), cupicef.Univ cupicef.prop + ), + Π + ( + R_27_ : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π + ( + __0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) B + ), cupicef.Univ cupicef.prop + ), cupicef.Univ cupicef.prop + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + R : Π + ( + __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + Π + ( + __0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) B + ), cupicef.Univ cupicef.prop + ), + λ + ( + R_27_ : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π + ( + __0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) B + ), cupicef.Univ cupicef.prop + ), + cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop))) + cupicef.I A + (λ + (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I B + (λ + ( + y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + B + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (R x y) + (λ (__ : cupicef.Term cupicef.prop (R x y)), R_27_ x y))); +symbol unique + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ cupicef.prop + ), + Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ cupicef.prop + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + and (P x) + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x_27_ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (P x_27_) + (λ (__ : cupicef.Term cupicef.prop (P x_27_)), + eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_))); +symbol uniqueness + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ cupicef.prop + ), cupicef.Univ cupicef.prop + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop))) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop)))) + cupicef.I A + (λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop))) + cupicef.I A + (λ + (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.prod cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I (P x) + (λ (__ : cupicef.Term cupicef.prop (P x)), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (P y) + (λ (__0 : cupicef.Term cupicef.prop (P y)), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y)))); +opaque symbol unique__existence + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ cupicef.prop + ), + cupicef.Term cupicef.prop + (iff + (and + (ex A + (λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + P x)) + (uniqueness A P)) + (ex A + (unique A + (λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + P x)))) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (and + (ex A + (λ + (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + P x)) + (uniqueness A P)) + (λ + ( + __ : cupicef.Term cupicef.prop + (and + (ex A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x)) + (uniqueness A P)) + ), + ex A + (unique A + (λ + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x)))) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (ex A + (unique A + (λ + (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + P x))) + (λ + ( + __ : cupicef.Term cupicef.prop + (ex A + (unique A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x))) + ), + and + (ex A + (λ + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x)) + (uniqueness A P))) + (λ + ( + H : cupicef.Term cupicef.prop + (and + (ex A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x)) + (uniqueness A P)) + ), + match____and cupicef.prop + (ex A + (λ + (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x)) + (uniqueness A P) + (λ + ( + __ : cupicef.Term cupicef.prop + (and + (ex A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x)) + (uniqueness A P)) + ), + ex A + (unique A + (λ + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), P x))) + (λ + ( + H0 : cupicef.Term cupicef.prop + (ex A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x)) + ), + match____ex cupicef.prop A + (λ + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), P x) + (λ + ( + __ : cupicef.Term cupicef.prop + (ex A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x)) + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (uniqueness A P) + (λ (__0 : cupicef.Term cupicef.prop (uniqueness A P)), + ex A + (unique A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x)))) + (λ + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + λ (Hx : cupicef.Term cupicef.prop (P x)), + λ (Huni : cupicef.Term cupicef.prop (uniqueness A P)), + ex__intro A + (unique A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x0)) + x + (cupicef.cast cupicef.prop cupicef.prop + (and (P x) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x_27_ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I (P x_27_) + (λ (__ : cupicef.Term cupicef.prop (P x_27_)), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_)))) + (unique A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x0) + x) + cupicef.I + (conj (P x) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x_27_ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I (P x_27_) + (λ (__ : cupicef.Term cupicef.prop (P x_27_)), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_))) + Hx + (λ + ( + x_27_ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + λ (H1 : cupicef.Term cupicef.prop (P x_27_)), + Huni x x_27_ Hx H1)))) + H0) + H) + (λ + ( + H : cupicef.Term cupicef.prop + (ex A + (unique A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x))) + ), + match____ex cupicef.prop A + (unique A + (λ + (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x)) + (λ + ( + __ : cupicef.Term cupicef.prop + (ex A + (unique A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x))) + ), + and + (ex A + (λ + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), P x)) + (uniqueness A P)) + (λ + (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + H0 : cupicef.Term cupicef.prop + (unique A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x0) + x) + ), + match____and cupicef.prop + ((λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x0) + x) + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x_27_ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + ((λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x0) + x_27_) + (λ + ( + __ : cupicef.Term cupicef.prop + ((λ + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), P x0) + x_27_) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_))) + (λ + ( + __ : cupicef.Term cupicef.prop + (and (P x) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x_27_ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I (P x_27_) + (λ + (__ : cupicef.Term cupicef.prop (P x_27_)), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_)))) + ), + and + (ex A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x0)) + (uniqueness A P)) + (λ (Hx : cupicef.Term cupicef.prop (P x)), + λ + ( + Huni : Π + ( + x_27_ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + Π + (__ : cupicef.Term cupicef.prop (P x_27_)), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_) + ), + conj + (ex A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x0)) + (uniqueness A P) + (ex__intro A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x0) + x Hx) + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop)))) + cupicef.prop + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop)))) + cupicef.I A + (λ + ( + x_27_ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop))) + cupicef.I A + (λ + ( + x_27__27_ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.prod cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I (P x_27_) + (λ + ( + Hx_27_ : cupicef.Term cupicef.prop + (P x_27_) + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I (P x_27__27_) + (λ + ( + Hx_27__27_ : cupicef.Term cupicef.prop + (P x_27__27_) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x_27_ x_27__27_))))) + (uniqueness A P) cupicef.I + (λ + ( + x_27_ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + x_27__27_ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + λ + (Hx_27_ : cupicef.Term cupicef.prop (P x_27_)), + λ + ( + Hx_27__27_ : cupicef.Term cupicef.prop + (P x_27__27_) + ), + eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x_27_ x x_27__27_ + (eq__sym + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_ (Huni x_27_ Hx_27_)) + (Huni x_27__27_ Hx_27__27_)))) + H0) + H); +symbol let__H + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ cupicef.prop + ), + Π + ( + H : cupicef.Term cupicef.prop + (ex A + (unique A + (λ + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), P x))) + ), + Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + H0 : cupicef.Term cupicef.prop + (unique A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x0) + x) + ), + Π (Hp : cupicef.Term cupicef.prop (P x)), + Π + ( + Huniq : Π + ( + x_27_ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + Π (__ : cupicef.Term cupicef.prop (P x_27_)), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_) + ), + Π + ( + Q : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + Π + ( + H1 : cupicef.Term cupicef.prop + (ex A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), and (P x0) (Q x0))) + ), + Π + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π + (H2 : cupicef.Term cupicef.prop (and (P x0) (Q x0)) + ), + Π (HPx0 : cupicef.Term cupicef.prop (P x0)), + Π (HQx0 : cupicef.Term cupicef.prop (Q x0)), + Π + ( + x1 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + Π (HPx1 : cupicef.Term cupicef.prop (P x1)), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 x1) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + λ + ( + H : cupicef.Term cupicef.prop + (ex A + (unique A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x))) + ), + λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + H0 : cupicef.Term cupicef.prop + (unique A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x0) + x) + ), + λ (Hp : cupicef.Term cupicef.prop (P x)), + λ + ( + Huniq : Π + ( + x_27_ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + Π (__ : cupicef.Term cupicef.prop (P x_27_)), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_) + ), + λ + ( + Q : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + λ + ( + H1 : cupicef.Term cupicef.prop + (ex A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), and (P x0) (Q x0))) + ), + λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + H2 : cupicef.Term cupicef.prop + (and (P x0) (Q x0)) + ), + λ (HPx0 : cupicef.Term cupicef.prop (P x0)), + λ (HQx0 : cupicef.Term cupicef.prop (Q x0)), + λ + ( + x1 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + λ + (HPx1 : cupicef.Term cupicef.prop (P x1)), + eq__trans + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 x x1 + (eq__sym + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x0 (Huniq x0 HPx0)) + (Huniq x1 HPx1); +opaque symbol forall__exists__unique__domain__coincide + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ cupicef.prop + ), + Π + ( + __ : cupicef.Term cupicef.prop + (ex A + (unique A + (λ + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), P x))) + ), + Π + ( + Q : Π + ( + __0 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), cupicef.Univ cupicef.prop + ), + cupicef.Term cupicef.prop + (iff + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I ( + P x) (λ (__0 : cupicef.Term cupicef.prop (P x)), Q x))) + (ex A + (λ + (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), and (P x) (Q x)))) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + λ + ( + H : cupicef.Term cupicef.prop + (ex A + (unique A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x))) + ), + match____ex cupicef.prop A + (unique A + (λ + (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + P x)) + (λ + ( + __ : cupicef.Term cupicef.prop + (ex A + (unique A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x))) + ), + cupicef.prod + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.prop + (cupicef.{|rule|} + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I)) + (λ + ( + Q : Π + ( + __0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + iff + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (P x) (λ (__0 : cupicef.Term cupicef.prop (P x)), Q x))) + (ex A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), and (P x) (Q x))))) + (λ + (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + H0 : cupicef.Term cupicef.prop + (unique A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x0) + x) + ), + match____and cupicef.prop + ((λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x0) + x) + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x_27_ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + ((λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x0) + x_27_) + (λ + ( + __ : cupicef.Term cupicef.prop + ((λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), P x0) + x_27_) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_))) + (λ + ( + __ : cupicef.Term cupicef.prop + (and (P x) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x_27_ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I (P x_27_) + (λ (__ : cupicef.Term cupicef.prop (P x_27_)), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_)))) + ), + cupicef.prod + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.prop + (cupicef.{|rule|} + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I)) + (λ + ( + Q : Π + ( + __0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + iff + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I (P x0) + (λ (__0 : cupicef.Term cupicef.prop (P x0)), Q x0))) + (ex A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), and (P x0) (Q x0))))) + (λ (Hp : cupicef.Term cupicef.prop (P x)), + λ + ( + Huniq : Π + ( + x_27_ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + Π + (__ : cupicef.Term cupicef.prop (P x_27_)), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_) + ), + λ + ( + Q : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + conj + (cupicef.prod + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.prop + (cupicef.{|rule|} + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.prop) + cupicef.I + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I (P x0) + (λ (__ : cupicef.Term cupicef.prop (P x0)), + Q x0))) + (λ + ( + __ : Π + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + Π + (__ : cupicef.Term cupicef.prop (P x0)), + cupicef.Term cupicef.prop (Q x0) + ), + ex A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), and (P x0) (Q x0)))) + (cupicef.prod cupicef.prop + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop))) + cupicef.I + (ex A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), and (P x0) (Q x0))) + (λ + ( + __ : cupicef.Term cupicef.prop + (ex A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), and (P x0) (Q x0))) + ), + cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I (P x0) + (λ (__0 : cupicef.Term cupicef.prop (P x0)), + Q x0)))) + (λ + ( + H1 : Π + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + Π (__ : cupicef.Term cupicef.prop (P x0)), + cupicef.Term cupicef.prop (Q x0) + ), + ex__intro A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), and (P x0) (Q x0)) + x (conj (P x) (Q x) Hp (H1 x Hp))) + (λ + ( + H1 : cupicef.Term cupicef.prop + (ex A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), and (P x0) (Q x0))) + ), + match____ex cupicef.prop A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), and (P x0) (Q x0)) + (λ + ( + __ : cupicef.Term cupicef.prop + (ex A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), and (P x0) (Q x0))) + ), + cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I (P x0) + (λ + (__0 : cupicef.Term cupicef.prop (P x0)), + Q x0))) + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + H2 : cupicef.Term cupicef.prop + (and (P x0) (Q x0)) + ), + match____and cupicef.prop (P x0) (Q x0) + (λ + ( + __ : cupicef.Term cupicef.prop + (and (P x0) (Q x0)) + ), + cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + cupicef.prop)) + cupicef.I A + (λ + ( + x1 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop + cupicef.prop) + cupicef.I (P x1) + (λ + ( + __0 : cupicef.Term cupicef.prop + (P x1) + ), Q x1))) + (λ + (HPx0 : cupicef.Term cupicef.prop (P x0)), + λ + (HQx0 : cupicef.Term cupicef.prop (Q x0) + ), + λ + ( + x1 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + λ + ( + HPx1 : cupicef.Term cupicef.prop + (P x1) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 + (λ + ( + y : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + λ + ( + __ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x0 y) + ), + cupicef.prod cupicef.prop + cupicef.prop + (cupicef.{|rule|} cupicef.prop + cupicef.prop) + cupicef.I (P y) + (λ + ( + HPx2 : cupicef.Term + cupicef.prop ( + P y) + ), Q y)) + (λ + ( + HPx2 : cupicef.Term cupicef.prop + (P x0) + ), HQx0) + x1 + (let__H A P H x H0 Hp Huniq Q H1 x0 + H2 HPx0 HQx0 x1 HPx1) + HPx1) + H2) + H1)) + H0) + H; +symbol let__i + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ cupicef.prop + ), + Π + ( + H : Π + ( + Q : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + cupicef.Term cupicef.prop + (iff + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (P x) (λ (__ : cupicef.Term cupicef.prop (P x)), Q x))) + (ex A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), and (P x) (Q x)))) + ), + cupicef.Term cupicef.prop + (iff + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I ( + P x) (λ (__ : cupicef.Term cupicef.prop (P x)), P x))) + (ex A + (λ + (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + and (P x) (P x)))) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + λ + ( + H : Π + ( + Q : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + cupicef.Term cupicef.prop + (iff + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (P x) (λ (__ : cupicef.Term cupicef.prop (P x)), Q x))) + (ex A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), and (P x) (Q x)))) + ), H P; +symbol let__i0 + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ cupicef.prop + ), + Π + ( + H : Π + ( + Q : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + cupicef.Term cupicef.prop + (iff + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (P x) (λ (__ : cupicef.Term cupicef.prop (P x)), Q x))) + (ex A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), and (P x) (Q x)))) + ), + Π + ( + H0 : Π + ( + __ : Π + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π (__ : cupicef.Term cupicef.prop (P x)), + cupicef.Term cupicef.prop (P x) + ), + cupicef.Term cupicef.prop + (ex A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), and (P x) (P x))) + ), + Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (H1 : cupicef.Term cupicef.prop (and (P x) (P x))), + Π (Hx : cupicef.Term cupicef.prop (P x)), + Π (__tmp : cupicef.Term cupicef.prop (P x)), + Π + ( + __tmp0 : Π + ( + __ : cupicef.Term cupicef.prop + (ex A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), and (P x0) (P x0))) + ), + Π + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + Π + (__0 : cupicef.Term cupicef.prop (P x0)), + cupicef.Term cupicef.prop (P x0) + ), + cupicef.Term cupicef.prop + (iff + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I (P x0) + (λ (__ : cupicef.Term cupicef.prop (P x0)), + (λ + ( + x_27_ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_) + x0))) + (ex A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + and (P x0) + ((λ + ( + x_27_ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_) + x0)))) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + λ + ( + H : Π + ( + Q : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + cupicef.Term cupicef.prop + (iff + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (P x) (λ (__ : cupicef.Term cupicef.prop (P x)), Q x))) + (ex A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), and (P x) (Q x)))) + ), + λ + ( + H0 : Π + ( + __ : Π + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π (__ : cupicef.Term cupicef.prop (P x)), + cupicef.Term cupicef.prop (P x) + ), + cupicef.Term cupicef.prop + (ex A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), and (P x) (P x))) + ), + λ + (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ (H1 : cupicef.Term cupicef.prop (and (P x) (P x))), + λ (Hx : cupicef.Term cupicef.prop (P x)), + λ (__tmp : cupicef.Term cupicef.prop (P x)), + λ + ( + __tmp0 : Π + ( + __ : cupicef.Term cupicef.prop + (ex A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), and (P x0) (P x0))) + ), + Π + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + Π + ( + __0 : cupicef.Term cupicef.prop (P x0) + ), cupicef.Term cupicef.prop (P x0) + ), + H + (λ + ( + x_27_ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_); +opaque symbol forall__exists__coincide__unique__domain + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ cupicef.prop + ), + Π + ( + __ : Π + ( + Q : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + cupicef.Term cupicef.prop + (iff + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (P x) (λ (__ : cupicef.Term cupicef.prop (P x)), Q x))) + (ex A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), and (P x) (Q x)))) + ), + cupicef.Term cupicef.prop + (ex A + (unique A + (λ + (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + P x))) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + λ + ( + H : Π + ( + Q : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + cupicef.Term cupicef.prop + (iff + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (P x) (λ (__ : cupicef.Term cupicef.prop (P x)), Q x))) + (ex A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), and (P x) (Q x)))) + ), + match____and cupicef.prop + (cupicef.prod + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.prop + (cupicef.{|rule|} + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I ( + P x) (λ (__ : cupicef.Term cupicef.prop (P x)), P x))) + (λ + ( + __ : Π + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π (__ : cupicef.Term cupicef.prop (P x)), + cupicef.Term cupicef.prop (P x) + ), + ex A + (λ + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), and (P x) (P x)))) + (cupicef.prod cupicef.prop + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop))) + cupicef.I + (ex A + (λ + (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), and (P x) (P x))) + (λ + ( + __ : cupicef.Term cupicef.prop + (ex A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), and (P x) (P x))) + ), + cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (P x) (λ (__0 : cupicef.Term cupicef.prop (P x)), P x)))) + (λ + ( + __ : cupicef.Term cupicef.prop + (and + (cupicef.prod + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.prop + (cupicef.{|rule|} + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.prop) + cupicef.I + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I (P x) + (λ (__ : cupicef.Term cupicef.prop (P x)), P x))) + (λ + ( + __ : Π + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + Π (__ : cupicef.Term cupicef.prop (P x)), + cupicef.Term cupicef.prop (P x) + ), + ex A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), and (P x) (P x)))) + (cupicef.prod cupicef.prop + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop))) + cupicef.I + (ex A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), and (P x) (P x))) + (λ + ( + __ : cupicef.Term cupicef.prop + (ex A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), and (P x) (P x))) + ), + cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I (P x) + (λ (__0 : cupicef.Term cupicef.prop (P x)), P x))))) + ), + ex A + (unique A + (λ + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), P x))) + (λ + ( + H0 : Π + ( + __ : Π + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π (__ : cupicef.Term cupicef.prop (P x)), + cupicef.Term cupicef.prop (P x) + ), + cupicef.Term cupicef.prop + (ex A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), and (P x) (P x))) + ), + match____ex cupicef.prop A + (λ + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), and (P x) (P x)) + (λ + ( + __ : cupicef.Term cupicef.prop + (ex A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), and (P x) (P x))) + ), + cupicef.prod + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.{|rule|} + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop))) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop + (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop))) + cupicef.I + (ex A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), and (P x) (P x))) + (λ + ( + __0 : cupicef.Term cupicef.prop + (ex A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), and (P x) (P x))) + ), + cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I (P x) + (λ (__1 : cupicef.Term cupicef.prop (P x)), P x)))) + (λ + ( + __0 : Π + ( + __0 : cupicef.Term cupicef.prop + (ex A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), and (P x) (P x))) + ), + Π + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π (__1 : cupicef.Term cupicef.prop (P x)), + cupicef.Term cupicef.prop (P x) + ), + ex A + (unique A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x)))) + (λ + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ (H1 : cupicef.Term cupicef.prop (and (P x) (P x))), + match____and cupicef.prop (P x) (P x) + (λ (__ : cupicef.Term cupicef.prop (and (P x) (P x))), + cupicef.prod + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop))) + cupicef.prop + (cupicef.{|rule|} + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop))) + cupicef.prop) + cupicef.I + (cupicef.prod cupicef.prop + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop))) + cupicef.I + (ex A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), and (P x0) (P x0))) + (λ + ( + __0 : cupicef.Term cupicef.prop + (ex A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), and (P x0) (P x0))) + ), + cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I (P x0) + (λ (__1 : cupicef.Term cupicef.prop (P x0)), + P x0)))) + (λ + ( + __0 : Π + ( + __0 : cupicef.Term cupicef.prop + (ex A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), and (P x0) (P x0))) + ), + Π + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + Π + (__1 : cupicef.Term cupicef.prop (P x0) + ), cupicef.Term cupicef.prop (P x0) + ), + ex A + (unique A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x0)))) + (λ (Hx : cupicef.Term cupicef.prop (P x)), + λ (__tmp : cupicef.Term cupicef.prop (P x)), + λ + ( + __tmp0 : Π + ( + __ : cupicef.Term cupicef.prop + (ex A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), and (P x0) (P x0))) + ), + Π + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + Π + ( + __0 : cupicef.Term cupicef.prop + (P x0) + ), cupicef.Term cupicef.prop (P x0) + ), + ex__intro A + (unique A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), P x0)) + x + (conj (P x) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x_27_ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I (P x_27_) + (λ + (__ : cupicef.Term cupicef.prop (P x_27_)), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_))) + Hx + (match____and cupicef.prop + (cupicef.prod + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.prop + (cupicef.{|rule|} + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.prop) + cupicef.I + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I (P x0) + (λ + (__ : cupicef.Term cupicef.prop (P x0)), + (λ + ( + x_27_ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_) + x0))) + (λ + ( + __ : Π + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + Π + ( + __ : cupicef.Term cupicef.prop + (P x0) + ), + cupicef.Term cupicef.prop + ((λ + ( + x_27_ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_) + x0) + ), + ex A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + and (P x0) + ((λ + ( + x_27_ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_) + x0)))) + (cupicef.prod cupicef.prop + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop))) + cupicef.I + (ex A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + and (P x0) + ((λ + ( + x_27_ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_) + x0))) + (λ + ( + __ : cupicef.Term cupicef.prop + (ex A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + and (P x0) + ((λ + ( + x_27_ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_) + x0))) + ), + cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop + cupicef.prop) + cupicef.I (P x0) + (λ + ( + __0 : cupicef.Term cupicef.prop (P x0) + ), + (λ + ( + x_27_ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_) + x0)))) + (λ + ( + __ : cupicef.Term cupicef.prop + (and + (cupicef.prod + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + cupicef.prop)) + cupicef.prop + (cupicef.{|rule|} + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + cupicef.prop)) + cupicef.prop) + cupicef.I + (cupicef.prod + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + cupicef.prop)) + cupicef.I A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.prod cupicef.prop + cupicef.prop + (cupicef.{|rule|} cupicef.prop + cupicef.prop) + cupicef.I (P x0) + (λ + ( + __ : cupicef.Term cupicef.prop + (P x0) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x0))) + (λ + ( + __ : Π + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + Π + ( + __ : cupicef.Term + cupicef.prop ( + P x0) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x0) + ), + ex A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + and (P x0) + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x0)))) + (cupicef.prod cupicef.prop + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + cupicef.prop)) + (cupicef.{|rule|} cupicef.prop + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + cupicef.prop))) + cupicef.I + (ex A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + and (P x0) + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x0))) + (λ + ( + __ : cupicef.Term cupicef.prop + (ex A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + and (P x0) + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x0))) + ), + cupicef.prod + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop + cupicef.prop)) + cupicef.I A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.prod cupicef.prop + cupicef.prop + (cupicef.{|rule|} cupicef.prop + cupicef.prop) + cupicef.I (P x0) + (λ + ( + __0 : cupicef.Term + cupicef.prop (P x0) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x0))))) + ), + cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} cupicef.prop cupicef.prop)) + cupicef.I A + (λ + ( + x_27_ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop + cupicef.prop) + cupicef.I (P x_27_) + (λ + ( + __0 : cupicef.Term cupicef.prop + (P x_27_) + ), + eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x_27_))) + (λ + ( + __tmp1 : Π + ( + __ : Π + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + Π + ( + __ : cupicef.Term + cupicef.prop ( + P x0) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A) + x x0) + ), + cupicef.Term cupicef.prop + (ex A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + and (P x0) + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x0))) + ), + λ + ( + Huniq : Π + ( + __ : cupicef.Term cupicef.prop + (ex A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s + cupicef.z)) + A + ), + and (P x0) + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s + cupicef.z)))) + cupicef.I) + cupicef.I A) + x x0))) + ), + Π + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + Π + ( + __0 : cupicef.Term + cupicef.prop (P x0) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x0) + ), + Huniq + (ex__intro A + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + and (P x0) + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x0)) + x + (conj (P x) + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x x) + Hx + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)))) + (let__i0 A P H H0 x H1 Hx __tmp __tmp0)))) + H1) + (H0 + (λ + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), λ (H1 : cupicef.Term cupicef.prop (P x)), H1))) + (let__i A P H); +symbol inhabited + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + cupicef.Univ cupicef.prop; +symbol inhabits + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Term cupicef.prop (inhabited A); +rule cupicef.code_guarded? _ (inhabits _ _) ↪ cupicef.guarded; +symbol match____inhabited + : Π (s : cupicef.Sort), + Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (P : cupicef.Term cupicef.prop (inhabited A) → cupicef.Univ s), + Π + ( + case__inhabits : Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), cupicef.Term s (P (inhabits A __)) + ), + Π (x : cupicef.Term cupicef.prop (inhabited A)), + cupicef.Term s (P x); +rule match____inhabited $s $A $P $case__inhabits (inhabits $A $__) ↪ +$case__inhabits $__; +rule match____inhabited $s $A + (λ x, cupicef.uncode $_.[] (cupicef.code (cupicef.cU $s'.[]) $P.[x])) ↪ +match____inhabited $s' $A (λ x, $P.[x]); +symbol inhabited__ind + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (P : cupicef.Univ cupicef.prop), + Π + ( + f : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Term cupicef.prop P + ), + Π (i : cupicef.Term cupicef.prop (inhabited A)), + cupicef.Term cupicef.prop P + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (P : cupicef.Univ cupicef.prop), + λ + ( + f : Π + ( + __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), cupicef.Term cupicef.prop P + ), + λ (i : cupicef.Term cupicef.prop (inhabited A)), + match____inhabited cupicef.prop A + (λ (__ : cupicef.Term cupicef.prop (inhabited A)), P) f i; +opaque symbol exists__inhabited + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ cupicef.prop + ), + Π + ( + __ : cupicef.Term cupicef.prop + (ex A + (λ + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), P x)) + ), cupicef.Term cupicef.prop (inhabited A) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + λ + ( + H : cupicef.Term cupicef.prop + (ex A + (λ + ( + x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), P x)) + ), + match____ex cupicef.prop A + (λ + (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + P x) + (λ + ( + __ : cupicef.Term cupicef.prop + (ex A + (λ + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), P x)) + ), inhabited A) + (λ + (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ (H0 : cupicef.Term cupicef.prop (P x)), inhabits A x) + H; +opaque symbol inhabited__covariant + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + __ : Π + ( + __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) B + ), + Π (__0 : cupicef.Term cupicef.prop (inhabited A)), + cupicef.Term cupicef.prop (inhabited B) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + f : Π + ( + __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) B + ), + λ (H : cupicef.Term cupicef.prop (inhabited A)), + match____inhabited cupicef.prop A + (λ (__ : cupicef.Term cupicef.prop (inhabited A)), inhabited B) + (λ + (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), inhabits B (f x)) + H; +opaque symbol eq__stepl + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (z : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + __ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + Π + ( + __0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x z) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + z y) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ (z : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + H1 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + λ + ( + H2 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x z) + ), + eq__ind + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + z0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + eq + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + z0 y) + H1 z H2; +opaque symbol iff__stepl + : Π (A : cupicef.Univ cupicef.prop), + Π (B : cupicef.Univ cupicef.prop), + Π (C : cupicef.Univ cupicef.prop), + Π (__ : cupicef.Term cupicef.prop (iff A B)), + Π (__0 : cupicef.Term cupicef.prop (iff A C)), + cupicef.Term cupicef.prop (iff C B) + ≔ λ (A : cupicef.Univ cupicef.prop), + λ (B : cupicef.Univ cupicef.prop), + λ (C : cupicef.Univ cupicef.prop), + λ (H : cupicef.Term cupicef.prop (iff A B)), + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), A)) + (λ + ( + __ : cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), A))) + ), + cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + (iff A C) + (λ (__0 : cupicef.Term cupicef.prop (iff A C)), iff C B)) + (λ + ( + H0 : Π (__ : cupicef.Term cupicef.prop A), + cupicef.Term cupicef.prop B + ), + λ + ( + H1 : Π (__ : cupicef.Term cupicef.prop B), + cupicef.Term cupicef.prop A + ), + λ (H2 : cupicef.Term cupicef.prop (iff A C)), + match____and cupicef.prop + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + A (λ (__ : cupicef.Term cupicef.prop A), C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I + C (λ (__ : cupicef.Term cupicef.prop C), A)) + (λ + ( + __ : cupicef.Term cupicef.prop + (and + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I A + (λ (__ : cupicef.Term cupicef.prop A), C)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I C + (λ (__ : cupicef.Term cupicef.prop C), A))) + ), iff C B) + (λ + ( + H3 : Π (__ : cupicef.Term cupicef.prop A), + cupicef.Term cupicef.prop C + ), + λ + ( + H4 : Π (__ : cupicef.Term cupicef.prop C), + cupicef.Term cupicef.prop A + ), + conj + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I C + (λ (__ : cupicef.Term cupicef.prop C), B)) + (cupicef.prod cupicef.prop cupicef.prop + (cupicef.{|rule|} cupicef.prop cupicef.prop) + cupicef.I B + (λ (__ : cupicef.Term cupicef.prop B), C)) + (λ (H5 : cupicef.Term cupicef.prop C), + H0 (H1 (H0 (H4 H5)))) + (λ (H5 : cupicef.Term cupicef.prop B), H3 (H1 H5))) + H2) + H; +opaque symbol eq__ex__uncurried + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ cupicef.prop + ), + Π (u1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (v1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (u2 : cupicef.Term cupicef.prop (P u1)), + Π (v2 : cupicef.Term cupicef.prop (P v1)), + Π + ( + pq : cupicef.Term cupicef.prop + (ex + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1)) + (λ + ( + p : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + ), + eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2)) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex A P)) + (ex__intro A P u1 u2) (ex__intro A P v1 v2)) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + λ (u1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + (v1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ (u2 : cupicef.Term cupicef.prop (P u1)), + λ (v2 : cupicef.Term cupicef.prop (P v1)), + λ + ( + pq : cupicef.Term cupicef.prop + (ex + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1)) + (λ + ( + p : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + ), + eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2)) + ), + match____ex cupicef.prop + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1)) + (λ + ( + p : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + ), + eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) + (λ + ( + __ : cupicef.Term cupicef.prop + (ex + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1)) + (λ + ( + p : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + ), + eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2)) + ), + eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex A P)) + (ex__intro A P u1 u2) (ex__intro A P v1 v2)) + (λ + ( + p : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + ), + λ + ( + q : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + (λ (y : cupicef.Term cupicef.prop (P v1)), + λ + ( + __ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + y) + ), + eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex A P)) + (ex__intro A P u1 u2) (ex__intro A P v1 y)) + (cupicef.cast cupicef.prop cupicef.prop + (eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex A P)) + (ex__intro A P u1 u2) + (ex__intro A P v1 + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p))) + (eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex A P)) + (ex__intro A P u1 u2) + (ex__intro A P v1 + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p))) + cupicef.I + (match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (λ + ( + y : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 y) + ), + eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex A P)) + (ex__intro A P u1 u2) + (ex__intro A P y + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 y e))) + (eq__refl + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex A P)) + (ex__intro A P u1 + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 u1 + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1)))) + v1 p)) + v2 q) + pq; +symbol eq__ex + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ cupicef.prop + ), + Π (u1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (v1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (u2 : cupicef.Term cupicef.prop (P u1)), + Π (v2 : cupicef.Term cupicef.prop (P v1)), + Π + ( + p : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + ), + Π + ( + q : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex A P)) + (ex__intro A P u1 u2) (ex__intro A P v1 v2)) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + λ (u1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + (v1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ (u2 : cupicef.Term cupicef.prop (P u1)), + λ (v2 : cupicef.Term cupicef.prop (P v1)), + λ + ( + p : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + ), + λ + ( + q : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) + ), + eq__ex__uncurried A P u1 v1 u2 v2 + (ex__intro + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1)) + (λ + ( + p0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + ), + eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p0) + v2) + p q); +symbol eq__ex__hprop + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ cupicef.prop + ), + Π + ( + P__hprop : Π + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π (p : cupicef.Term cupicef.prop (P x)), + Π (q : cupicef.Term cupicef.prop (P x)), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + p q) + ), + Π (u1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + (v1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (u2 : cupicef.Term cupicef.prop (P u1)), + Π (v2 : cupicef.Term cupicef.prop (P v1)), + Π + ( + p : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex A P)) + (ex__intro A P u1 u2) (ex__intro A P v1 v2)) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + λ + ( + P__hprop : Π + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π (p : cupicef.Term cupicef.prop (P x)), + Π (q : cupicef.Term cupicef.prop (P x)), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + p q) + ), + λ + (u1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + (v1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ (u2 : cupicef.Term cupicef.prop (P u1)), + λ (v2 : cupicef.Term cupicef.prop (P v1)), + λ + ( + p : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + ), + eq__ex A P u1 v1 u2 v2 p + (P__hprop v1 + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2); +opaque symbol rew__ex + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) + ), + Π + ( + Q : Π + ( + a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) (P a) + ), cupicef.Univ cupicef.prop + ), + Π + ( + u : cupicef.Term cupicef.prop + (ex (P x) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) (P x) + ), Q x p)) + ), + Π + (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + H : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (ex (P y) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) ( + P y) + ), Q y p))) + (eq__rect + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + a : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (λ + ( + a : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + ex (P a) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) ( + P a) + ), Q a p))) + u y H) + (match____ex cupicef.prop (P x) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) ( + P x) + ), Q x p) + (λ + ( + u0 : cupicef.Term cupicef.prop + (ex (P x) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), Q x p)) + ), + ex (P y) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) ( + P y) + ), Q y p)) + (λ + ( + u1 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) ( + P x) + ), + λ (u2 : cupicef.Term cupicef.prop (Q x u1)), + ex__intro (P y) (Q y) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P u1 y H) + (match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + a : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + x0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x a) + ), + Q a + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P u1 a x0)) + u2 y H)) + u)) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + P : Π + ( + __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) + ), + λ + ( + Q : Π + ( + a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) ( + P a) + ), cupicef.Univ cupicef.prop + ), + λ + ( + u : cupicef.Term cupicef.prop + (ex (P x) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) ( + P x) + ), Q x p)) + ), + λ + ( + y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + H : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + y0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) + ), + eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (ex (P y0) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P y0) + ), Q y0 p))) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + a : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (λ + ( + a : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + ex (P a) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P a) + ), Q a p))) + u y0 e) + (match____ex cupicef.prop (P x) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), Q x p) + (λ + ( + u0 : cupicef.Term cupicef.prop + (ex (P x) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (P x) + ), Q x p)) + ), + ex (P y0) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P y0) + ), Q y0 p)) + (λ + ( + u1 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), + λ (u2 : cupicef.Term cupicef.prop (Q x u1)), + ex__intro (P y0) (Q y0) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P u1 y0 e) + (match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + a : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + λ + ( + x0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x a) + ), + Q a + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P u1 a x0)) + u2 y0 e)) + u)) + (match____ex cupicef.prop (P x) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) ( + P x) + ), Q x p) + (λ + ( + e : cupicef.Term cupicef.prop + (ex (P x) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), Q x p)) + ), + eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (ex (P x) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), Q x p))) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + a : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (λ + ( + a : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + ex (P a) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P a) + ), Q a p))) + e x + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + (match____ex cupicef.prop (P x) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), Q x p) + (λ + ( + u0 : cupicef.Term cupicef.prop + (ex (P x) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (P x) + ), Q x p)) + ), + ex (P x) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), Q x p)) + (λ + ( + u1 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), + λ (u2 : cupicef.Term cupicef.prop (Q x u1)), + ex__intro (P x) (Q x) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P u1 x + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + u2) + e)) + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) ( + P x) + ), + λ (q : cupicef.Term cupicef.prop (Q x x0)), + eq__refl + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (ex (P x) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), Q x p))) + (ex__intro (P x) (Q x) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P x0 x + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + q)) + u) + y H; +opaque symbol eq__ex2__uncurried + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ cupicef.prop + ), + Π + ( + Q : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + Π (u1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + (v1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (u2 : cupicef.Term cupicef.prop (P u1)), + Π (v2 : cupicef.Term cupicef.prop (P v1)), + Π (u3 : cupicef.Term cupicef.prop (Q u1)), + Π (v3 : cupicef.Term cupicef.prop (Q v1)), + Π + ( + pq : cupicef.Term cupicef.prop + (ex2 + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1)) + (λ + ( + p : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + ), + eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) + (λ + ( + p : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + ), + eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + v3)) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex2 A P Q)) + (ex__intro2 A P Q u1 u2 u3) + (ex__intro2 A P Q v1 v2 v3)) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + λ + ( + Q : Π + ( + __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), cupicef.Univ cupicef.prop + ), + λ + (u1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + (v1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ (u2 : cupicef.Term cupicef.prop (P u1)), + λ (v2 : cupicef.Term cupicef.prop (P v1)), + λ (u3 : cupicef.Term cupicef.prop (Q u1)), + λ (v3 : cupicef.Term cupicef.prop (Q v1)), + λ + ( + pq : cupicef.Term cupicef.prop + (ex2 + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1)) + (λ + ( + p : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + ), + eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) + (λ + ( + p : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + ), + eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + v3)) + ), + match____ex2 cupicef.prop + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1)) + (λ + ( + p : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + ), + eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) + (λ + ( + p : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + ), + eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + v3) + (λ + ( + __ : cupicef.Term cupicef.prop + (ex2 + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1)) + (λ + ( + p : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + ), + eq + (cupicef.cast + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) + (λ + ( + p : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + ), + eq + (cupicef.cast + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + v3)) + ), + eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex2 A P Q)) + (ex__intro2 A P Q u1 u2 u3) + (ex__intro2 A P Q v1 v2 v3)) + (λ + ( + p : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + ), + λ + ( + q : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) + ), + λ + ( + r : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + v3) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + (λ (y : cupicef.Term cupicef.prop (Q v1)), + λ + ( + __ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + y) + ), + eq + (cupicef.cast + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex2 A P Q)) + (ex__intro2 A P Q u1 u2 u3) + (ex__intro2 A P Q v1 v2 y)) + (match____eq cupicef.prop + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + (λ (y : cupicef.Term cupicef.prop (P v1)), + λ + ( + __ : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + y) + ), + eq + (cupicef.cast + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex2 A P Q)) + (ex__intro2 A P Q u1 u2 u3) + (ex__intro2 A P Q v1 y + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __0 : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p))) + (match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (λ + ( + y : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + λ + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 y) + ), + eq + (cupicef.cast + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex2 A P Q)) + (ex__intro2 A P Q u1 u2 u3) + (ex__intro2 A P Q y + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 y e) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 y e))) + (cupicef.cast cupicef.prop cupicef.prop + (eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex2 A P Q)) + (ex__intro2 A P Q u1 u2 u3) + (ex__intro2 A P Q u1 u2 u3)) + (eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex2 A P Q)) + (ex__intro2 A P Q u1 u2 u3) + (ex__intro2 A P Q u1 + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 u1 + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 u1 + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1)))) + cupicef.I + (eq__refl + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex2 A P Q)) + (ex__intro2 A P Q u1 u2 u3))) + v1 p) + v2 q) + v3 r) + pq; +symbol eq__ex2 + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ cupicef.prop + ), + Π + ( + Q : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + Π (u1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + (v1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π (u2 : cupicef.Term cupicef.prop (P u1)), + Π (v2 : cupicef.Term cupicef.prop (P v1)), + Π (u3 : cupicef.Term cupicef.prop (Q u1)), + Π (v3 : cupicef.Term cupicef.prop (Q v1)), + Π + ( + p : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + ), + Π + ( + q : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) + ), + Π + ( + r : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + v3) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex2 A P Q)) + (ex__intro2 A P Q u1 u2 u3) + (ex__intro2 A P Q v1 v2 v3)) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + λ + ( + Q : Π + ( + __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), cupicef.Univ cupicef.prop + ), + λ + (u1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + (v1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ (u2 : cupicef.Term cupicef.prop (P u1)), + λ (v2 : cupicef.Term cupicef.prop (P v1)), + λ (u3 : cupicef.Term cupicef.prop (Q u1)), + λ (v3 : cupicef.Term cupicef.prop (Q v1)), + λ + ( + p : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + ), + λ + ( + q : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) + ), + λ + ( + r : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + v3) + ), + eq__ex2__uncurried A P Q u1 v1 u2 v2 u3 v3 + (ex__intro2 + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + cupicef.I + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1)) + (λ + ( + p0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + ), + eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p0) + v2) + (λ + ( + p0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + ), + eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q v1)) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} + (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p0) + v3) + p q r); +symbol eq__ex2__hprop + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + cupicef.Univ cupicef.prop + ), + Π + ( + Q : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + Π + ( + P__hprop : Π + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π (p : cupicef.Term cupicef.prop (P x)), + Π (q : cupicef.Term cupicef.prop (P x)), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + p q) + ), + Π + ( + Q__hprop : Π + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π (p : cupicef.Term cupicef.prop (Q x)), + Π (q : cupicef.Term cupicef.prop (Q x)), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q x)) + p q) + ), + Π + (u1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π + ( + v1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + Π (u2 : cupicef.Term cupicef.prop (P u1)), + Π (v2 : cupicef.Term cupicef.prop (P v1)), + Π (u3 : cupicef.Term cupicef.prop (Q u1)), + Π (v3 : cupicef.Term cupicef.prop (Q v1)), + Π + ( + p : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (ex2 A P Q)) + (ex__intro2 A P Q u1 u2 u3) + (ex__intro2 A P Q v1 v2 v3)) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ cupicef.prop + ), + λ + ( + Q : Π + ( + __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), cupicef.Univ cupicef.prop + ), + λ + ( + P__hprop : Π + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π (p : cupicef.Term cupicef.prop (P x)), + Π (q : cupicef.Term cupicef.prop (P x)), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (P x)) + p q) + ), + λ + ( + Q__hprop : Π + ( + x : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π (p : cupicef.Term cupicef.prop (Q x)), + Π (q : cupicef.Term cupicef.prop (Q x)), + cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I (Q x)) + p q) + ), + λ + ( + u1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + λ + ( + v1 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ (u2 : cupicef.Term cupicef.prop (P u1)), + λ (v2 : cupicef.Term cupicef.prop (P v1)), + λ (u3 : cupicef.Term cupicef.prop (Q u1)), + λ (v3 : cupicef.Term cupicef.prop (Q v1)), + λ + ( + p : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 v1) + ), + eq__ex2 A P Q u1 v1 u2 v2 u3 v3 p + (P__hprop v1 + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I P) + u2 v1 p) + v2) + (Q__hprop v1 + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + u1 + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I Q) + u3 v1 p) + v3); +opaque symbol rew__ex2 + : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + Π + ( + P : Π + (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) + ), + Π + ( + Q : Π + ( + a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) (P a) + ), cupicef.Univ cupicef.prop + ), + Π + ( + R : Π + ( + a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) ( + P a) + ), cupicef.Univ cupicef.prop + ), + Π + ( + u : cupicef.Term cupicef.prop + (ex2 (P x) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) ( + P x) + ), Q x p) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) ( + P x) + ), R x p)) + ), + Π + ( + y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π + ( + H : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + cupicef.Term cupicef.prop + (eq + (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (ex2 (P y) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) ( + P y) + ), Q y p) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) ( + P y) + ), R y p))) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + a : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I)) + (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (λ + ( + a : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + ex2 (P a) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P a) + ), Q a p) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P a) + ), R a p))) + u y H) + (match____ex2 cupicef.prop (P x) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) ( + P x) + ), Q x p) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) ( + P x) + ), R x p) + (λ + ( + u0 : cupicef.Term cupicef.prop + (ex2 (P x) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), Q x p) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), R x p)) + ), + ex2 (P y) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P y) + ), Q y p) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P y) + ), R y p)) + (λ + ( + u1 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) ( + P x) + ), + λ (u2 : cupicef.Term cupicef.prop (Q x u1)), + λ (u3 : cupicef.Term cupicef.prop (R x u1)), + ex__intro2 (P y) (Q y) (R y) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P u1 y H) + (match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + a : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + λ + ( + x0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x a) + ), + Q a + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P u1 a x0)) + u2 y H) + (match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + a : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + λ + ( + x0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x a) + ), + R a + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P u1 a x0)) + u3 y H)) + u)) + ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), + λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), + λ + ( + P : Π + ( + __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) + ), + λ + ( + Q : Π + ( + a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) ( + P a) + ), cupicef.Univ cupicef.prop + ), + λ + ( + R : Π + ( + a : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + Π + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) ( + P a) + ), cupicef.Univ cupicef.prop + ), + λ + ( + u : cupicef.Term cupicef.prop + (ex2 (P x) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) ( + P x) + ), Q x p) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) ( + P x) + ), R x p)) + ), + λ + ( + y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + λ + ( + H : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y) + ), + match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + y0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + λ + ( + e : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x y0) + ), + eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (ex2 (P y0) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P y0) + ), Q y0 p) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P y0) + ), R y0 p))) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + a : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (λ + ( + a : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + ex2 (P a) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (P a) + ), Q a p) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (P a) + ), R a p))) + u y0 e) + (match____ex2 cupicef.prop (P x) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), Q x p) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), R x p) + (λ + ( + u0 : cupicef.Term cupicef.prop + (ex2 (P x) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (P x) + ), Q x p) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (P x) + ), R x p)) + ), + ex2 (P y0) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P y0) + ), Q y0 p) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P y0) + ), R y0 p)) + (λ + ( + u1 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), + λ (u2 : cupicef.Term cupicef.prop (Q x u1)), + λ + (u3 : cupicef.Term cupicef.prop (R x u1)), + ex__intro2 (P y0) (Q y0) (R y0) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P u1 y0 e) + (match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + a : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + λ + ( + x0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x a) + ), + Q a + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P u1 a x0)) + u2 y0 e) + (match____eq cupicef.prop + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (λ + ( + a : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + A + ), + λ + ( + x0 : cupicef.Term cupicef.prop + (eq + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x a) + ), + R a + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s + (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P u1 a x0)) + u3 y0 e)) + u)) + (match____ex2 cupicef.prop (P x) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) ( + P x) + ), Q x p) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) ( + P x) + ), R x p) + (λ + ( + e : cupicef.Term cupicef.prop + (ex2 (P x) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), Q x p) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), R x p)) + ), + eq + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (ex2 (P x) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), Q x p) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), R x p))) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x + (cupicef.cast + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I A + (λ + ( + a : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I)) + (cupicef.prod + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.{|rule|} + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z)))) + cupicef.I A + (λ + ( + __ : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + A + ), + cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I)) + cupicef.I + (λ + ( + a : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) A + ), + ex2 (P a) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P a) + ), Q a p) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P a) + ), R a p))) + e x + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + (match____ex2 cupicef.prop (P x) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), Q x p) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), R x p) + (λ + ( + u0 : cupicef.Term cupicef.prop + (ex2 (P x) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (P x) + ), Q x p) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (P x) + ), R x p)) + ), + ex2 (P x) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), Q x p) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), R x p)) + (λ + ( + u1 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) + (P x) + ), + λ (u2 : cupicef.Term cupicef.prop (Q x u1)), + λ (u3 : cupicef.Term cupicef.prop (R x u1)), + ex__intro2 (P x) (Q x) (R x) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P u1 x + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + u2 u3) + e)) + (λ + ( + x0 : cupicef.Term + (cupicef.{|type|} (cupicef.s cupicef.z)) ( + P x) + ), + λ (q : cupicef.Term cupicef.prop (Q x x0)), + λ (r : cupicef.Term cupicef.prop (R x x0)), + eq__refl + (cupicef.cast + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ cupicef.prop + (cupicef.{|type|} (cupicef.s cupicef.z)) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I + (ex2 (P x) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (P x) + ), Q x p) + (λ + ( + p : cupicef.Term + (cupicef.{|type|} + (cupicef.s cupicef.z)) + (P x) + ), R x p))) + (ex__intro2 (P x) (Q x) (R x) + (eq__rect + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x P x0 x + (eq__refl + (cupicef.cast + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + (cupicef.univ + (cupicef.{|type|} (cupicef.s cupicef.z)) + (cupicef.axiom + (cupicef.{|type|} (cupicef.s cupicef.z))) + cupicef.I) + (cupicef.univ + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z))) + (cupicef.axiom + (cupicef.{|type|} + (cupicef.s (cupicef.s cupicef.z)))) + cupicef.I) + cupicef.I A) + x)) + q r)) + u) + y H; diff --git a/out/Makefile b/out/Makefile deleted file mode 100644 index 103ed004a..000000000 --- a/out/Makefile +++ /dev/null @@ -1,4 +0,0 @@ -all: - dune exec lambdapi -- check --lib-root=. -c cupicef.lp - dune exec lambdapi -- check --lib-root=. -c Coq__Init__Logic.dk - dune exec lambdapi -- check --lib-root=. -c Coq__Init__Datatypes.dk \ No newline at end of file diff --git a/out/compile.sh b/out/compile.sh new file mode 100755 index 000000000..1c6eaca5c --- /dev/null +++ b/out/compile.sh @@ -0,0 +1,9 @@ +#!/bin/sh +set -e +set -x + +export PATH=$PWD/_build/install/default/bin:$PATH +cd out +lambdapi check --lib-root . -c coq/cupicef.lp +lambdapi check --lib-root . -c Coq__Init__Logic.lp +lambdapi check --lib-root . -c Coq__Init__Datatypes.dk \ No newline at end of file diff --git a/out/cupicef.lp b/out/coq/cupicef.lp similarity index 100% rename from out/cupicef.lp rename to out/coq/cupicef.lp diff --git a/out/lambdapi.pkg b/out/lambdapi.pkg deleted file mode 100644 index 64a65dab8..000000000 --- a/out/lambdapi.pkg +++ /dev/null @@ -1,2 +0,0 @@ -package_name = coq -root_path = coq From 2f8e741c9fc4c4aa9b779663d3e32d7dc19e487e Mon Sep 17 00:00:00 2001 From: Claudio Sacerdoti Coen Date: Fri, 27 Jan 2023 17:24:15 +0100 Subject: [PATCH 05/25] Generate lp files with requires automatically --- out/Coq__Arith__PeanoNat.dk | 0 out/Coq__Init__Datatypes.dk | 3 - out/Coq__Init__Logic.dk | 2 - out/Coq__Init__Logic.lp | 30148 ------------------ out/add_requires.sh | 3 + out/compile.sh | 27 +- out/{coq/cupicef.lp => cupicef.lp.handmade} | 23 +- out/header | 1 + out/lambdapi.pkg | 2 + 9 files changed, 43 insertions(+), 30166 deletions(-) create mode 100644 out/Coq__Arith__PeanoNat.dk delete mode 100644 out/Coq__Init__Logic.lp create mode 100755 out/add_requires.sh rename out/{coq/cupicef.lp => cupicef.lp.handmade} (94%) create mode 100644 out/header create mode 100644 out/lambdapi.pkg diff --git a/out/Coq__Arith__PeanoNat.dk b/out/Coq__Arith__PeanoNat.dk new file mode 100644 index 000000000..e69de29bb diff --git a/out/Coq__Init__Datatypes.dk b/out/Coq__Init__Datatypes.dk index 386b9d26f..9d3df8e57 100644 --- a/out/Coq__Init__Datatypes.dk +++ b/out/Coq__Init__Datatypes.dk @@ -1,9 +1,6 @@ (; This file was automatically generated by Coqine. ;) (; The encoding used was: "template_cic". ;) -#REQUIRE cupicef. -#REQUIRE Coq__Init__Logic. - def Empty__set : cupicef.Univ cupicef.set. diff --git a/out/Coq__Init__Logic.dk b/out/Coq__Init__Logic.dk index 35d8c288f..3887ebe72 100644 --- a/out/Coq__Init__Logic.dk +++ b/out/Coq__Init__Logic.dk @@ -1,8 +1,6 @@ (; This file was automatically generated by Coqine. ;) (; The encoding used was: "template_cic". ;) -#REQUIRE cupicef. - def True : cupicef.Univ cupicef.prop. diff --git a/out/Coq__Init__Logic.lp b/out/Coq__Init__Logic.lp deleted file mode 100644 index 18cd3f66e..000000000 --- a/out/Coq__Init__Logic.lp +++ /dev/null @@ -1,30148 +0,0 @@ -flag "eta_equality" on; -require coq.cupicef as cupicef; -symbol True : cupicef.Univ cupicef.prop; -symbol I : cupicef.Term cupicef.prop True; -rule cupicef.code_guarded? _ I ↪ cupicef.guarded; -symbol match____True - : Π (s : cupicef.Sort), - Π (P : cupicef.Term cupicef.prop True → cupicef.Univ s), - Π (case__I : cupicef.Term s (P I)), - Π (x : cupicef.Term cupicef.prop True), cupicef.Term s (P x); -rule match____True $s $P $case__I I ↪ $case__I; -rule match____True $s - (λ x, cupicef.uncode $_.[] (cupicef.code (cupicef.cU $s'.[]) $P.[x])) ↪ -match____True $s' (λ x, $P.[x]); -symbol True__rect - : Π (P : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (f : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) P), - Π (t : cupicef.Term cupicef.prop True), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) P - ≔ λ (P : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (f : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) P), - λ (t : cupicef.Term cupicef.prop True), - match____True (cupicef.{|type|} (cupicef.s cupicef.z)) - (λ (__ : cupicef.Term cupicef.prop True), P) f t; -symbol True__ind - : Π (P : cupicef.Univ cupicef.prop), - Π (f : cupicef.Term cupicef.prop P), - Π (t : cupicef.Term cupicef.prop True), cupicef.Term cupicef.prop P - ≔ λ (P : cupicef.Univ cupicef.prop), - λ (f : cupicef.Term cupicef.prop P), - λ (t : cupicef.Term cupicef.prop True), - match____True cupicef.prop - (λ (__ : cupicef.Term cupicef.prop True), P) f t; -symbol True__rec - : Π (P : cupicef.Univ cupicef.set), - Π (f : cupicef.Term cupicef.set P), - Π (t : cupicef.Term cupicef.prop True), cupicef.Term cupicef.set P - ≔ λ (P : cupicef.Univ cupicef.set), - True__rect - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) cupicef.I) - cupicef.I P); -symbol False : cupicef.Univ cupicef.prop; -symbol match____False - : Π (s : cupicef.Sort), - Π (P : cupicef.Term cupicef.prop False → cupicef.Univ s), - Π (x : cupicef.Term cupicef.prop False), cupicef.Term s (P x); -rule match____False $s - (λ x, cupicef.uncode $_.[] (cupicef.code (cupicef.cU $s'.[]) $P.[x])) ↪ -match____False $s' (λ x, $P.[x]); -symbol False__rect - : Π (P : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (f : cupicef.Term cupicef.prop False), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) P - ≔ λ (P : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (f : cupicef.Term cupicef.prop False), - match____False (cupicef.{|type|} (cupicef.s cupicef.z)) - (λ (__ : cupicef.Term cupicef.prop False), P) f; -symbol False__ind - : Π (P : cupicef.Univ cupicef.prop), - Π (f : cupicef.Term cupicef.prop False), cupicef.Term cupicef.prop P - ≔ λ (P : cupicef.Univ cupicef.prop), - λ (f : cupicef.Term cupicef.prop False), - match____False cupicef.prop - (λ (__ : cupicef.Term cupicef.prop False), P) f; -symbol False__rec - : Π (P : cupicef.Univ cupicef.set), - Π (f : cupicef.Term cupicef.prop False), cupicef.Term cupicef.set P - ≔ λ (P : cupicef.Univ cupicef.set), - False__rect - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) cupicef.I) - cupicef.I P); -symbol not : Π (A : cupicef.Univ cupicef.prop), cupicef.Univ cupicef.prop - ≔ λ (A : cupicef.Univ cupicef.prop), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), False); -symbol and - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), cupicef.Univ cupicef.prop; -symbol conj - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (__ : cupicef.Term cupicef.prop A), - Π (__0 : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (and A B); -rule cupicef.code_guarded? _ (conj _ _ _ _) ↪ cupicef.guarded; -symbol match____and - : Π (s : cupicef.Sort), - Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (P : cupicef.Term cupicef.prop (and A B) → cupicef.Univ s), - Π - ( - case__conj : Π (__ : cupicef.Term cupicef.prop A), - Π (__0 : cupicef.Term cupicef.prop B), - cupicef.Term s (P (conj A B __ __0)) - ), - Π (x : cupicef.Term cupicef.prop (and A B)), - cupicef.Term s (P x); -rule match____and $s $A $B $P $case__conj (conj $A $B $__ $__0) ↪ $case__conj - $__ $__0; -rule match____and $s $A $B - (λ x, cupicef.uncode $_.[] (cupicef.code (cupicef.cU $s'.[]) $P.[x])) ↪ -match____and $s' $A $B (λ x, $P.[x]); -symbol and__rect - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (P : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - f : Π (__ : cupicef.Term cupicef.prop A), - Π (__0 : cupicef.Term cupicef.prop B), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) P - ), - Π (a : cupicef.Term cupicef.prop (and A B)), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) P - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (P : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - f : Π (__ : cupicef.Term cupicef.prop A), - Π (__0 : cupicef.Term cupicef.prop B), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) P - ), - λ (a : cupicef.Term cupicef.prop (and A B)), - match____and (cupicef.{|type|} (cupicef.s cupicef.z)) A B - (λ (__ : cupicef.Term cupicef.prop (and A B)), P) f a; -symbol and__ind - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (P : cupicef.Univ cupicef.prop), - Π - ( - f : Π (__ : cupicef.Term cupicef.prop A), - Π (__0 : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop P - ), - Π (a : cupicef.Term cupicef.prop (and A B)), - cupicef.Term cupicef.prop P - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (P : cupicef.Univ cupicef.prop), - λ - ( - f : Π (__ : cupicef.Term cupicef.prop A), - Π (__0 : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop P - ), - λ (a : cupicef.Term cupicef.prop (and A B)), - match____and cupicef.prop A B - (λ (__ : cupicef.Term cupicef.prop (and A B)), P) f a; -symbol and__rec - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (P : cupicef.Univ cupicef.set), - Π - ( - f : Π (__ : cupicef.Term cupicef.prop A), - Π (__0 : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.set P - ), - Π (a : cupicef.Term cupicef.prop (and A B)), - cupicef.Term cupicef.set P - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (P : cupicef.Univ cupicef.set), - and__rect A B - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I P); -opaque symbol proj1 - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (__ : cupicef.Term cupicef.prop (and A B)), - cupicef.Term cupicef.prop A - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (H : cupicef.Term cupicef.prop (and A B)), - match____and cupicef.prop A B - (λ (__ : cupicef.Term cupicef.prop (and A B)), A) - (λ (H0 : cupicef.Term cupicef.prop A), - λ (H1 : cupicef.Term cupicef.prop B), H0) - H; -opaque symbol proj2 - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (__ : cupicef.Term cupicef.prop (and A B)), - cupicef.Term cupicef.prop B - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (H : cupicef.Term cupicef.prop (and A B)), - match____and cupicef.prop A B - (λ (__ : cupicef.Term cupicef.prop (and A B)), B) - (λ (H0 : cupicef.Term cupicef.prop A), - λ (H1 : cupicef.Term cupicef.prop B), H1) - H; -symbol or - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), cupicef.Univ cupicef.prop; -symbol or__introl - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (__ : cupicef.Term cupicef.prop A), - cupicef.Term cupicef.prop (or A B); -symbol or__intror - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (or A B); -rule cupicef.code_guarded? _ (or__introl _ _ _) ↪ cupicef.guarded; -rule cupicef.code_guarded? _ (or__intror _ _ _) ↪ cupicef.guarded; -symbol match____or - : Π (s : cupicef.Sort), - Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (P : cupicef.Term cupicef.prop (or A B) → cupicef.Univ s), - Π - ( - case__or__introl : Π (__ : cupicef.Term cupicef.prop A), - cupicef.Term s (P (or__introl A B __)) - ), - Π - ( - case__or__intror : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term s (P (or__intror A B __)) - ), - Π (x : cupicef.Term cupicef.prop (or A B)), - cupicef.Term s (P x); -rule match____or $s $A $B $P $case__or__introl $case__or__intror - (or__introl $A $B $__) ↪ $case__or__introl $__; -rule match____or $s $A $B $P $case__or__introl $case__or__intror - (or__intror $A $B $__) ↪ $case__or__intror $__; -rule match____or $s $A $B - (λ x, cupicef.uncode $_.[] (cupicef.code (cupicef.cU $s'.[]) $P.[x])) ↪ -match____or $s' $A $B (λ x, $P.[x]); -symbol or__ind - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (P : cupicef.Univ cupicef.prop), - Π - ( - f : Π (__ : cupicef.Term cupicef.prop A), - cupicef.Term cupicef.prop P - ), - Π - ( - f0 : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop P - ), - Π (o : cupicef.Term cupicef.prop (or A B)), - cupicef.Term cupicef.prop P - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (P : cupicef.Univ cupicef.prop), - λ - ( - f : Π (__ : cupicef.Term cupicef.prop A), - cupicef.Term cupicef.prop P - ), - λ - ( - f0 : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop P - ), - λ (o : cupicef.Term cupicef.prop (or A B)), - match____or cupicef.prop A B - (λ (__ : cupicef.Term cupicef.prop (or A B)), P) f f0 o; -symbol iff - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), cupicef.Univ cupicef.prop - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), A)); -opaque symbol iff__refl - : Π (A : cupicef.Univ cupicef.prop), cupicef.Term cupicef.prop (iff A A) - ≔ λ (A : cupicef.Univ cupicef.prop), - conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), A)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), A)) - (λ (H : cupicef.Term cupicef.prop A), H) - (λ (H : cupicef.Term cupicef.prop A), H); -opaque symbol iff__trans - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - Π (__ : cupicef.Term cupicef.prop (iff A B)), - Π (__0 : cupicef.Term cupicef.prop (iff B C)), - cupicef.Term cupicef.prop (iff A C) - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - λ (H : cupicef.Term cupicef.prop (iff A B)), - match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), A)) - (λ - ( - __ : cupicef.Term cupicef.prop - (and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), A))) - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (iff B C) - (λ (__0 : cupicef.Term cupicef.prop (iff B C)), iff A C)) - (λ - ( - H1 : Π (__ : cupicef.Term cupicef.prop A), - cupicef.Term cupicef.prop B - ), - λ - ( - H2 : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop A - ), - λ (H0 : cupicef.Term cupicef.prop (iff B C)), - match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - B (λ (__ : cupicef.Term cupicef.prop B), C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - C (λ (__ : cupicef.Term cupicef.prop C), B)) - (λ - ( - __ : cupicef.Term cupicef.prop - (and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I C - (λ (__ : cupicef.Term cupicef.prop C), B))) - ), iff A C) - (λ - ( - H3 : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop C - ), - λ - ( - H4 : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop B - ), - conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I C - (λ (__ : cupicef.Term cupicef.prop C), A)) - (λ (H5 : cupicef.Term cupicef.prop A), H3 (H1 H5)) - (λ (H5 : cupicef.Term cupicef.prop C), - H2 (H1 (H2 (H4 H5))))) - H0) - H; -opaque symbol iff__sym - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (__ : cupicef.Term cupicef.prop (iff A B)), - cupicef.Term cupicef.prop (iff B A) - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (H : cupicef.Term cupicef.prop (iff A B)), - match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), A)) - (λ - ( - __ : cupicef.Term cupicef.prop - (and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - A (λ (__ : cupicef.Term cupicef.prop A), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - B (λ (__ : cupicef.Term cupicef.prop B), A))) - ), iff B A) - (λ - ( - H1 : Π (__ : cupicef.Term cupicef.prop A), - cupicef.Term cupicef.prop B - ), - λ - ( - H2 : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop A - ), - conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), A)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), B)) - H2 H1) - H; -opaque symbol and__iff__compat__l - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - Π (__ : cupicef.Term cupicef.prop (iff B C)), - cupicef.Term cupicef.prop (iff (and A B) (and A C)) - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - λ (H : cupicef.Term cupicef.prop (iff B C)), - match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I C - (λ (__ : cupicef.Term cupicef.prop C), B)) - (λ - ( - __ : cupicef.Term cupicef.prop - (and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I C - (λ (__ : cupicef.Term cupicef.prop C), B))) - ), iff (and A B) (and A C)) - (λ - ( - Hl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop C - ), - λ - ( - Hr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop B - ), - conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (and A B) - (λ (__ : cupicef.Term cupicef.prop (and A B)), and A C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (and A C) - (λ (__ : cupicef.Term cupicef.prop (and A C)), and A B)) - (λ (H0 : cupicef.Term cupicef.prop (and A B)), - match____and cupicef.prop A B - (λ (__ : cupicef.Term cupicef.prop (and A B)), and A C) - (λ (H1 : cupicef.Term cupicef.prop A), - λ (H2 : cupicef.Term cupicef.prop B), - conj A C H1 (Hl H2)) - H0) - (λ (H0 : cupicef.Term cupicef.prop (and A C)), - match____and cupicef.prop A C - (λ (__ : cupicef.Term cupicef.prop (and A C)), and A B) - (λ (H1 : cupicef.Term cupicef.prop A), - λ (H2 : cupicef.Term cupicef.prop C), - conj A B H1 (Hr H2)) - H0)) - H; -opaque symbol and__iff__compat__r - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - Π (__ : cupicef.Term cupicef.prop (iff B C)), - cupicef.Term cupicef.prop (iff (and B A) (and C A)) - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - λ (H : cupicef.Term cupicef.prop (iff B C)), - match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I C - (λ (__ : cupicef.Term cupicef.prop C), B)) - (λ - ( - __ : cupicef.Term cupicef.prop - (and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I C - (λ (__ : cupicef.Term cupicef.prop C), B))) - ), iff (and B A) (and C A)) - (λ - ( - Hl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop C - ), - λ - ( - Hr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop B - ), - conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (and B A) - (λ (__ : cupicef.Term cupicef.prop (and B A)), and C A)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (and C A) - (λ (__ : cupicef.Term cupicef.prop (and C A)), and B A)) - (λ (H0 : cupicef.Term cupicef.prop (and B A)), - match____and cupicef.prop B A - (λ (__ : cupicef.Term cupicef.prop (and B A)), and C A) - (λ (H1 : cupicef.Term cupicef.prop B), - λ (H2 : cupicef.Term cupicef.prop A), - conj C A (Hl H1) H2) - H0) - (λ (H0 : cupicef.Term cupicef.prop (and C A)), - match____and cupicef.prop C A - (λ (__ : cupicef.Term cupicef.prop (and C A)), and B A) - (λ (H1 : cupicef.Term cupicef.prop C), - λ (H2 : cupicef.Term cupicef.prop A), - conj B A (Hr H1) H2) - H0)) - H; -opaque symbol or__iff__compat__l - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - Π (__ : cupicef.Term cupicef.prop (iff B C)), - cupicef.Term cupicef.prop (iff (or A B) (or A C)) - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - λ (H : cupicef.Term cupicef.prop (iff B C)), - match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I C - (λ (__ : cupicef.Term cupicef.prop C), B)) - (λ - ( - __ : cupicef.Term cupicef.prop - (and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I C - (λ (__ : cupicef.Term cupicef.prop C), B))) - ), iff (or A B) (or A C)) - (λ - ( - Hl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop C - ), - λ - ( - Hr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop B - ), - conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (or A B) - (λ (__ : cupicef.Term cupicef.prop (or A B)), or A C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (or A C) - (λ (__ : cupicef.Term cupicef.prop (or A C)), or A B)) - (λ (H0 : cupicef.Term cupicef.prop (or A B)), - match____or cupicef.prop A B - (λ (__ : cupicef.Term cupicef.prop (or A B)), or A C) - (λ (H1 : cupicef.Term cupicef.prop A), - or__introl A C H1) - (λ (H1 : cupicef.Term cupicef.prop B), - or__intror A C (Hl H1)) - H0) - (λ (H0 : cupicef.Term cupicef.prop (or A C)), - match____or cupicef.prop A C - (λ (__ : cupicef.Term cupicef.prop (or A C)), or A B) - (λ (H1 : cupicef.Term cupicef.prop A), - or__introl A B H1) - (λ (H1 : cupicef.Term cupicef.prop C), - or__intror A B (Hr H1)) - H0)) - H; -opaque symbol or__iff__compat__r - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - Π (__ : cupicef.Term cupicef.prop (iff B C)), - cupicef.Term cupicef.prop (iff (or B A) (or C A)) - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - λ (H : cupicef.Term cupicef.prop (iff B C)), - match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I C - (λ (__ : cupicef.Term cupicef.prop C), B)) - (λ - ( - __ : cupicef.Term cupicef.prop - (and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I C - (λ (__ : cupicef.Term cupicef.prop C), B))) - ), iff (or B A) (or C A)) - (λ - ( - Hl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop C - ), - λ - ( - Hr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop B - ), - conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (or B A) - (λ (__ : cupicef.Term cupicef.prop (or B A)), or C A)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (or C A) - (λ (__ : cupicef.Term cupicef.prop (or C A)), or B A)) - (λ (H0 : cupicef.Term cupicef.prop (or B A)), - match____or cupicef.prop B A - (λ (__ : cupicef.Term cupicef.prop (or B A)), or C A) - (λ (H1 : cupicef.Term cupicef.prop B), - or__introl C A (Hl H1)) - (λ (H1 : cupicef.Term cupicef.prop A), - or__intror C A H1) - H0) - (λ (H0 : cupicef.Term cupicef.prop (or C A)), - match____or cupicef.prop C A - (λ (__ : cupicef.Term cupicef.prop (or C A)), or B A) - (λ (H1 : cupicef.Term cupicef.prop C), - or__introl B A (Hr H1)) - (λ (H1 : cupicef.Term cupicef.prop A), - or__intror B A H1) - H0)) - H; -opaque symbol imp__iff__compat__l - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - Π (__ : cupicef.Term cupicef.prop (iff B C)), - cupicef.Term cupicef.prop - (iff - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__0 : cupicef.Term cupicef.prop A), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__0 : cupicef.Term cupicef.prop A), C))) - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - λ (H : cupicef.Term cupicef.prop (iff B C)), - match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I C - (λ (__ : cupicef.Term cupicef.prop C), B)) - (λ - ( - __ : cupicef.Term cupicef.prop - (and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I C - (λ (__ : cupicef.Term cupicef.prop C), B))) - ), - iff - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__0 : cupicef.Term cupicef.prop A), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__0 : cupicef.Term cupicef.prop A), C))) - (λ - ( - Hl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop C - ), - λ - ( - Hr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop B - ), - conj - (cupicef.prod (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), B)) - (λ - ( - __ : Π (__ : cupicef.Term cupicef.prop A), - cupicef.Term cupicef.prop B - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - A (λ (__0 : cupicef.Term cupicef.prop A), C))) - (cupicef.prod (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), C)) - (λ - ( - __ : Π (__ : cupicef.Term cupicef.prop A), - cupicef.Term cupicef.prop C - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - A (λ (__0 : cupicef.Term cupicef.prop A), B))) - (λ - ( - H0 : Π (__ : cupicef.Term cupicef.prop A), - cupicef.Term cupicef.prop B - ), λ (H1 : cupicef.Term cupicef.prop A), Hl (H0 H1)) - (λ - ( - H0 : Π (__ : cupicef.Term cupicef.prop A), - cupicef.Term cupicef.prop C - ), λ (H1 : cupicef.Term cupicef.prop A), Hr (H0 H1))) - H; -opaque symbol imp__iff__compat__r - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - Π (__ : cupicef.Term cupicef.prop (iff B C)), - cupicef.Term cupicef.prop - (iff - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__0 : cupicef.Term cupicef.prop B), A)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I C - (λ (__0 : cupicef.Term cupicef.prop C), A))) - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - λ (H : cupicef.Term cupicef.prop (iff B C)), - match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I C - (λ (__ : cupicef.Term cupicef.prop C), B)) - (λ - ( - __ : cupicef.Term cupicef.prop - (and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I C - (λ (__ : cupicef.Term cupicef.prop C), B))) - ), - iff - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__0 : cupicef.Term cupicef.prop B), A)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I C - (λ (__0 : cupicef.Term cupicef.prop C), A))) - (λ - ( - Hl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop C - ), - λ - ( - Hr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop B - ), - conj - (cupicef.prod (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), A)) - (λ - ( - __ : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - C (λ (__0 : cupicef.Term cupicef.prop C), A))) - (cupicef.prod (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I C - (λ (__ : cupicef.Term cupicef.prop C), A)) - (λ - ( - __ : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - B (λ (__0 : cupicef.Term cupicef.prop B), A))) - (λ - ( - H0 : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop A - ), λ (H1 : cupicef.Term cupicef.prop C), H0 (Hr H1)) - (λ - ( - H0 : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop A - ), λ (H1 : cupicef.Term cupicef.prop B), H0 (Hl H1))) - H; -opaque symbol not__iff__compat - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (__ : cupicef.Term cupicef.prop (iff A B)), - cupicef.Term cupicef.prop (iff (not A) (not B)) - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (H : cupicef.Term cupicef.prop (iff A B)), - imp__iff__compat__r False A B H; -opaque symbol neg__false - : Π (A : cupicef.Univ cupicef.prop), - cupicef.Term cupicef.prop (iff (not A) (iff A False)) - ≔ λ (A : cupicef.Univ cupicef.prop), - cupicef.cast cupicef.prop cupicef.prop - (and - (cupicef.prod (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.prop - (cupicef.{|rule|} (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.prop) - cupicef.I - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), False)) - (λ - ( - __ : Π (__ : cupicef.Term cupicef.prop A), - cupicef.Term cupicef.prop False - ), iff A False)) - (cupicef.prod cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I (iff A False) - (λ (__ : cupicef.Term cupicef.prop (iff A False)), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__0 : cupicef.Term cupicef.prop A), False)))) - (iff (not A) (iff A False)) cupicef.I - (conj - (cupicef.prod (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.prop - (cupicef.{|rule|} (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.prop) - cupicef.I - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), False)) - (λ - ( - __ : Π (__ : cupicef.Term cupicef.prop A), - cupicef.Term cupicef.prop False - ), iff A False)) - (cupicef.prod cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I (iff A False) - (λ (__ : cupicef.Term cupicef.prop (iff A False)), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__0 : cupicef.Term cupicef.prop A), False))) - (λ - ( - H : Π (__ : cupicef.Term cupicef.prop A), - cupicef.Term cupicef.prop False - ), - conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), False)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I False - (λ (__ : cupicef.Term cupicef.prop False), A)) - H (λ (H1 : cupicef.Term cupicef.prop False), False__ind A H1)) - (λ (H0 : cupicef.Term cupicef.prop (iff A False)), - match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), False)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I False - (λ (__ : cupicef.Term cupicef.prop False), A)) - (λ - ( - __ : cupicef.Term cupicef.prop - (and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - A (λ (__ : cupicef.Term cupicef.prop A), False)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - False (λ (__ : cupicef.Term cupicef.prop False), A))) - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__0 : cupicef.Term cupicef.prop A), False)) - (λ - ( - H : Π (__ : cupicef.Term cupicef.prop A), - cupicef.Term cupicef.prop False - ), - λ - ( - __tmp : Π (__ : cupicef.Term cupicef.prop False), - cupicef.Term cupicef.prop A - ), H) - H0)); -symbol let__H0 - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - Π - ( - Hl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop A - ), - Π - ( - Hr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop A - ), - Π (H : cupicef.Term cupicef.prop (iff (and A B) (and A C))), - Π - ( - HypL : Π (__ : cupicef.Term cupicef.prop (and A B)), - cupicef.Term cupicef.prop (and A C) - ), - Π - ( - HypR : Π (__ : cupicef.Term cupicef.prop (and A C)), - cupicef.Term cupicef.prop (and A B) - ), - Π (H0 : cupicef.Term cupicef.prop B), - Π (__ : cupicef.Term cupicef.prop (and A B)), - cupicef.Term cupicef.prop C - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - λ - ( - Hl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop A - ), - λ - ( - Hr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop A - ), - λ (H : cupicef.Term cupicef.prop (iff (and A B) (and A C))), - λ - ( - HypL : Π (__ : cupicef.Term cupicef.prop (and A B)), - cupicef.Term cupicef.prop (and A C) - ), - λ - ( - HypR : Π (__ : cupicef.Term cupicef.prop (and A C)), - cupicef.Term cupicef.prop (and A B) - ), - λ (H0 : cupicef.Term cupicef.prop B), - λ (__ : cupicef.Term cupicef.prop (and A B)), - match____and cupicef.prop A C - (λ (__0 : cupicef.Term cupicef.prop (and A C)), C) - (λ (__0 : cupicef.Term cupicef.prop A), - λ (__1 : cupicef.Term cupicef.prop C), __1) - (HypL __); -symbol let__H1 - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - Π - ( - Hl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop A - ), - Π - ( - Hr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop A - ), - Π (H : cupicef.Term cupicef.prop (iff (and A B) (and A C))), - Π - ( - HypL : Π (__ : cupicef.Term cupicef.prop (and A B)), - cupicef.Term cupicef.prop (and A C) - ), - Π - ( - HypR : Π (__ : cupicef.Term cupicef.prop (and A C)), - cupicef.Term cupicef.prop (and A B) - ), - Π (H0 : cupicef.Term cupicef.prop C), - Π (__ : cupicef.Term cupicef.prop (and A C)), - cupicef.Term cupicef.prop B - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - λ - ( - Hl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop A - ), - λ - ( - Hr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop A - ), - λ (H : cupicef.Term cupicef.prop (iff (and A B) (and A C))), - λ - ( - HypL : Π (__ : cupicef.Term cupicef.prop (and A B)), - cupicef.Term cupicef.prop (and A C) - ), - λ - ( - HypR : Π (__ : cupicef.Term cupicef.prop (and A C)), - cupicef.Term cupicef.prop (and A B) - ), - λ (H0 : cupicef.Term cupicef.prop C), - λ (__ : cupicef.Term cupicef.prop (and A C)), - match____and cupicef.prop A B - (λ (__0 : cupicef.Term cupicef.prop (and A B)), B) - (λ (__0 : cupicef.Term cupicef.prop A), - λ (__1 : cupicef.Term cupicef.prop B), __1) - (HypR __); -opaque symbol and__cancel__l - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - Π - ( - __ : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop A - ), - Π - ( - __0 : Π (__0 : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop A - ), - cupicef.Term cupicef.prop - (iff (iff (and A B) (and A C)) (iff B C)) - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - λ - ( - Hl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop A - ), - λ - ( - Hr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop A - ), - conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (iff (and A B) (and A C)) - (λ - (__ : cupicef.Term cupicef.prop (iff (and A B) (and A C)) - ), iff B C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (iff B C) - (λ (__ : cupicef.Term cupicef.prop (iff B C)), - iff (and A B) (and A C))) - (λ - (H : cupicef.Term cupicef.prop (iff (and A B) (and A C))), - match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (and A B) - (λ (__ : cupicef.Term cupicef.prop (and A B)), and A C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (and A C) - (λ (__ : cupicef.Term cupicef.prop (and A C)), and A B)) - (λ - ( - __ : cupicef.Term cupicef.prop - (and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I (and A B) - (λ (__ : cupicef.Term cupicef.prop (and A B)), - and A C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I (and A C) - (λ (__ : cupicef.Term cupicef.prop (and A C)), - and A B))) - ), iff B C) - (λ - ( - HypL : Π (__ : cupicef.Term cupicef.prop (and A B)), - cupicef.Term cupicef.prop (and A C) - ), - λ - ( - HypR : Π (__ : cupicef.Term cupicef.prop (and A C)), - cupicef.Term cupicef.prop (and A B) - ), - conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I C - (λ (__ : cupicef.Term cupicef.prop C), B)) - (λ (H0 : cupicef.Term cupicef.prop B), - let__H0 A B C Hl Hr H HypL HypR H0 - (conj A B (Hl H0) H0)) - (λ (H0 : cupicef.Term cupicef.prop C), - let__H1 A B C Hl Hr H HypL HypR H0 - (conj A C (Hr H0) H0))) - H) - (and__iff__compat__l A B C); -symbol let__H2 - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - Π - ( - Hl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop A - ), - Π - ( - Hr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop A - ), - Π (H : cupicef.Term cupicef.prop (iff (and B A) (and C A))), - Π - ( - HypL : Π (__ : cupicef.Term cupicef.prop (and B A)), - cupicef.Term cupicef.prop (and C A) - ), - Π - ( - HypR : Π (__ : cupicef.Term cupicef.prop (and C A)), - cupicef.Term cupicef.prop (and B A) - ), - Π (H0 : cupicef.Term cupicef.prop B), - Π (__ : cupicef.Term cupicef.prop (and B A)), - cupicef.Term cupicef.prop C - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - λ - ( - Hl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop A - ), - λ - ( - Hr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop A - ), - λ (H : cupicef.Term cupicef.prop (iff (and B A) (and C A))), - λ - ( - HypL : Π (__ : cupicef.Term cupicef.prop (and B A)), - cupicef.Term cupicef.prop (and C A) - ), - λ - ( - HypR : Π (__ : cupicef.Term cupicef.prop (and C A)), - cupicef.Term cupicef.prop (and B A) - ), - λ (H0 : cupicef.Term cupicef.prop B), - λ (__ : cupicef.Term cupicef.prop (and B A)), - match____and cupicef.prop C A - (λ (__0 : cupicef.Term cupicef.prop (and C A)), C) - (λ (__0 : cupicef.Term cupicef.prop C), - λ (__1 : cupicef.Term cupicef.prop A), __0) - (HypL __); -symbol let__H3 - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - Π - ( - Hl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop A - ), - Π - ( - Hr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop A - ), - Π (H : cupicef.Term cupicef.prop (iff (and B A) (and C A))), - Π - ( - HypL : Π (__ : cupicef.Term cupicef.prop (and B A)), - cupicef.Term cupicef.prop (and C A) - ), - Π - ( - HypR : Π (__ : cupicef.Term cupicef.prop (and C A)), - cupicef.Term cupicef.prop (and B A) - ), - Π (H0 : cupicef.Term cupicef.prop C), - Π (__ : cupicef.Term cupicef.prop (and C A)), - cupicef.Term cupicef.prop B - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - λ - ( - Hl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop A - ), - λ - ( - Hr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop A - ), - λ (H : cupicef.Term cupicef.prop (iff (and B A) (and C A))), - λ - ( - HypL : Π (__ : cupicef.Term cupicef.prop (and B A)), - cupicef.Term cupicef.prop (and C A) - ), - λ - ( - HypR : Π (__ : cupicef.Term cupicef.prop (and C A)), - cupicef.Term cupicef.prop (and B A) - ), - λ (H0 : cupicef.Term cupicef.prop C), - λ (__ : cupicef.Term cupicef.prop (and C A)), - match____and cupicef.prop B A - (λ (__0 : cupicef.Term cupicef.prop (and B A)), B) - (λ (__0 : cupicef.Term cupicef.prop B), - λ (__1 : cupicef.Term cupicef.prop A), __0) - (HypR __); -opaque symbol and__cancel__r - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - Π - ( - __ : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop A - ), - Π - ( - __0 : Π (__0 : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop A - ), - cupicef.Term cupicef.prop - (iff (iff (and B A) (and C A)) (iff B C)) - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - λ - ( - Hl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop A - ), - λ - ( - Hr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop A - ), - conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (iff (and B A) (and C A)) - (λ - (__ : cupicef.Term cupicef.prop (iff (and B A) (and C A)) - ), iff B C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (iff B C) - (λ (__ : cupicef.Term cupicef.prop (iff B C)), - iff (and B A) (and C A))) - (λ - (H : cupicef.Term cupicef.prop (iff (and B A) (and C A))), - match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (and B A) - (λ (__ : cupicef.Term cupicef.prop (and B A)), and C A)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (and C A) - (λ (__ : cupicef.Term cupicef.prop (and C A)), and B A)) - (λ - ( - __ : cupicef.Term cupicef.prop - (and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I (and B A) - (λ (__ : cupicef.Term cupicef.prop (and B A)), - and C A)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I (and C A) - (λ (__ : cupicef.Term cupicef.prop (and C A)), - and B A))) - ), iff B C) - (λ - ( - HypL : Π (__ : cupicef.Term cupicef.prop (and B A)), - cupicef.Term cupicef.prop (and C A) - ), - λ - ( - HypR : Π (__ : cupicef.Term cupicef.prop (and C A)), - cupicef.Term cupicef.prop (and B A) - ), - conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I C - (λ (__ : cupicef.Term cupicef.prop C), B)) - (λ (H0 : cupicef.Term cupicef.prop B), - let__H2 A B C Hl Hr H HypL HypR H0 - (conj B A H0 (Hl H0))) - (λ (H0 : cupicef.Term cupicef.prop C), - let__H3 A B C Hl Hr H HypL HypR H0 - (conj C A H0 (Hr H0)))) - H) - (and__iff__compat__r A B C); -opaque symbol and__comm - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - cupicef.Term cupicef.prop (iff (and A B) (and B A)) - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I (and A B) - (λ (__ : cupicef.Term cupicef.prop (and A B)), and B A)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I (and B A) - (λ (__ : cupicef.Term cupicef.prop (and B A)), and A B)) - (λ (H : cupicef.Term cupicef.prop (and A B)), - match____and cupicef.prop A B - (λ (__ : cupicef.Term cupicef.prop (and A B)), and B A) - (λ (H0 : cupicef.Term cupicef.prop A), - λ (H1 : cupicef.Term cupicef.prop B), conj B A H1 H0) - H) - (λ (H : cupicef.Term cupicef.prop (and B A)), - match____and cupicef.prop B A - (λ (__ : cupicef.Term cupicef.prop (and B A)), and A B) - (λ (H0 : cupicef.Term cupicef.prop B), - λ (H1 : cupicef.Term cupicef.prop A), conj A B H1 H0) - H); -opaque symbol and__assoc - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - cupicef.Term cupicef.prop (iff (and (and A B) C) (and A (and B C))) - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (and (and A B) C) - (λ (__ : cupicef.Term cupicef.prop (and (and A B) C)), - and A (and B C))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (and A (and B C)) - (λ (__ : cupicef.Term cupicef.prop (and A (and B C))), - and (and A B) C)) - (λ (H : cupicef.Term cupicef.prop (and (and A B) C)), - match____and cupicef.prop (and A B) C - (λ (__ : cupicef.Term cupicef.prop (and (and A B) C)), - and A (and B C)) - (λ (H0 : cupicef.Term cupicef.prop (and A B)), - match____and cupicef.prop A B - (λ (__ : cupicef.Term cupicef.prop (and A B)), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I C - (λ (__0 : cupicef.Term cupicef.prop C), and A (and B C))) - (λ (H1 : cupicef.Term cupicef.prop A), - λ (H2 : cupicef.Term cupicef.prop B), - λ (H3 : cupicef.Term cupicef.prop C), - conj A (and B C) H1 (conj B C H2 H3)) - H0) - H) - (λ (H : cupicef.Term cupicef.prop (and A (and B C))), - match____and cupicef.prop A (and B C) - (λ (__ : cupicef.Term cupicef.prop (and A (and B C))), - and (and A B) C) - (λ (H0 : cupicef.Term cupicef.prop A), - λ (H1 : cupicef.Term cupicef.prop (and B C)), - match____and cupicef.prop B C - (λ (__ : cupicef.Term cupicef.prop (and B C)), - and (and A B) C) - (λ (H2 : cupicef.Term cupicef.prop B), - λ (H3 : cupicef.Term cupicef.prop C), - conj (and A B) C (conj A B H0 H2) H3) - H1) - H); -symbol let__o - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - Π - ( - Fl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (not A) - ), - Π - ( - Fr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop (not A) - ), - Π (H : cupicef.Term cupicef.prop (iff (or A B) (or A C))), - Π - ( - Hl : Π (__ : cupicef.Term cupicef.prop (or A B)), - cupicef.Term cupicef.prop (or A C) - ), - Π - ( - Hr : Π (__ : cupicef.Term cupicef.prop (or A C)), - cupicef.Term cupicef.prop (or A B) - ), - Π (H0 : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (or A C) - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - λ - ( - Fl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (not A) - ), - λ - ( - Fr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop (not A) - ), - λ (H : cupicef.Term cupicef.prop (iff (or A B) (or A C))), - λ - ( - Hl : Π (__ : cupicef.Term cupicef.prop (or A B)), - cupicef.Term cupicef.prop (or A C) - ), - λ - ( - Hr : Π (__ : cupicef.Term cupicef.prop (or A C)), - cupicef.Term cupicef.prop (or A B) - ), - λ (H0 : cupicef.Term cupicef.prop B), - Hl (or__intror A B H0); -symbol let__n - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - Π - ( - Fl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (not A) - ), - Π - ( - Fr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop (not A) - ), - Π (H : cupicef.Term cupicef.prop (iff (or A B) (or A C))), - Π - ( - Hl : Π (__ : cupicef.Term cupicef.prop (or A B)), - cupicef.Term cupicef.prop (or A C) - ), - Π - ( - Hr : Π (__ : cupicef.Term cupicef.prop (or A C)), - cupicef.Term cupicef.prop (or A B) - ), - Π (H0 : cupicef.Term cupicef.prop B), - Π (H1 : cupicef.Term cupicef.prop A), - cupicef.Term cupicef.prop False - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - λ - ( - Fl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (not A) - ), - λ - ( - Fr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop (not A) - ), - λ (H : cupicef.Term cupicef.prop (iff (or A B) (or A C))), - λ - ( - Hl : Π (__ : cupicef.Term cupicef.prop (or A B)), - cupicef.Term cupicef.prop (or A C) - ), - λ - ( - Hr : Π (__ : cupicef.Term cupicef.prop (or A C)), - cupicef.Term cupicef.prop (or A B) - ), - λ (H0 : cupicef.Term cupicef.prop B), - λ (H1 : cupicef.Term cupicef.prop A), Fl H0 H1; -symbol let__o0 - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - Π - ( - Fl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (not A) - ), - Π - ( - Fr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop (not A) - ), - Π (H : cupicef.Term cupicef.prop (iff (or A B) (or A C))), - Π - ( - Hl : Π (__ : cupicef.Term cupicef.prop (or A B)), - cupicef.Term cupicef.prop (or A C) - ), - Π - ( - Hr : Π (__ : cupicef.Term cupicef.prop (or A C)), - cupicef.Term cupicef.prop (or A B) - ), - Π (H0 : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop (or A B) - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - λ - ( - Fl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (not A) - ), - λ - ( - Fr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop (not A) - ), - λ (H : cupicef.Term cupicef.prop (iff (or A B) (or A C))), - λ - ( - Hl : Π (__ : cupicef.Term cupicef.prop (or A B)), - cupicef.Term cupicef.prop (or A C) - ), - λ - ( - Hr : Π (__ : cupicef.Term cupicef.prop (or A C)), - cupicef.Term cupicef.prop (or A B) - ), - λ (H0 : cupicef.Term cupicef.prop C), - Hr (or__intror A C H0); -symbol let__n0 - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - Π - ( - Fl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (not A) - ), - Π - ( - Fr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop (not A) - ), - Π (H : cupicef.Term cupicef.prop (iff (or A B) (or A C))), - Π - ( - Hl : Π (__ : cupicef.Term cupicef.prop (or A B)), - cupicef.Term cupicef.prop (or A C) - ), - Π - ( - Hr : Π (__ : cupicef.Term cupicef.prop (or A C)), - cupicef.Term cupicef.prop (or A B) - ), - Π (H0 : cupicef.Term cupicef.prop C), - Π (H1 : cupicef.Term cupicef.prop A), - cupicef.Term cupicef.prop False - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - λ - ( - Fl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (not A) - ), - λ - ( - Fr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop (not A) - ), - λ (H : cupicef.Term cupicef.prop (iff (or A B) (or A C))), - λ - ( - Hl : Π (__ : cupicef.Term cupicef.prop (or A B)), - cupicef.Term cupicef.prop (or A C) - ), - λ - ( - Hr : Π (__ : cupicef.Term cupicef.prop (or A C)), - cupicef.Term cupicef.prop (or A B) - ), - λ (H0 : cupicef.Term cupicef.prop C), - λ (H1 : cupicef.Term cupicef.prop A), Fr H0 H1; -opaque symbol or__cancel__l - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - Π - ( - __ : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (not A) - ), - Π - ( - __0 : Π (__0 : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop (not A) - ), - cupicef.Term cupicef.prop - (iff (iff (or A B) (or A C)) (iff B C)) - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - λ - ( - Fl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (not A) - ), - λ - ( - Fr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop (not A) - ), - conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (iff (or A B) (or A C)) - (λ - (__ : cupicef.Term cupicef.prop (iff (or A B) (or A C))), - iff B C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (iff B C) - (λ (__ : cupicef.Term cupicef.prop (iff B C)), - iff (or A B) (or A C))) - (λ (H : cupicef.Term cupicef.prop (iff (or A B) (or A C))), - match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (or A B) - (λ (__ : cupicef.Term cupicef.prop (or A B)), or A C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (or A C) - (λ (__ : cupicef.Term cupicef.prop (or A C)), or A B)) - (λ - ( - __ : cupicef.Term cupicef.prop - (and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I (or A B) - (λ (__ : cupicef.Term cupicef.prop (or A B)), - or A C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I (or A C) - (λ (__ : cupicef.Term cupicef.prop (or A C)), - or A B))) - ), iff B C) - (λ - ( - Hl : Π (__ : cupicef.Term cupicef.prop (or A B)), - cupicef.Term cupicef.prop (or A C) - ), - λ - ( - Hr : Π (__ : cupicef.Term cupicef.prop (or A C)), - cupicef.Term cupicef.prop (or A B) - ), - conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I C - (λ (__ : cupicef.Term cupicef.prop C), B)) - (λ (H0 : cupicef.Term cupicef.prop B), - match____or cupicef.prop A C - (λ (__ : cupicef.Term cupicef.prop (or A C)), C) - (λ (H1 : cupicef.Term cupicef.prop A), - match____False cupicef.prop - (λ (__ : cupicef.Term cupicef.prop False), C) - (let__n A B C Fl Fr H Hl Hr H0 H1)) - (λ (H1 : cupicef.Term cupicef.prop C), H1) - (let__o A B C Fl Fr H Hl Hr H0)) - (λ (H0 : cupicef.Term cupicef.prop C), - match____or cupicef.prop A B - (λ (__ : cupicef.Term cupicef.prop (or A B)), B) - (λ (H1 : cupicef.Term cupicef.prop A), - match____False cupicef.prop - (λ (__ : cupicef.Term cupicef.prop False), B) - (let__n0 A B C Fl Fr H Hl Hr H0 H1)) - (λ (H1 : cupicef.Term cupicef.prop B), H1) - (let__o0 A B C Fl Fr H Hl Hr H0))) - H) - (or__iff__compat__l A B C); -symbol let__o1 - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - Π - ( - Fl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (not A) - ), - Π - ( - Fr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop (not A) - ), - Π (H : cupicef.Term cupicef.prop (iff (or B A) (or C A))), - Π - ( - Hl : Π (__ : cupicef.Term cupicef.prop (or B A)), - cupicef.Term cupicef.prop (or C A) - ), - Π - ( - Hr : Π (__ : cupicef.Term cupicef.prop (or C A)), - cupicef.Term cupicef.prop (or B A) - ), - Π (H0 : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (or C A) - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - λ - ( - Fl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (not A) - ), - λ - ( - Fr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop (not A) - ), - λ (H : cupicef.Term cupicef.prop (iff (or B A) (or C A))), - λ - ( - Hl : Π (__ : cupicef.Term cupicef.prop (or B A)), - cupicef.Term cupicef.prop (or C A) - ), - λ - ( - Hr : Π (__ : cupicef.Term cupicef.prop (or C A)), - cupicef.Term cupicef.prop (or B A) - ), - λ (H0 : cupicef.Term cupicef.prop B), - Hl (or__introl B A H0); -symbol let__n1 - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - Π - ( - Fl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (not A) - ), - Π - ( - Fr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop (not A) - ), - Π (H : cupicef.Term cupicef.prop (iff (or B A) (or C A))), - Π - ( - Hl : Π (__ : cupicef.Term cupicef.prop (or B A)), - cupicef.Term cupicef.prop (or C A) - ), - Π - ( - Hr : Π (__ : cupicef.Term cupicef.prop (or C A)), - cupicef.Term cupicef.prop (or B A) - ), - Π (H0 : cupicef.Term cupicef.prop B), - Π (H1 : cupicef.Term cupicef.prop A), - cupicef.Term cupicef.prop False - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - λ - ( - Fl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (not A) - ), - λ - ( - Fr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop (not A) - ), - λ (H : cupicef.Term cupicef.prop (iff (or B A) (or C A))), - λ - ( - Hl : Π (__ : cupicef.Term cupicef.prop (or B A)), - cupicef.Term cupicef.prop (or C A) - ), - λ - ( - Hr : Π (__ : cupicef.Term cupicef.prop (or C A)), - cupicef.Term cupicef.prop (or B A) - ), - λ (H0 : cupicef.Term cupicef.prop B), - λ (H1 : cupicef.Term cupicef.prop A), Fl H0 H1; -symbol let__o2 - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - Π - ( - Fl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (not A) - ), - Π - ( - Fr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop (not A) - ), - Π (H : cupicef.Term cupicef.prop (iff (or B A) (or C A))), - Π - ( - Hl : Π (__ : cupicef.Term cupicef.prop (or B A)), - cupicef.Term cupicef.prop (or C A) - ), - Π - ( - Hr : Π (__ : cupicef.Term cupicef.prop (or C A)), - cupicef.Term cupicef.prop (or B A) - ), - Π (H0 : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop (or B A) - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - λ - ( - Fl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (not A) - ), - λ - ( - Fr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop (not A) - ), - λ (H : cupicef.Term cupicef.prop (iff (or B A) (or C A))), - λ - ( - Hl : Π (__ : cupicef.Term cupicef.prop (or B A)), - cupicef.Term cupicef.prop (or C A) - ), - λ - ( - Hr : Π (__ : cupicef.Term cupicef.prop (or C A)), - cupicef.Term cupicef.prop (or B A) - ), - λ (H0 : cupicef.Term cupicef.prop C), - Hr (or__introl C A H0); -symbol let__n2 - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - Π - ( - Fl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (not A) - ), - Π - ( - Fr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop (not A) - ), - Π (H : cupicef.Term cupicef.prop (iff (or B A) (or C A))), - Π - ( - Hl : Π (__ : cupicef.Term cupicef.prop (or B A)), - cupicef.Term cupicef.prop (or C A) - ), - Π - ( - Hr : Π (__ : cupicef.Term cupicef.prop (or C A)), - cupicef.Term cupicef.prop (or B A) - ), - Π (H0 : cupicef.Term cupicef.prop C), - Π (H1 : cupicef.Term cupicef.prop A), - cupicef.Term cupicef.prop False - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - λ - ( - Fl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (not A) - ), - λ - ( - Fr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop (not A) - ), - λ (H : cupicef.Term cupicef.prop (iff (or B A) (or C A))), - λ - ( - Hl : Π (__ : cupicef.Term cupicef.prop (or B A)), - cupicef.Term cupicef.prop (or C A) - ), - λ - ( - Hr : Π (__ : cupicef.Term cupicef.prop (or C A)), - cupicef.Term cupicef.prop (or B A) - ), - λ (H0 : cupicef.Term cupicef.prop C), - λ (H1 : cupicef.Term cupicef.prop A), Fr H0 H1; -opaque symbol or__cancel__r - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - Π - ( - __ : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (not A) - ), - Π - ( - __0 : Π (__0 : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop (not A) - ), - cupicef.Term cupicef.prop - (iff (iff (or B A) (or C A)) (iff B C)) - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - λ - ( - Fl : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (not A) - ), - λ - ( - Fr : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop (not A) - ), - conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (iff (or B A) (or C A)) - (λ - (__ : cupicef.Term cupicef.prop (iff (or B A) (or C A))), - iff B C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (iff B C) - (λ (__ : cupicef.Term cupicef.prop (iff B C)), - iff (or B A) (or C A))) - (λ (H : cupicef.Term cupicef.prop (iff (or B A) (or C A))), - match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (or B A) - (λ (__ : cupicef.Term cupicef.prop (or B A)), or C A)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (or C A) - (λ (__ : cupicef.Term cupicef.prop (or C A)), or B A)) - (λ - ( - __ : cupicef.Term cupicef.prop - (and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I (or B A) - (λ (__ : cupicef.Term cupicef.prop (or B A)), - or C A)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I (or C A) - (λ (__ : cupicef.Term cupicef.prop (or C A)), - or B A))) - ), iff B C) - (λ - ( - Hl : Π (__ : cupicef.Term cupicef.prop (or B A)), - cupicef.Term cupicef.prop (or C A) - ), - λ - ( - Hr : Π (__ : cupicef.Term cupicef.prop (or C A)), - cupicef.Term cupicef.prop (or B A) - ), - conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I C - (λ (__ : cupicef.Term cupicef.prop C), B)) - (λ (H0 : cupicef.Term cupicef.prop B), - match____or cupicef.prop C A - (λ (__ : cupicef.Term cupicef.prop (or C A)), C) - (λ (H1 : cupicef.Term cupicef.prop C), H1) - (λ (H1 : cupicef.Term cupicef.prop A), - match____False cupicef.prop - (λ (__ : cupicef.Term cupicef.prop False), C) - (let__n1 A B C Fl Fr H Hl Hr H0 H1)) - (let__o1 A B C Fl Fr H Hl Hr H0)) - (λ (H0 : cupicef.Term cupicef.prop C), - match____or cupicef.prop B A - (λ (__ : cupicef.Term cupicef.prop (or B A)), B) - (λ (H1 : cupicef.Term cupicef.prop B), H1) - (λ (H1 : cupicef.Term cupicef.prop A), - match____False cupicef.prop - (λ (__ : cupicef.Term cupicef.prop False), B) - (let__n2 A B C Fl Fr H Hl Hr H0 H1)) - (let__o2 A B C Fl Fr H Hl Hr H0))) - H) - (or__iff__compat__r A B C); -opaque symbol or__comm - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - cupicef.Term cupicef.prop (iff (or A B) (or B A)) - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I (or A B) - (λ (__ : cupicef.Term cupicef.prop (or A B)), or B A)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I (or B A) - (λ (__ : cupicef.Term cupicef.prop (or B A)), or A B)) - (λ (H : cupicef.Term cupicef.prop (or A B)), - match____or cupicef.prop A B - (λ (__ : cupicef.Term cupicef.prop (or A B)), or B A) - (λ (H0 : cupicef.Term cupicef.prop A), or__intror B A H0) - (λ (H0 : cupicef.Term cupicef.prop B), or__introl B A H0) H) - (λ (H : cupicef.Term cupicef.prop (or B A)), - match____or cupicef.prop B A - (λ (__ : cupicef.Term cupicef.prop (or B A)), or A B) - (λ (H0 : cupicef.Term cupicef.prop B), or__intror A B H0) - (λ (H0 : cupicef.Term cupicef.prop A), or__introl A B H0) H); -opaque symbol or__assoc - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - cupicef.Term cupicef.prop (iff (or (or A B) C) (or A (or B C))) - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (or (or A B) C) - (λ (__ : cupicef.Term cupicef.prop (or (or A B) C)), - or A (or B C))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (or A (or B C)) - (λ (__ : cupicef.Term cupicef.prop (or A (or B C))), - or (or A B) C)) - (λ (H : cupicef.Term cupicef.prop (or (or A B) C)), - match____or cupicef.prop (or A B) C - (λ (__ : cupicef.Term cupicef.prop (or (or A B) C)), - or A (or B C)) - (λ (H0 : cupicef.Term cupicef.prop (or A B)), - match____or cupicef.prop A B - (λ (__ : cupicef.Term cupicef.prop (or A B)), - or A (or B C)) - (λ (H1 : cupicef.Term cupicef.prop A), - or__introl A (or B C) H1) - (λ (H1 : cupicef.Term cupicef.prop B), - or__intror A (or B C) (or__introl B C H1)) - H0) - (λ (H0 : cupicef.Term cupicef.prop C), - or__intror A (or B C) (or__intror B C H0)) - H) - (λ (H : cupicef.Term cupicef.prop (or A (or B C))), - match____or cupicef.prop A (or B C) - (λ (__ : cupicef.Term cupicef.prop (or A (or B C))), - or (or A B) C) - (λ (H0 : cupicef.Term cupicef.prop A), - or__introl (or A B) C (or__introl A B H0)) - (λ (H0 : cupicef.Term cupicef.prop (or B C)), - match____or cupicef.prop B C - (λ (__ : cupicef.Term cupicef.prop (or B C)), - or (or A B) C) - (λ (H1 : cupicef.Term cupicef.prop B), - or__introl (or A B) C (or__intror A B H1)) - (λ (H1 : cupicef.Term cupicef.prop C), - or__intror (or A B) C H1) - H0) - H); -opaque symbol iff__and - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (__ : cupicef.Term cupicef.prop (iff A B)), - cupicef.Term cupicef.prop - (and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__0 : cupicef.Term cupicef.prop A), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__0 : cupicef.Term cupicef.prop B), A))) - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (H : cupicef.Term cupicef.prop (iff A B)), - match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), A)) - (λ - ( - __ : cupicef.Term cupicef.prop - (and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - A (λ (__ : cupicef.Term cupicef.prop A), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - B (λ (__ : cupicef.Term cupicef.prop B), A))) - ), - and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__0 : cupicef.Term cupicef.prop A), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__0 : cupicef.Term cupicef.prop B), A))) - (λ - ( - H0 : Π (__ : cupicef.Term cupicef.prop A), - cupicef.Term cupicef.prop B - ), - λ - ( - H1 : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop A - ), - conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), A)) - H0 H1) - H; -opaque symbol iff__to__and - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - cupicef.Term cupicef.prop - (iff (iff A B) - (and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), A)))) - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I (iff A B) - (λ (__ : cupicef.Term cupicef.prop (iff A B)), - and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__0 : cupicef.Term cupicef.prop A), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__0 : cupicef.Term cupicef.prop B), A)))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), A))) - (λ - ( - __ : cupicef.Term cupicef.prop - (and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), A))) - ), iff A B)) - (λ (H : cupicef.Term cupicef.prop (iff A B)), - match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), A)) - (λ - ( - __ : cupicef.Term cupicef.prop - (and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - A (λ (__ : cupicef.Term cupicef.prop A), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - B (λ (__ : cupicef.Term cupicef.prop B), A))) - ), - and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__0 : cupicef.Term cupicef.prop A), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__0 : cupicef.Term cupicef.prop B), A))) - (λ - ( - Hl : Π (__ : cupicef.Term cupicef.prop A), - cupicef.Term cupicef.prop B - ), - λ - ( - Hr : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop A - ), - conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), A)) - (λ (H0 : cupicef.Term cupicef.prop A), Hl H0) - (λ (H0 : cupicef.Term cupicef.prop B), Hr H0)) - H) - (λ - ( - H : cupicef.Term cupicef.prop - (and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), A))) - ), - match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), A)) - (λ - ( - __ : cupicef.Term cupicef.prop - (and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - A (λ (__ : cupicef.Term cupicef.prop A), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - B (λ (__ : cupicef.Term cupicef.prop B), A))) - ), iff A B) - (λ - ( - Hl : Π (__ : cupicef.Term cupicef.prop A), - cupicef.Term cupicef.prop B - ), - λ - ( - Hr : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop A - ), - conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), A)) - (λ (H0 : cupicef.Term cupicef.prop A), Hl H0) - (λ (H0 : cupicef.Term cupicef.prop B), Hr H0)) - H); -symbol IF__then__else - : Π (P : cupicef.Univ cupicef.prop), - Π (Q : cupicef.Univ cupicef.prop), - Π (R : cupicef.Univ cupicef.prop), cupicef.Univ cupicef.prop - ≔ λ (P : cupicef.Univ cupicef.prop), - λ (Q : cupicef.Univ cupicef.prop), - λ (R : cupicef.Univ cupicef.prop), or (and P Q) (and (not P) R); -symbol ex - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ cupicef.prop - ), cupicef.Univ cupicef.prop; -symbol ex__intro - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ cupicef.prop - ), - Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (__ : cupicef.Term cupicef.prop (P x)), - cupicef.Term cupicef.prop (ex A P); -rule cupicef.code_guarded? _ (ex__intro _ _ _ _) ↪ cupicef.guarded; -symbol match____ex - : Π (s : cupicef.Sort), - Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - Π (P0 : cupicef.Term cupicef.prop (ex A P) → cupicef.Univ s), - Π - ( - case__ex__intro : Π - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - Π (__ : cupicef.Term cupicef.prop (P x0)), - cupicef.Term s (P0 (ex__intro A P x0 __)) - ), - Π (x : cupicef.Term cupicef.prop (ex A P)), - cupicef.Term s (P0 x); -rule match____ex $s $A $P $P0 $case__ex__intro (ex__intro $A $P $x0 $__) ↪ -$case__ex__intro $x0 $__; -rule match____ex $s $A $P - (λ x, cupicef.uncode $_.[] (cupicef.code (cupicef.cU $s'.[]) $P0.[x])) ↪ -match____ex $s' $A $P (λ x, $P0.[x]); -symbol ex__ind - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ cupicef.prop - ), - Π (P0 : cupicef.Univ cupicef.prop), - Π - ( - f : Π - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π (__ : cupicef.Term cupicef.prop (P x)), - cupicef.Term cupicef.prop P0 - ), - Π (e : cupicef.Term cupicef.prop (ex A P)), - cupicef.Term cupicef.prop P0 - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - λ (P0 : cupicef.Univ cupicef.prop), - λ - ( - f : Π - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - Π (__ : cupicef.Term cupicef.prop (P x)), - cupicef.Term cupicef.prop P0 - ), - λ (e : cupicef.Term cupicef.prop (ex A P)), - match____ex cupicef.prop A P - (λ (__ : cupicef.Term cupicef.prop (ex A P)), P0) f e; -symbol ex2 - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ cupicef.prop - ), - Π - ( - Q : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), cupicef.Univ cupicef.prop; -symbol ex__intro2 - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ cupicef.prop - ), - Π - ( - Q : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (__ : cupicef.Term cupicef.prop (P x)), - Π (__0 : cupicef.Term cupicef.prop (Q x)), - cupicef.Term cupicef.prop (ex2 A P Q); -rule cupicef.code_guarded? _ (ex__intro2 _ _ _ _ _ _) ↪ cupicef.guarded; -symbol match____ex2 - : Π (s : cupicef.Sort), - Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - Π - ( - Q : Π - ( - __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), cupicef.Univ cupicef.prop - ), - Π - (P0 : cupicef.Term cupicef.prop (ex2 A P Q) → cupicef.Univ s), - Π - ( - case__ex__intro2 : Π - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - Π - (__ : cupicef.Term cupicef.prop (P x0) - ), - Π - ( - __0 : cupicef.Term cupicef.prop - (Q x0) - ), - cupicef.Term s - (P0 (ex__intro2 A P Q x0 __ __0)) - ), - Π (x : cupicef.Term cupicef.prop (ex2 A P Q)), - cupicef.Term s (P0 x); -rule match____ex2 $s $A $P $Q $P0 $case__ex__intro2 - (ex__intro2 $A $P $Q $x0 $__ $__0) ↪ $case__ex__intro2 $x0 $__ $__0; -rule match____ex2 $s $A $P $Q - (λ x, cupicef.uncode $_.[] (cupicef.code (cupicef.cU $s'.[]) $P0.[x])) ↪ -match____ex2 $s' $A $P $Q (λ x, $P0.[x]); -symbol ex2__ind - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ cupicef.prop - ), - Π - ( - Q : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - Π (P0 : cupicef.Univ cupicef.prop), - Π - ( - f : Π - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - Π (__ : cupicef.Term cupicef.prop (P x)), - Π (__0 : cupicef.Term cupicef.prop (Q x)), - cupicef.Term cupicef.prop P0 - ), - Π (e : cupicef.Term cupicef.prop (ex2 A P Q)), - cupicef.Term cupicef.prop P0 - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - λ - ( - Q : Π - ( - __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), cupicef.Univ cupicef.prop - ), - λ (P0 : cupicef.Univ cupicef.prop), - λ - ( - f : Π - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π (__ : cupicef.Term cupicef.prop (P x)), - Π (__0 : cupicef.Term cupicef.prop (Q x)), - cupicef.Term cupicef.prop P0 - ), - λ (e : cupicef.Term cupicef.prop (ex2 A P Q)), - match____ex2 cupicef.prop A P Q - (λ (__ : cupicef.Term cupicef.prop (ex2 A P Q)), P0) f e; -symbol all - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ cupicef.prop - ), cupicef.Univ cupicef.prop - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.prop - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.prop) - cupicef.I A - (λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - P x); -opaque symbol inst - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ cupicef.prop - ), - Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - __ : cupicef.Term cupicef.prop - (all A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x0)) - ), cupicef.Term cupicef.prop (P x) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - cupicef.cast - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.prop) - cupicef.prop)) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.prop) - cupicef.prop) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.prop) - cupicef.prop)) - cupicef.I A - (λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.prod - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.prop) - cupicef.prop - (cupicef.{|rule|} - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.prop) - cupicef.prop) - cupicef.I - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.prop - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.prop) - cupicef.I A - (λ - ( - x0 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), P x0)) - (λ - ( - H : Π - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Term cupicef.prop (P x0) - ), P x))) - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (all A - (λ - ( - x0 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), P x0)) - (λ - ( - __ : cupicef.Term cupicef.prop - (all A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x0)) - ), P x))) - cupicef.I - (λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - H : Π - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Term cupicef.prop (P x0) - ), H x); -opaque symbol gen - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ cupicef.prop - ), - Π (B : cupicef.Univ cupicef.prop), - Π - ( - f : Π - ( - y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (P y) - ), - Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (all A P) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - cupicef.cast - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.prop)))) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - (cupicef.{|rule|} cupicef.prop cupicef.prop))) - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.prop))) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.prop)))) - cupicef.I - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (λ (B : cupicef.Univ cupicef.prop), - cupicef.prod - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.prop)) - (cupicef.{|rule|} - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.prop))) - cupicef.I - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), P y))) - (λ - ( - f : Π - ( - y : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (P y) - ), - cupicef.prod cupicef.prop - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.prop) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.prop)) - cupicef.I B - (λ (H : cupicef.Term cupicef.prop B), - cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.prop - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.prop) - cupicef.I A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x))))) - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - (cupicef.{|rule|} cupicef.prop cupicef.prop))) - cupicef.I - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (λ (B : cupicef.Univ cupicef.prop), - cupicef.prod - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), P y))) - (λ - ( - f : Π - ( - y : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (P y) - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), all A P)))) - cupicef.I - (λ (B : cupicef.Univ cupicef.prop), - λ - ( - f : Π - ( - y : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop (P y) - ), - λ (H : cupicef.Term cupicef.prop B), - λ - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), f x H); -symbol eq - : Π - (A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))), - Π - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - A - ), - Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), cupicef.Univ cupicef.prop; -symbol eq__refl - : Π - (A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))), - Π - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - A - ), cupicef.Term cupicef.prop (eq A x x); -rule cupicef.code_guarded? _ (eq__refl _ _) ↪ cupicef.guarded; -symbol match____eq - : Π (s : cupicef.Sort), - Π - ( - A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - ), - Π - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - Π - ( - P : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), cupicef.Term cupicef.prop (eq A x __) → cupicef.Univ s - ), - Π (case__eq__refl : cupicef.Term s (P x (eq__refl A x))), - Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - Π (x0 : cupicef.Term cupicef.prop (eq A x __)), - cupicef.Term s (P __ x0); -rule match____eq $s $A $x $P $case__eq__refl $x (eq__refl $A $x) ↪ $case__eq__refl; -rule match____eq $s $A $x - (λ __, - λ x, - cupicef.uncode $_.[] (cupicef.code (cupicef.cU $s'.[]) $P.[__; x])) ↪ -match____eq $s' $A $x (λ __, λ x, $P.[__; x]); -symbol eq__rect - : Π - (A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))), - Π - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - A - ), - Π - ( - P : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) - ), - Π - (f : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) (P x)), - Π - ( - y : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - Π (e : cupicef.Term cupicef.prop (eq A x y)), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) (P y) - ≔ λ - ( - A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - ), - λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - λ - ( - P : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) - ), - λ - ( - f : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) (P x) - ), - λ - ( - y : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - λ (e : cupicef.Term cupicef.prop (eq A x y)), - match____eq (cupicef.{|type|} (cupicef.s cupicef.z)) A x - (λ - ( - y0 : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - A - ), λ (__ : cupicef.Term cupicef.prop (eq A x y0)), P y0) - f y e; -symbol eq__ind - : Π - (A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))), - Π - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - A - ), - Π - ( - P : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), cupicef.Univ cupicef.prop - ), - Π (f : cupicef.Term cupicef.prop (P x)), - Π - ( - y : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - Π (e : cupicef.Term cupicef.prop (eq A x y)), - cupicef.Term cupicef.prop (P y) - ≔ λ - ( - A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - ), - λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - λ - ( - P : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), cupicef.Univ cupicef.prop - ), - λ (f : cupicef.Term cupicef.prop (P x)), - λ - ( - y : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - λ (e : cupicef.Term cupicef.prop (eq A x y)), - match____eq cupicef.prop A x - (λ - ( - y0 : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - A - ), λ (__ : cupicef.Term cupicef.prop (eq A x y0)), P y0) - f y e; -symbol eq__rec - : Π - (A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))), - Π - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - A - ), - Π - ( - P : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), cupicef.Univ cupicef.set - ), - Π (f : cupicef.Term cupicef.set (P x)), - Π - ( - y : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - Π (e : cupicef.Term cupicef.prop (eq A x y)), - cupicef.Term cupicef.set (P y) - ≔ λ - ( - A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - ), - λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - λ - ( - P : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), cupicef.Univ cupicef.set - ), - eq__rect A x - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - cupicef.univ cupicef.set - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P); -symbol let__f - : Π (A : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - Π (h1 : cupicef.Term cupicef.prop A), - Π - ( - h2 : Π (__ : cupicef.Term cupicef.prop A), - cupicef.Term cupicef.prop False - ), cupicef.Term cupicef.prop False - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - λ (h1 : cupicef.Term cupicef.prop A), - λ - ( - h2 : Π (__ : cupicef.Term cupicef.prop A), - cupicef.Term cupicef.prop False - ), h2 h1; -opaque symbol absurd - : Π (A : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - Π (__ : cupicef.Term cupicef.prop A), - Π (__0 : cupicef.Term cupicef.prop (not A)), - cupicef.Term cupicef.prop C - ≔ cupicef.cast - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.prop)))) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop)))) - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.prop))) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.prop)))) - cupicef.I - (cupicef.univ cupicef.prop (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (λ (A : cupicef.Univ cupicef.prop), - cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.prop)) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.prop))) - cupicef.I - (cupicef.univ cupicef.prop (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (λ (C : cupicef.Univ cupicef.prop), - cupicef.prod cupicef.prop - (cupicef.{|rule|} (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.prop) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.prop)) - cupicef.I A - (λ (h1 : cupicef.Term cupicef.prop A), - cupicef.prod (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.prop - (cupicef.{|rule|} - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.prop) - cupicef.I - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), False)) - (λ - ( - h2 : Π (__ : cupicef.Term cupicef.prop A), - cupicef.Term cupicef.prop False - ), C))))) - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop))) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop)))) - cupicef.I - (cupicef.univ cupicef.prop (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (λ (A : cupicef.Univ cupicef.prop), - cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop))) - cupicef.I - (cupicef.univ cupicef.prop (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (λ (C : cupicef.Univ cupicef.prop), - cupicef.prod cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (not A) (λ (__0 : cupicef.Term cupicef.prop (not A)), C))))) - cupicef.I - (λ (A : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - λ (h1 : cupicef.Term cupicef.prop A), - λ - ( - h2 : Π (__ : cupicef.Term cupicef.prop A), - cupicef.Term cupicef.prop False - ), - match____False cupicef.prop - (λ (__ : cupicef.Term cupicef.prop False), C) - (let__f A C h1 h2)); -symbol eq__sym - : Π - (A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))), - Π - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - A - ), - Π - ( - y : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - Π (__ : cupicef.Term cupicef.prop (eq A x y)), - cupicef.Term cupicef.prop (eq A y x) - ≔ λ - ( - A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - ), - λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - λ - ( - y : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - λ (H : cupicef.Term cupicef.prop (eq A x y)), - match____eq cupicef.prop A x - (λ - ( - y0 : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - λ (__ : cupicef.Term cupicef.prop (eq A x y0)), eq A y0 x) - (eq__refl A x) y H; -symbol eq__trans - : Π - (A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))), - Π - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - A - ), - Π - ( - y : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - Π - ( - z : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - Π (__ : cupicef.Term cupicef.prop (eq A x y)), - Π (__0 : cupicef.Term cupicef.prop (eq A y z)), - cupicef.Term cupicef.prop (eq A x z) - ≔ λ - ( - A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - ), - λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - λ - ( - y : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - λ - ( - z : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - λ (H : cupicef.Term cupicef.prop (eq A x y)), - λ (H0 : cupicef.Term cupicef.prop (eq A y z)), - match____eq cupicef.prop A y - (λ - ( - y0 : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - A - ), - λ (__ : cupicef.Term cupicef.prop (eq A y y0)), - eq A x y0) - H z H0; -symbol f__equal - : Π - (A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))), - Π - ( - B : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - ), - Π - ( - f : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) B - ), - Π - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - Π - ( - y : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - Π (__ : cupicef.Term cupicef.prop (eq A x y)), - cupicef.Term cupicef.prop (eq B (f x) (f y)) - ≔ λ - ( - A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - ), - λ - ( - B : cupicef.Univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - ), - λ - ( - f : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) B - ), - λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - λ - ( - y : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - λ (H : cupicef.Term cupicef.prop (eq A x y)), - match____eq cupicef.prop A x - (λ - ( - y0 : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - A - ), - λ (__ : cupicef.Term cupicef.prop (eq A x y0)), - eq B (f x) (f y0)) - (eq__refl B (f x)) y H; -opaque symbol not__eq__sym - : Π - (A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))), - Π - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - A - ), - Π - ( - y : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - Π (__ : cupicef.Term cupicef.prop (not (eq A x y))), - cupicef.Term cupicef.prop (not (eq A y x)) - ≔ λ - ( - A : cupicef.Univ (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - ), - λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - λ - ( - y : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) A - ), - cupicef.cast - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.prod cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I (not (eq A x y)) - (λ (h1 : cupicef.Term cupicef.prop (not (eq A x y))), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (eq A y x) - (λ (h2 : cupicef.Term cupicef.prop (eq A y x)), False))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (not (eq A x y)) - (λ (__ : cupicef.Term cupicef.prop (not (eq A x y))), - not (eq A y x))) - cupicef.I - (λ (h1 : cupicef.Term cupicef.prop (not (eq A x y))), - λ (h2 : cupicef.Term cupicef.prop (eq A y x)), - h1 - (match____eq cupicef.prop A y - (λ - ( - y0 : cupicef.Term - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - A - ), - λ (__ : cupicef.Term cupicef.prop (eq A y y0)), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (not (eq A y0 y)) - (λ - (h3 : cupicef.Term cupicef.prop (not (eq A y0 y))), - eq A y0 y)) - (λ (h3 : cupicef.Term cupicef.prop (not (eq A y y))), - eq__refl A y) - x h2 h1)); -symbol eq__ind__r - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - Π (__ : cupicef.Term cupicef.prop (P x)), - Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - __0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y x) - ), cupicef.Term cupicef.prop (P y) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - P : Π - ( - __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), cupicef.Univ cupicef.prop - ), - λ (H : cupicef.Term cupicef.prop (P x)), - λ - (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - H0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y x) - ), - eq__ind - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - y0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P y0) - H y - (eq__sym - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y x H0); -symbol eq__rec__r - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.set - ), - Π (__ : cupicef.Term cupicef.set (P x)), - Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - __0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y x) - ), cupicef.Term cupicef.set (P y) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - P : Π - ( - __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), cupicef.Univ cupicef.set - ), - λ (H : cupicef.Term cupicef.set (P x)), - λ - (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - H0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y x) - ), - eq__rec - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - y0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P y0) - H y - (eq__sym - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y x H0); -symbol eq__rect__r - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) - ), - Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) (P x) - ), - Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - __0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y x) - ), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) (P y) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - P : Π - ( - __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) - ), - λ - ( - H : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) (P x) - ), - λ - (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - H0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y x) - ), - eq__rect - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - y0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P y0) - H y - (eq__sym - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y x H0); -symbol rew__opp__r - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) - ), - Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - H : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - Π - ( - a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - (P y) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P y)) - (eq__rect - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P (eq__rect__r A y P a x H) y H) - a) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) - ), - λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - H : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - λ - ( - a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - (P y) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - y0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0) - ), - cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.prop - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.prop) - cupicef.I (P y0) - (λ - ( - a0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P y0) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P y0)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P (eq__rect__r A y0 P a0 x e) y0 e) - a0)) - (λ - ( - a0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) (P x) - ), - eq__refl - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P x)) - a0) - y H a; -symbol rew__opp__l - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) - ), - Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - H : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - Π - ( - a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P x)) - (eq__rect__r A y P - (eq__rect - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P a y H) - x H) - a) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) - ), - λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - H : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - λ - ( - a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - y0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P x)) - (eq__rect__r A y0 P - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P a y0 e) - x e) - a) - (eq__refl - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P x)) - a) - y H; -opaque symbol f__equal2 - : Π (A1 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (A2 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - f : Π - ( - __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A1 - ), - Π - ( - __0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A2 - ), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) B - ), - Π - (x1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A1), - Π - ( - y1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A1 - ), - Π - ( - x2 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A2 - ), - Π - ( - y2 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A2 - ), - Π - ( - __ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A1) - x1 y1) - ), - Π - ( - __0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A2) - x2 y2) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x1 x2) (f y1 y2)) - ≔ λ (A1 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (A2 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - f : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A1 - ), - Π - ( - __0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A2 - ), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) B - ), - λ - ( - x1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A1 - ), - λ - ( - y1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A1 - ), - λ - ( - x2 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A2 - ), - λ - ( - y2 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A2 - ), - λ - ( - H : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A1) - x1 y1) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A1) - x1 - (λ - ( - y : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A1 - ), - λ - ( - __ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A1) - x1 y) - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A2) - x2 y2) - (λ - ( - __0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A2) - x2 y2) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x1 x2) (f y y2))) - (λ - ( - H0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A2) - x2 y2) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A2) - x2 - (λ - ( - y : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A2 - ), - λ - ( - __ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A2) - x2 y) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x1 x2) (f x1 y)) - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x1 x2)) - y2 H0) - y1 H; -opaque symbol f__equal3 - : Π (A1 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (A2 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (A3 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - f : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A1 - ), - Π - ( - __0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A2 - ), - Π - ( - __1 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A3 - ), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - B - ), - Π - ( - x1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A1 - ), - Π - ( - y1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A1 - ), - Π - ( - x2 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A2 - ), - Π - ( - y2 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A2 - ), - Π - ( - x3 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A3 - ), - Π - ( - y3 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A3 - ), - Π - ( - __ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A1) - x1 y1) - ), - Π - ( - __0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A2) - x2 y2) - ), - Π - ( - __1 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A3) - x3 y3) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x1 x2 x3) (f y1 y2 y3)) - ≔ λ (A1 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (A2 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (A3 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - f : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A1 - ), - Π - ( - __0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A2 - ), - Π - ( - __1 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A3 - ), - cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) B - ), - λ - ( - x1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A1 - ), - λ - ( - y1 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A1 - ), - λ - ( - x2 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A2 - ), - λ - ( - y2 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A2 - ), - λ - ( - x3 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A3 - ), - λ - ( - y3 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A3 - ), - λ - ( - H : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A1) - x1 y1) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A1) - x1 - (λ - ( - y : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A1 - ), - λ - ( - __ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A1) - x1 y) - ), - cupicef.prod cupicef.prop - (cupicef.{|rule|} cupicef.prop - cupicef.prop) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop - cupicef.prop)) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A2) - x2 y2) - (λ - ( - __0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A2) - x2 y2) - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop - cupicef.prop) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A3) - x3 y3) - (λ - ( - __1 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A3) - x3 y3) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x1 x2 x3) (f y y2 y3)))) - (λ - ( - H0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A2) - x2 y2) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A2) - x2 - (λ - ( - y : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A2 - ), - λ - ( - __ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A2) - x2 y) - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop - cupicef.prop) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A3) - x3 y3) - (λ - ( - __0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A3) - x3 y3) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x1 x2 x3) (f x1 y y3))) - (λ - ( - H1 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A3) - x3 y3) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A3) - x3 - (λ - ( - y : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A3 - ), - λ - ( - __ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A3) - x3 y) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x1 x2 x3) (f x1 x2 y)) - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x1 x2 x3)) - y3 H1) - y2 H0) - y1 H; -opaque symbol f__equal4 - : Π (A1 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (A2 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (A3 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (A4 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - f : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A1 - ), - Π - ( - __0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A2 - ), - Π - ( - __1 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A3 - ), - Π - ( - __2 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A4 - ), - cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) B - ), - Π - ( - x1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A1 - ), - Π - ( - y1 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A1 - ), - Π - ( - x2 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A2 - ), - Π - ( - y2 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A2 - ), - Π - ( - x3 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A3 - ), - Π - ( - y3 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A3 - ), - Π - ( - x4 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A4 - ), - Π - ( - y4 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A4 - ), - Π - ( - __ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A1) - x1 y1) - ), - Π - ( - __0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A2) - x2 y2) - ), - Π - ( - __1 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A3) - x3 y3) - ), - Π - ( - __2 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A4) - x4 y4) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x1 x2 x3 x4) (f y1 y2 y3 y4)) - ≔ λ (A1 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (A2 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (A3 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (A4 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - f : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A1 - ), - Π - ( - __0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A2 - ), - Π - ( - __1 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A3 - ), - Π - ( - __2 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A4 - ), - cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) B - ), - λ - ( - x1 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A1 - ), - λ - ( - y1 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A1 - ), - λ - ( - x2 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A2 - ), - λ - ( - y2 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A2 - ), - λ - ( - x3 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A3 - ), - λ - ( - y3 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A3 - ), - λ - ( - x4 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A4 - ), - λ - ( - y4 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A4 - ), - λ - ( - H : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A1) - x1 y1) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A1) - x1 - (λ - ( - y : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A1 - ), - λ - ( - __ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A1) - x1 y) - ), - cupicef.prod cupicef.prop - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop - cupicef.prop)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop - cupicef.prop))) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A2) - x2 y2) - (λ - ( - __0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A2) - x2 y2) - ), - cupicef.prod cupicef.prop - (cupicef.{|rule|} cupicef.prop - cupicef.prop) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop - cupicef.prop)) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A3) - x3 y3) - (λ - ( - __1 : cupicef.Term - cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A3) - x3 y3) - ), - cupicef.prod cupicef.prop - cupicef.prop - (cupicef.{|rule|} cupicef.prop - cupicef.prop) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A4) - x4 y4) - (λ - ( - __2 : cupicef.Term - cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A4) - x4 y4) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x1 x2 x3 x4) - (f y y2 y3 y4))))) - (λ - ( - H0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A2) - x2 y2) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A2) - x2 - (λ - ( - y : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A2 - ), - λ - ( - __ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A2) - x2 y) - ), - cupicef.prod cupicef.prop - (cupicef.{|rule|} cupicef.prop - cupicef.prop) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop - cupicef.prop)) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A3) - x3 y3) - (λ - ( - __0 : cupicef.Term - cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A3) - x3 y3) - ), - cupicef.prod cupicef.prop - cupicef.prop - (cupicef.{|rule|} cupicef.prop - cupicef.prop) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A4) - x4 y4) - (λ - ( - __1 : cupicef.Term - cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A4) - x4 y4) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x1 x2 x3 x4) - (f x1 y y3 y4)))) - (λ - ( - H1 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A3) - x3 y3) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A3) - x3 - (λ - ( - y : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A3 - ), - λ - ( - __ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A3) - x3 y) - ), - cupicef.prod cupicef.prop - cupicef.prop - (cupicef.{|rule|} cupicef.prop - cupicef.prop) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A4) - x4 y4) - (λ - ( - __0 : cupicef.Term - cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A4) - x4 y4) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x1 x2 x3 x4) - (f x1 x2 y y4))) - (λ - ( - H2 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A4) - x4 y4) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A4) - x4 - (λ - ( - y : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A4 - ), - λ - ( - __ : cupicef.Term - cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A4) - x4 y) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x1 x2 x3 x4) - (f x1 x2 x3 y)) - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x1 x2 x3 x4)) - y4 H2) - y3 H1) - y2 H0) - y1 H; -opaque symbol f__equal5 - : Π (A1 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (A2 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (A3 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (A4 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (A5 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - f : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A1 - ), - Π - ( - __0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A2 - ), - Π - ( - __1 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A3 - ), - Π - ( - __2 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A4 - ), - Π - ( - __3 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A5 - ), - cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) B - ), - Π - ( - x1 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A1 - ), - Π - ( - y1 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A1 - ), - Π - ( - x2 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A2 - ), - Π - ( - y2 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A2 - ), - Π - ( - x3 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A3 - ), - Π - ( - y3 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A3 - ), - Π - ( - x4 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A4 - ), - Π - ( - y4 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A4 - ), - Π - ( - x5 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A5 - ), - Π - ( - y5 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A5 - ), - Π - ( - __ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A1) - x1 y1) - ), - Π - ( - __0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A2) - x2 y2) - ), - Π - ( - __1 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A3) - x3 y3) - ), - Π - ( - __2 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A4) - x4 y4) - ), - Π - ( - __3 : cupicef.Term - cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A5) - x5 y5) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x1 x2 x3 x4 x5) - (f y1 y2 y3 y4 y5)) - ≔ λ (A1 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (A2 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (A3 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (A4 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - (A5 : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - f : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A1 - ), - Π - ( - __0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A2 - ), - Π - ( - __1 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A3 - ), - Π - ( - __2 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A4 - ), - Π - ( - __3 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A5 - ), - cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) B - ), - λ - ( - x1 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A1 - ), - λ - ( - y1 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A1 - ), - λ - ( - x2 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A2 - ), - λ - ( - y2 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A2 - ), - λ - ( - x3 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A3 - ), - λ - ( - y3 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A3 - ), - λ - ( - x4 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A4 - ), - λ - ( - y4 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A4 - ), - λ - ( - x5 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A5 - ), - λ - ( - y5 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A5 - ), - λ - ( - H : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A1) - x1 y1) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A1) - x1 - (λ - ( - y : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A1 - ), - λ - ( - __ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A1) - x1 y) - ), - cupicef.prod cupicef.prop - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} - cupicef.prop - (cupicef.{|rule|} - cupicef.prop cupicef.prop))) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} - cupicef.prop - (cupicef.{|rule|} - cupicef.prop - (cupicef.{|rule|} - cupicef.prop cupicef.prop)))) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A2) - x2 y2) - (λ - ( - __0 : cupicef.Term - cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A2) - x2 y2) - ), - cupicef.prod cupicef.prop - (cupicef.{|rule|} - cupicef.prop - (cupicef.{|rule|} - cupicef.prop cupicef.prop)) - (cupicef.{|rule|} - cupicef.prop - (cupicef.{|rule|} - cupicef.prop - (cupicef.{|rule|} - cupicef.prop - cupicef.prop))) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A3) - x3 y3) - (λ - ( - __1 : cupicef.Term - cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A3) - x3 y3) - ), - cupicef.prod - cupicef.prop - (cupicef.{|rule|} - cupicef.prop - cupicef.prop) - (cupicef.{|rule|} - cupicef.prop - (cupicef.{|rule|} - cupicef.prop - cupicef.prop)) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A4) - x4 y4) - (λ - ( - __2 : cupicef.Term - cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - ( - cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A4) - x4 y4) - ), - cupicef.prod - cupicef.prop - cupicef.prop - (cupicef.{|rule|} - cupicef.prop - cupicef.prop) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A5) - x5 y5) - (λ - ( - __3 : cupicef.Term - cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - ( - cupicef.{|type|} - ( - cupicef.s - cupicef.z))) - (cupicef.axiom - ( - cupicef.{|type|} - ( - cupicef.s - ( - cupicef.s - cupicef.z)))) - (cupicef.univ - ( - cupicef.{|type|} - ( - cupicef.s - cupicef.z)) - ( - cupicef.axiom - ( - cupicef.{|type|} - ( - cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - ( - cupicef.{|type|} - ( - cupicef.s - ( - cupicef.s - cupicef.z))) - ( - cupicef.axiom - ( - cupicef.{|type|} - ( - cupicef.s - ( - cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I - A5) - x5 y5) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x1 x2 x3 x4 x5) - (f y y2 y3 y4 y5)))))) - (λ - ( - H0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A2) - x2 y2) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A2) - x2 - (λ - ( - y : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A2 - ), - λ - ( - __ : cupicef.Term - cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A2) - x2 y) - ), - cupicef.prod cupicef.prop - (cupicef.{|rule|} - cupicef.prop - (cupicef.{|rule|} - cupicef.prop cupicef.prop)) - (cupicef.{|rule|} - cupicef.prop - (cupicef.{|rule|} - cupicef.prop - (cupicef.{|rule|} - cupicef.prop - cupicef.prop))) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A3) - x3 y3) - (λ - ( - __0 : cupicef.Term - cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A3) - x3 y3) - ), - cupicef.prod - cupicef.prop - (cupicef.{|rule|} - cupicef.prop - cupicef.prop) - (cupicef.{|rule|} - cupicef.prop - (cupicef.{|rule|} - cupicef.prop - cupicef.prop)) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A4) - x4 y4) - (λ - ( - __1 : cupicef.Term - cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - ( - cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A4) - x4 y4) - ), - cupicef.prod - cupicef.prop - cupicef.prop - (cupicef.{|rule|} - cupicef.prop - cupicef.prop) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A5) - x5 y5) - (λ - ( - __2 : cupicef.Term - cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - ( - cupicef.{|type|} - ( - cupicef.s - cupicef.z))) - (cupicef.axiom - ( - cupicef.{|type|} - ( - cupicef.s - ( - cupicef.s - cupicef.z)))) - (cupicef.univ - ( - cupicef.{|type|} - ( - cupicef.s - cupicef.z)) - ( - cupicef.axiom - ( - cupicef.{|type|} - ( - cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - ( - cupicef.{|type|} - ( - cupicef.s - ( - cupicef.s - cupicef.z))) - ( - cupicef.axiom - ( - cupicef.{|type|} - ( - cupicef.s - ( - cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I - A5) - x5 y5) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x1 x2 x3 x4 x5) - (f x1 y y3 y4 y5))))) - (λ - ( - H1 : cupicef.Term - cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A3) - x3 y3) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A3) - x3 - (λ - ( - y : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A3 - ), - λ - ( - __ : cupicef.Term - cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A3) - x3 y) - ), - cupicef.prod - cupicef.prop - (cupicef.{|rule|} - cupicef.prop - cupicef.prop) - (cupicef.{|rule|} - cupicef.prop - (cupicef.{|rule|} - cupicef.prop - cupicef.prop)) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A4) - x4 y4) - (λ - ( - __0 : cupicef.Term - cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - ( - cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A4) - x4 y4) - ), - cupicef.prod - cupicef.prop - cupicef.prop - (cupicef.{|rule|} - cupicef.prop - cupicef.prop) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A5) - x5 y5) - (λ - ( - __1 : cupicef.Term - cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - ( - cupicef.{|type|} - ( - cupicef.s - cupicef.z))) - (cupicef.axiom - ( - cupicef.{|type|} - ( - cupicef.s - ( - cupicef.s - cupicef.z)))) - (cupicef.univ - ( - cupicef.{|type|} - ( - cupicef.s - cupicef.z)) - ( - cupicef.axiom - ( - cupicef.{|type|} - ( - cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - ( - cupicef.{|type|} - ( - cupicef.s - ( - cupicef.s - cupicef.z))) - ( - cupicef.axiom - ( - cupicef.{|type|} - ( - cupicef.s - ( - cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I - A5) - x5 y5) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x1 x2 x3 x4 x5) - (f x1 x2 y y4 y5)))) - (λ - ( - H2 : cupicef.Term - cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A4) - x4 y4) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A4) - x4 - (λ - ( - y : cupicef.Term - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - A4 - ), - λ - ( - __ : cupicef.Term - cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A4) - x4 y) - ), - cupicef.prod - cupicef.prop - cupicef.prop - (cupicef.{|rule|} - cupicef.prop - cupicef.prop) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A5) - x5 y5) - (λ - ( - __0 : cupicef.Term - cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - ( - cupicef.{|type|} - ( - cupicef.s - cupicef.z))) - (cupicef.axiom - ( - cupicef.{|type|} - ( - cupicef.s - ( - cupicef.s - cupicef.z)))) - (cupicef.univ - ( - cupicef.{|type|} - ( - cupicef.s - cupicef.z)) - ( - cupicef.axiom - ( - cupicef.{|type|} - ( - cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - ( - cupicef.{|type|} - ( - cupicef.s - ( - cupicef.s - cupicef.z))) - ( - cupicef.axiom - ( - cupicef.{|type|} - ( - cupicef.s - ( - cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I - A5) - x5 y5) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x1 x2 x3 x4 x5) - (f x1 x2 x3 y y5))) - (λ - ( - H3 : cupicef.Term - cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A5) - x5 y5) - ), - match____eq - cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A5) - x5 - (λ - ( - y : cupicef.Term - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - A5 - ), - λ - ( - __ : cupicef.Term - cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - ( - cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - ( - cupicef.s - ( - cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - ( - cupicef.s - cupicef.z)) - (cupicef.axiom - ( - cupicef.{|type|} - ( - cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - ( - cupicef.s - ( - cupicef.s - cupicef.z))) - (cupicef.axiom - ( - cupicef.{|type|} - ( - cupicef.s - ( - cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I - A5) - x5 y) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x1 x2 x3 x4 x5) - (f x1 x2 x3 x4 y)) - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x1 x2 x3 x4 x5)) - y5 H3) - y4 H2) - y3 H1) - y2 H0) - y1 H; -symbol f__equal__compose - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (C : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (b : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - f : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) B - ), - Π - ( - g : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) B - ), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - C - ), - Π - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a b) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I C) - (g (f a)) (g (f b)))) - (f__equal - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I C) - g (f a) (f b) - (f__equal - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - f a b e)) - (f__equal - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I C) - (λ - ( - a0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), g (f a0)) - a b e)) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (C : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - (b : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - f : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - B - ), - λ - ( - g : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) B - ), - cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) C - ), - λ - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a b) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a - (λ - ( - y : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - e0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a y) - ), - eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I C) - (g (f a)) (g (f y)))) - (f__equal - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I C) - g (f a) (f y) - (f__equal - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - f a y e0)) - (f__equal - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I C) - (λ - ( - a0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), g (f a0)) - a y e0)) - (eq__refl - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I C) - (g (f a)) (g (f a)))) - (f__equal - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I C) - (λ - ( - a0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), g (f a0)) - a a - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a))) - b e; -symbol eq__trans__refl__l - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y)) - (eq__trans - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x y - (eq__refl - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x) - e) - e) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - y0 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - e0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0) - ), - eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0)) - (eq__trans - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x y0 - (eq__refl - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x) - e0) - e0) - (eq__refl - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x)) - (eq__refl - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x)) - y e; -symbol eq__trans__refl__r - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y)) - (eq__trans - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y y e - (eq__refl - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y)) - e) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - y0 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - e0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0) - ), - eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0)) - (eq__trans - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0 y0 e0 - (eq__refl - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y0)) - e0) - (eq__refl - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x)) - (eq__refl - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x)) - y e; -symbol eq__sym__involutive - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y)) - (eq__sym - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y x - (eq__sym - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y e)) - e) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - y0 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - e0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0) - ), - eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0)) - (eq__sym - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y0 x - (eq__sym - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0 e0)) - e0) - (eq__refl - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x)) - (eq__refl - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x)) - y e; -symbol eq__trans__sym__inv__l - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y y)) - (eq__trans - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y x y - (eq__sym - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y e) - e) - (eq__refl - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y)) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - y0 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - e0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0) - ), - eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y0 y0)) - (eq__trans - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y0 x y0 - (eq__sym - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0 e0) - e0) - (eq__refl - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y0)) - (eq__refl - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x)) - (eq__refl - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x)) - y e; -symbol eq__trans__sym__inv__r - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x)) - (eq__trans - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y x e - (eq__sym - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y e)) - (eq__refl - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x)) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - y0 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - e0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0) - ), - eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x)) - (eq__trans - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0 x e0 - (eq__sym - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0 e0)) - (eq__refl - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x)) - (eq__refl - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x)) - (eq__refl - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x)) - y e; -symbol eq__trans__assoc - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (z : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (t : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - Π - ( - e_27_ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y z) - ), - Π - ( - e_27__27_ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - z t) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x t)) - (eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y t e - (eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y z t e_27_ e_27__27_)) - (eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x z t - (eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y z e e_27_) - e_27__27_)) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ (z : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - (t : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - λ - ( - e_27_ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y z) - ), - λ - ( - e_27__27_ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - z t) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - z - (λ - ( - y0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - e0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - z y0) - ), - eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0)) - (eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y y0 e - (eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y z y0 e_27_ e0)) - (eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x z y0 - (eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y z e e_27_) - e0)) - (eq__refl - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x z)) - (eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x z z - (eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y z e e_27_) - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - z))) - t e_27__27_; -symbol let__e - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - f : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π - ( - Hf : Π - (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a (f a)) - ), - Π (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a (f a)) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - f : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - Hf : Π - ( - a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a (f a)) - ), - λ (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Hf a; -symbol let__a0 - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - f : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π - ( - Hf : Π - (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a (f a)) - ), - Π (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - f : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - Hf : Π - ( - a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a (f a)) - ), - λ (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - f a; -symbol let__e0 - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - f : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π - ( - Hf : Π - (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a (f a)) - ), - Π (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a (f a)) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - f : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - Hf : Π - ( - a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a (f a)) - ), - λ (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Hf a; -symbol let__a1 - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - f : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π - ( - Hf : Π - (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a (f a)) - ), - Π (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - f : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - Hf : Π - ( - a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a (f a)) - ), - λ (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - f a; -symbol eq__id__comm__l - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - f : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π - ( - Hf : Π - (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a (f a)) - ), - Π (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Term cupicef.prop - (eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) (f (f a)))) - (f__equal - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - f a (f a) (Hf a)) - (Hf (f a))) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - f : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - Hf : Π - ( - a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a (f a)) - ), - λ (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.cast cupicef.prop cupicef.prop - ((λ - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) (f a)) - ), - eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) (f (f a)))) - (match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a - (λ - ( - y : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - __ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a y) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) (f y)) - e (f a) (Hf a)) - (Hf (f a))) - (eq__refl - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a))) - (eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) (f (f a)))) - (f__equal - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - f a (f a) (Hf a)) - (Hf (f a))) - cupicef.I - (eq__ind - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) (f a))) - (eq__trans - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) a (f a) - (eq__sym - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a (f a) (Hf a)) - (Hf a)) - (λ - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) (f a)) - ), - eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) (f (f a)))) - (match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a - (λ - ( - y : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - __ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a y) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) (f y)) - e (f a) (Hf a)) - (Hf (f a))) - (match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a - (λ - ( - y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - λ - ( - e0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a y) - ), - eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y (f y))) - (match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a - (λ - ( - y0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - __ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a y0) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y (f y0)) - (eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y a (f a) - (eq__sym - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a y e0) - (Hf a)) - y e0) - (Hf y)) - (match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a - (λ - ( - y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - λ - ( - e0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a y) - ), - eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a y)) - (eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a a y - (eq__sym - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a a - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a)) - e0) - e0) - (eq__refl - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a a)) - (eq__refl - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a)) - (f a) (let__e0 A f Hf a)) - (f a) (let__e A f Hf a)) - (eq__refl - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a)) - (eq__trans__sym__inv__l A a (f a) (Hf a))); -symbol let__Hfsymf - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - f : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π - ( - Hf : Π - (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) a) - ), - Π (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - (a0 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - a0 (f a0)) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - f : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - Hf : Π - ( - a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) a) - ), - λ (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - (a0 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - eq__sym - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a0) a0 (Hf a0); -symbol let__e1 - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - f : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π - ( - Hf : Π - (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) a) - ), - Π (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Term cupicef.prop - (eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f (f a)) (f (f (f a))))) - (f__equal - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - f (f a) (f (f a)) (let__Hfsymf A f Hf a (f a))) - (let__Hfsymf A f Hf a (f (f a)))) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - f : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - Hf : Π - ( - a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) a) - ), - λ (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - eq__id__comm__l A f (let__Hfsymf A f Hf a) (f a); -symbol let__e2 - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - f : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π - ( - Hf : Π - (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) a) - ), - Π (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f (f a)) (f (f (f a)))) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - f : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - Hf : Π - ( - a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) a) - ), - λ (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - let__Hfsymf A f Hf a (f (f a)); -symbol let__e3 - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - f : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π - ( - Hf : Π - (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) a) - ), - Π (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Term cupicef.prop - (eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) (f (f a)))) - (f__equal - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - f a (f a) (let__Hfsymf A f Hf a a)) - (let__Hfsymf A f Hf a (f a))) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - f : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - Hf : Π - ( - a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) a) - ), - λ (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - eq__id__comm__l A f (let__Hfsymf A f Hf a) a; -symbol let__e4 - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - f : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π - ( - Hf : Π - (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) a) - ), - Π (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) (f (f a))) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - f : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - Hf : Π - ( - a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) a) - ), - λ (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - let__Hfsymf A f Hf a (f a); -symbol let__e5 - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - f : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π - ( - Hf : Π - (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) a) - ), - Π (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) a) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - f : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - Hf : Π - ( - a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) a) - ), - λ (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Hf a; -symbol eq__id__comm__r - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - f : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π - ( - Hf : Π - (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) a) - ), - Π (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Term cupicef.prop - (eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f (f a)) (f a))) - (f__equal - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - f (f a) a (Hf a)) - (Hf (f a))) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - f : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - Hf : Π - ( - a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) a) - ), - λ (a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.cast cupicef.prop cupicef.prop - ((λ - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f (f a)) (f (f a))) - ), - eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f (f a)) (f a))) - (match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f a) - (λ - ( - y : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - __ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - 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(cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f (f (f a))) (f (f a)))) - (Hf (f (f a))) - (λ - ( - e1 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f (f (f a))) (f (f a))) - ), - eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f (f (f a))) (f (f a)))) - e1 (Hf (f (f a)))) - (eq__refl - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f (f (f a))) (f (f a)))) - (Hf (f (f a)))) - (eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f (f (f a))) (f (f (f a))) (f (f a)) - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f (f (f a)))) - (Hf (f (f a)))) - (eq__trans__refl__l A (f (f (f a))) (f (f a)) - (Hf (f (f a)))))) - a (let__e5 A f Hf a))) - (let__Hfsymf A f Hf a (f a)) (let__e3 A f Hf a)) - (let__Hfsymf A f Hf a (f (f a))) (let__e1 A f Hf a))) - (eq__refl - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (f (f a))) - (eq__trans__sym__inv__l A (f (f (f a))) (f (f a)) - (Hf (f (f a))))); -opaque symbol eq__refl__map__distr - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - f : Π - ( - __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) B - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x) (f x))) - (f__equal - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - f x x - (eq__refl - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x)) - (eq__refl - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x))) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - f : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) B - ), - eq__refl - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x) (f x))) - (eq__refl - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x)); -symbol eq__trans__map__distr - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (z : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - f : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) B - ), - Π - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - Π - ( - e_27_ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y z) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x) (f z))) - (f__equal - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - f x z - (eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y z e e_27_)) - (eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x) (f y) (f z) - (f__equal - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - f x y e) - (f__equal - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - f y z e_27_))) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - (z : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - f : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - B - ), - λ - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - λ - ( - e_27_ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y z) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y - (λ - ( - y0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - e0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y y0) - ), - eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x) (f y0))) - (f__equal - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - f x y0 - (eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y y0 e e0)) - (eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x) (f y) (f y0) - (f__equal - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - f x y e) - (f__equal - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - f y y0 e0))) - (eq__refl - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x) (f y))) - (eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x) (f y) (f y) - (f__equal - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - f x y e) - (f__equal - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - f y y - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y)))) - z e_27_; -symbol eq__sym__map__distr - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - f : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) B - ), - Π - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f y) (f x))) - (eq__sym - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x) (f y) - (f__equal - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - f x y e)) - (f__equal - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - f y x - (eq__sym - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y e))) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - f : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) B - ), - λ - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - y0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - e0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0) - ), - eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f y0) (f x))) - (eq__sym - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x) (f y0) - (f__equal - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - f x y0 e0)) - (f__equal - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - f y0 x - (eq__sym - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0 e0))) - (eq__refl - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - (f x) (f x))) - (f__equal - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I B) - f x x - (eq__sym - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x)))) - y e; -symbol eq__trans__sym__distr - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (z : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - Π - ( - e_27_ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y z) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - z x)) - (eq__sym - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x z - (eq__trans - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y z e e_27_)) - (eq__trans - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - z y x - (eq__sym - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y z e_27_) - (eq__sym - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y e))) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ (z : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - y0 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - λ - ( - e0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0) - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y0 z) - (λ - ( - e_27_ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y0 z) - ), - eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - z x)) - (eq__sym - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x z - (eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0 z e0 e_27_)) - (eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - z y0 x - (eq__sym - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y0 z e_27_) - (eq__sym - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0 e0)))) - (λ - ( - e_27_ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x z) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - y0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - e0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0) - ), - eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y0 x)) - (eq__sym - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0 - (eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x y0 - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x) - e0)) - (eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y0 x x - (eq__sym - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0 e0) - (eq__sym - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x)))) - (eq__refl - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x)) - (eq__trans - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x x - (eq__sym - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x)) - (eq__sym - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x)))) - z e_27_) - y e; -opaque symbol eq__trans__rew__distr - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) - ), - Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (z : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - Π - ( - e_27_ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y z) - ), - Π - ( - k : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P z)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P k z - (eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y z e e_27_)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y P - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P k y e) - z e_27_)) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) - ), - λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - (z : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - y0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - e0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0) - ), - cupicef.prod cupicef.prop - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.prop) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.prop)) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y0 z) - (λ - ( - e_27_ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y0 z) - ), - cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.prop - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.prop) - cupicef.I (P x) - (λ - ( - k : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P z)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P k z - (eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0 z e0 e_27_)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - y0 P - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P k y0 e0) - z e_27_)))) - (λ - ( - e_27_ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x z) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - y0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - e0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0) - ), - cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.prop - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.prop) - cupicef.I (P x) - (λ - ( - k : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P y0)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P k y0 - (eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x y0 - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x) - e0)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P k x - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x)) - y0 e0))) - (λ - ( - k : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) ( - P x) - ), - eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P x)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P k x - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x)) - x - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x))) - z e_27_) - y e; -opaque symbol rew__const - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (P : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - Π - (k : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) P), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I P) - (eq__rect - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P) - k y e) - k) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (P : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - y0 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - λ - ( - e0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0) - ), - cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.prop - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.prop) - cupicef.I P - (λ - ( - k : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) P - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I P) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P) - k y0 e0) - k)) - (λ - ( - k : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - P - ), - eq__refl - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I P) - k) - y e; -symbol subrelation - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - R : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π - ( - __0 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - B - ), cupicef.Univ cupicef.prop - ), - Π - ( - R_27_ : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π - ( - __0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) B - ), cupicef.Univ cupicef.prop - ), cupicef.Univ cupicef.prop - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - R : Π - ( - __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - Π - ( - __0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) B - ), cupicef.Univ cupicef.prop - ), - λ - ( - R_27_ : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π - ( - __0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) B - ), cupicef.Univ cupicef.prop - ), - cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop))) - cupicef.I A - (λ - (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I B - (λ - ( - y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - B - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (R x y) - (λ (__ : cupicef.Term cupicef.prop (R x y)), R_27_ x y))); -symbol unique - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ cupicef.prop - ), - Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ cupicef.prop - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - and (P x) - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x_27_ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (P x_27_) - (λ (__ : cupicef.Term cupicef.prop (P x_27_)), - eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x_27_))); -symbol uniqueness - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ cupicef.prop - ), cupicef.Univ cupicef.prop - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop))) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop)))) - cupicef.I A - (λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop))) - cupicef.I A - (λ - (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.prod cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I (P x) - (λ (__ : cupicef.Term cupicef.prop (P x)), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (P y) - (λ (__0 : cupicef.Term cupicef.prop (P y)), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y)))); -opaque symbol unique__existence - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ cupicef.prop - ), - cupicef.Term cupicef.prop - (iff - (and - (ex A - (λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - P x)) - (uniqueness A P)) - (ex A - (unique A - (λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - P x)))) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (and - (ex A - (λ - (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - P x)) - (uniqueness A P)) - (λ - ( - __ : cupicef.Term cupicef.prop - (and - (ex A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x)) - (uniqueness A P)) - ), - ex A - (unique A - (λ - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x)))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (ex A - (unique A - (λ - (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - P x))) - (λ - ( - __ : cupicef.Term cupicef.prop - (ex A - (unique A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x))) - ), - and - (ex A - (λ - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x)) - (uniqueness A P))) - (λ - ( - H : cupicef.Term cupicef.prop - (and - (ex A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x)) - (uniqueness A P)) - ), - match____and cupicef.prop - (ex A - (λ - (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x)) - (uniqueness A P) - (λ - ( - __ : cupicef.Term cupicef.prop - (and - (ex A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x)) - (uniqueness A P)) - ), - ex A - (unique A - (λ - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), P x))) - (λ - ( - H0 : cupicef.Term cupicef.prop - (ex A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x)) - ), - match____ex cupicef.prop A - (λ - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), P x) - (λ - ( - __ : cupicef.Term cupicef.prop - (ex A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x)) - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (uniqueness A P) - (λ (__0 : cupicef.Term cupicef.prop (uniqueness A P)), - ex A - (unique A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x)))) - (λ - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - λ (Hx : cupicef.Term cupicef.prop (P x)), - λ (Huni : cupicef.Term cupicef.prop (uniqueness A P)), - ex__intro A - (unique A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x0)) - x - (cupicef.cast cupicef.prop cupicef.prop - (and (P x) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x_27_ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I (P x_27_) - (λ (__ : cupicef.Term cupicef.prop (P x_27_)), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x_27_)))) - (unique A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x0) - x) - cupicef.I - (conj (P x) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x_27_ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I (P x_27_) - (λ (__ : cupicef.Term cupicef.prop (P x_27_)), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x_27_))) - Hx - (λ - ( - x_27_ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - λ (H1 : cupicef.Term cupicef.prop (P x_27_)), - Huni x x_27_ Hx H1)))) - H0) - H) - (λ - ( - H : cupicef.Term cupicef.prop - (ex A - (unique A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x))) - ), - match____ex cupicef.prop A - (unique A - (λ - (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x)) - (λ - ( - __ : cupicef.Term cupicef.prop - (ex A - (unique A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x))) - ), - and - (ex A - (λ - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), P x)) - (uniqueness A P)) - (λ - (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - H0 : cupicef.Term cupicef.prop - (unique A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x0) - x) - ), - match____and cupicef.prop - ((λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x0) - x) - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x_27_ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - ((λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x0) - x_27_) - (λ - ( - __ : cupicef.Term cupicef.prop - ((λ - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), P x0) - x_27_) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x_27_))) - (λ - ( - __ : cupicef.Term cupicef.prop - (and (P x) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x_27_ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I (P x_27_) - (λ - (__ : cupicef.Term cupicef.prop (P x_27_)), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x_27_)))) - ), - and - (ex A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x0)) - (uniqueness A P)) - (λ (Hx : cupicef.Term cupicef.prop (P x)), - λ - ( - Huni : Π - ( - x_27_ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - Π - (__ : cupicef.Term cupicef.prop (P x_27_)), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x_27_) - ), - conj - (ex A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x0)) - (uniqueness A P) - (ex__intro A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x0) - x Hx) - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop)))) - cupicef.prop - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop)))) - cupicef.I A - (λ - ( - x_27_ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop))) - cupicef.I A - (λ - ( - x_27__27_ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.prod cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I (P x_27_) - (λ - ( - Hx_27_ : cupicef.Term cupicef.prop - (P x_27_) - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I (P x_27__27_) - (λ - ( - Hx_27__27_ : cupicef.Term cupicef.prop - (P x_27__27_) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x_27_ x_27__27_))))) - (uniqueness A P) cupicef.I - (λ - ( - x_27_ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - x_27__27_ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - λ - (Hx_27_ : cupicef.Term cupicef.prop (P x_27_)), - λ - ( - Hx_27__27_ : cupicef.Term cupicef.prop - (P x_27__27_) - ), - eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x_27_ x x_27__27_ - (eq__sym - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x_27_ (Huni x_27_ Hx_27_)) - (Huni x_27__27_ Hx_27__27_)))) - H0) - H); -symbol let__H - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ cupicef.prop - ), - Π - ( - H : cupicef.Term cupicef.prop - (ex A - (unique A - (λ - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), P x))) - ), - Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - H0 : cupicef.Term cupicef.prop - (unique A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x0) - x) - ), - Π (Hp : cupicef.Term cupicef.prop (P x)), - Π - ( - Huniq : Π - ( - x_27_ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - Π (__ : cupicef.Term cupicef.prop (P x_27_)), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x_27_) - ), - Π - ( - Q : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - Π - ( - H1 : cupicef.Term cupicef.prop - (ex A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), and (P x0) (Q x0))) - ), - Π - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π - (H2 : cupicef.Term cupicef.prop (and (P x0) (Q x0)) - ), - Π (HPx0 : cupicef.Term cupicef.prop (P x0)), - Π (HQx0 : cupicef.Term cupicef.prop (Q x0)), - Π - ( - x1 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - Π (HPx1 : cupicef.Term cupicef.prop (P x1)), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x0 x1) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - λ - ( - H : cupicef.Term cupicef.prop - (ex A - (unique A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x))) - ), - λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - H0 : cupicef.Term cupicef.prop - (unique A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x0) - x) - ), - λ (Hp : cupicef.Term cupicef.prop (P x)), - λ - ( - Huniq : Π - ( - x_27_ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - Π (__ : cupicef.Term cupicef.prop (P x_27_)), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x_27_) - ), - λ - ( - Q : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - λ - ( - H1 : cupicef.Term cupicef.prop - (ex A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), and (P x0) (Q x0))) - ), - λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - H2 : cupicef.Term cupicef.prop - (and (P x0) (Q x0)) - ), - λ (HPx0 : cupicef.Term cupicef.prop (P x0)), - λ (HQx0 : cupicef.Term cupicef.prop (Q x0)), - λ - ( - x1 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - λ - (HPx1 : cupicef.Term cupicef.prop (P x1)), - eq__trans - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x0 x x1 - (eq__sym - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x0 (Huniq x0 HPx0)) - (Huniq x1 HPx1); -opaque symbol forall__exists__unique__domain__coincide - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ cupicef.prop - ), - Π - ( - __ : cupicef.Term cupicef.prop - (ex A - (unique A - (λ - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), P x))) - ), - Π - ( - Q : Π - ( - __0 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), cupicef.Univ cupicef.prop - ), - cupicef.Term cupicef.prop - (iff - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I ( - P x) (λ (__0 : cupicef.Term cupicef.prop (P x)), Q x))) - (ex A - (λ - (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), and (P x) (Q x)))) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - λ - ( - H : cupicef.Term cupicef.prop - (ex A - (unique A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x))) - ), - match____ex cupicef.prop A - (unique A - (λ - (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - P x)) - (λ - ( - __ : cupicef.Term cupicef.prop - (ex A - (unique A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x))) - ), - cupicef.prod - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.prop - (cupicef.{|rule|} - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.prop) - cupicef.I - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I)) - (λ - ( - Q : Π - ( - __0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - iff - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (P x) (λ (__0 : cupicef.Term cupicef.prop (P x)), Q x))) - (ex A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), and (P x) (Q x))))) - (λ - (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - H0 : cupicef.Term cupicef.prop - (unique A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x0) - x) - ), - match____and cupicef.prop - ((λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x0) - x) - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x_27_ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - ((λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x0) - x_27_) - (λ - ( - __ : cupicef.Term cupicef.prop - ((λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), P x0) - x_27_) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x_27_))) - (λ - ( - __ : cupicef.Term cupicef.prop - (and (P x) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x_27_ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I (P x_27_) - (λ (__ : cupicef.Term cupicef.prop (P x_27_)), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x_27_)))) - ), - cupicef.prod - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.prop - (cupicef.{|rule|} - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.prop) - cupicef.I - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I)) - (λ - ( - Q : Π - ( - __0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - iff - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I (P x0) - (λ (__0 : cupicef.Term cupicef.prop (P x0)), Q x0))) - (ex A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), and (P x0) (Q x0))))) - (λ (Hp : cupicef.Term cupicef.prop (P x)), - λ - ( - Huniq : Π - ( - x_27_ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - Π - (__ : cupicef.Term cupicef.prop (P x_27_)), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x_27_) - ), - λ - ( - Q : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - conj - (cupicef.prod - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.prop - (cupicef.{|rule|} - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.prop) - cupicef.I - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I (P x0) - (λ (__ : cupicef.Term cupicef.prop (P x0)), - Q x0))) - (λ - ( - __ : Π - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - Π - (__ : cupicef.Term cupicef.prop (P x0)), - cupicef.Term cupicef.prop (Q x0) - ), - ex A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), and (P x0) (Q x0)))) - (cupicef.prod cupicef.prop - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop))) - cupicef.I - (ex A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), and (P x0) (Q x0))) - (λ - ( - __ : cupicef.Term cupicef.prop - (ex A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), and (P x0) (Q x0))) - ), - cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I (P x0) - (λ (__0 : cupicef.Term cupicef.prop (P x0)), - Q x0)))) - (λ - ( - H1 : Π - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - Π (__ : cupicef.Term cupicef.prop (P x0)), - cupicef.Term cupicef.prop (Q x0) - ), - ex__intro A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), and (P x0) (Q x0)) - x (conj (P x) (Q x) Hp (H1 x Hp))) - (λ - ( - H1 : cupicef.Term cupicef.prop - (ex A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), and (P x0) (Q x0))) - ), - match____ex cupicef.prop A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), and (P x0) (Q x0)) - (λ - ( - __ : cupicef.Term cupicef.prop - (ex A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), and (P x0) (Q x0))) - ), - cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I (P x0) - (λ - (__0 : cupicef.Term cupicef.prop (P x0)), - Q x0))) - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - H2 : cupicef.Term cupicef.prop - (and (P x0) (Q x0)) - ), - match____and cupicef.prop (P x0) (Q x0) - (λ - ( - __ : cupicef.Term cupicef.prop - (and (P x0) (Q x0)) - ), - cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - cupicef.prop)) - cupicef.I A - (λ - ( - x1 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop - cupicef.prop) - cupicef.I (P x1) - (λ - ( - __0 : cupicef.Term cupicef.prop - (P x1) - ), Q x1))) - (λ - (HPx0 : cupicef.Term cupicef.prop (P x0)), - λ - (HQx0 : cupicef.Term cupicef.prop (Q x0) - ), - λ - ( - x1 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - λ - ( - HPx1 : cupicef.Term cupicef.prop - (P x1) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x0 - (λ - ( - y : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - λ - ( - __ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x0 y) - ), - cupicef.prod cupicef.prop - cupicef.prop - (cupicef.{|rule|} cupicef.prop - cupicef.prop) - cupicef.I (P y) - (λ - ( - HPx2 : cupicef.Term - cupicef.prop ( - P y) - ), Q y)) - (λ - ( - HPx2 : cupicef.Term cupicef.prop - (P x0) - ), HQx0) - x1 - (let__H A P H x H0 Hp Huniq Q H1 x0 - H2 HPx0 HQx0 x1 HPx1) - HPx1) - H2) - H1)) - H0) - H; -symbol let__i - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ cupicef.prop - ), - Π - ( - H : Π - ( - Q : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - cupicef.Term cupicef.prop - (iff - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (P x) (λ (__ : cupicef.Term cupicef.prop (P x)), Q x))) - (ex A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), and (P x) (Q x)))) - ), - cupicef.Term cupicef.prop - (iff - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I ( - P x) (λ (__ : cupicef.Term cupicef.prop (P x)), P x))) - (ex A - (λ - (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - and (P x) (P x)))) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - λ - ( - H : Π - ( - Q : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - cupicef.Term cupicef.prop - (iff - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (P x) (λ (__ : cupicef.Term cupicef.prop (P x)), Q x))) - (ex A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), and (P x) (Q x)))) - ), H P; -symbol let__i0 - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ cupicef.prop - ), - Π - ( - H : Π - ( - Q : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - cupicef.Term cupicef.prop - (iff - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (P x) (λ (__ : cupicef.Term cupicef.prop (P x)), Q x))) - (ex A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), and (P x) (Q x)))) - ), - Π - ( - H0 : Π - ( - __ : Π - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π (__ : cupicef.Term cupicef.prop (P x)), - cupicef.Term cupicef.prop (P x) - ), - cupicef.Term cupicef.prop - (ex A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), and (P x) (P x))) - ), - Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (H1 : cupicef.Term cupicef.prop (and (P x) (P x))), - Π (Hx : cupicef.Term cupicef.prop (P x)), - Π (__tmp : cupicef.Term cupicef.prop (P x)), - Π - ( - __tmp0 : Π - ( - __ : cupicef.Term cupicef.prop - (ex A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), and (P x0) (P x0))) - ), - Π - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - Π - (__0 : cupicef.Term cupicef.prop (P x0)), - cupicef.Term cupicef.prop (P x0) - ), - cupicef.Term cupicef.prop - (iff - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I (P x0) - (λ (__ : cupicef.Term cupicef.prop (P x0)), - (λ - ( - x_27_ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x_27_) - x0))) - (ex A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - and (P x0) - ((λ - ( - x_27_ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x_27_) - x0)))) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - λ - ( - H : Π - ( - Q : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - cupicef.Term cupicef.prop - (iff - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (P x) (λ (__ : cupicef.Term cupicef.prop (P x)), Q x))) - (ex A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), and (P x) (Q x)))) - ), - λ - ( - H0 : Π - ( - __ : Π - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π (__ : cupicef.Term cupicef.prop (P x)), - cupicef.Term cupicef.prop (P x) - ), - cupicef.Term cupicef.prop - (ex A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), and (P x) (P x))) - ), - λ - (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ (H1 : cupicef.Term cupicef.prop (and (P x) (P x))), - λ (Hx : cupicef.Term cupicef.prop (P x)), - λ (__tmp : cupicef.Term cupicef.prop (P x)), - λ - ( - __tmp0 : Π - ( - __ : cupicef.Term cupicef.prop - (ex A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), and (P x0) (P x0))) - ), - Π - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - Π - ( - __0 : cupicef.Term cupicef.prop (P x0) - ), cupicef.Term cupicef.prop (P x0) - ), - H - (λ - ( - x_27_ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x_27_); -opaque symbol forall__exists__coincide__unique__domain - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ cupicef.prop - ), - Π - ( - __ : Π - ( - Q : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - cupicef.Term cupicef.prop - (iff - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (P x) (λ (__ : cupicef.Term cupicef.prop (P x)), Q x))) - (ex A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), and (P x) (Q x)))) - ), - cupicef.Term cupicef.prop - (ex A - (unique A - (λ - (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - P x))) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - λ - ( - H : Π - ( - Q : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - cupicef.Term cupicef.prop - (iff - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (P x) (λ (__ : cupicef.Term cupicef.prop (P x)), Q x))) - (ex A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), and (P x) (Q x)))) - ), - match____and cupicef.prop - (cupicef.prod - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.prop - (cupicef.{|rule|} - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.prop) - cupicef.I - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I ( - P x) (λ (__ : cupicef.Term cupicef.prop (P x)), P x))) - (λ - ( - __ : Π - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π (__ : cupicef.Term cupicef.prop (P x)), - cupicef.Term cupicef.prop (P x) - ), - ex A - (λ - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), and (P x) (P x)))) - (cupicef.prod cupicef.prop - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop))) - cupicef.I - (ex A - (λ - (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), and (P x) (P x))) - (λ - ( - __ : cupicef.Term cupicef.prop - (ex A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), and (P x) (P x))) - ), - cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (P x) (λ (__0 : cupicef.Term cupicef.prop (P x)), P x)))) - (λ - ( - __ : cupicef.Term cupicef.prop - (and - (cupicef.prod - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.prop - (cupicef.{|rule|} - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.prop) - cupicef.I - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I (P x) - (λ (__ : cupicef.Term cupicef.prop (P x)), P x))) - (λ - ( - __ : Π - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - Π (__ : cupicef.Term cupicef.prop (P x)), - cupicef.Term cupicef.prop (P x) - ), - ex A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), and (P x) (P x)))) - (cupicef.prod cupicef.prop - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop))) - cupicef.I - (ex A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), and (P x) (P x))) - (λ - ( - __ : cupicef.Term cupicef.prop - (ex A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), and (P x) (P x))) - ), - cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I (P x) - (λ (__0 : cupicef.Term cupicef.prop (P x)), P x))))) - ), - ex A - (unique A - (λ - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), P x))) - (λ - ( - H0 : Π - ( - __ : Π - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π (__ : cupicef.Term cupicef.prop (P x)), - cupicef.Term cupicef.prop (P x) - ), - cupicef.Term cupicef.prop - (ex A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), and (P x) (P x))) - ), - match____ex cupicef.prop A - (λ - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), and (P x) (P x)) - (λ - ( - __ : cupicef.Term cupicef.prop - (ex A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), and (P x) (P x))) - ), - cupicef.prod - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop))) - cupicef.prop - (cupicef.{|rule|} - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop))) - cupicef.prop) - cupicef.I - (cupicef.prod cupicef.prop - (cupicef.{|rule|} (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop))) - cupicef.I - (ex A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), and (P x) (P x))) - (λ - ( - __0 : cupicef.Term cupicef.prop - (ex A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), and (P x) (P x))) - ), - cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I (P x) - (λ (__1 : cupicef.Term cupicef.prop (P x)), P x)))) - (λ - ( - __0 : Π - ( - __0 : cupicef.Term cupicef.prop - (ex A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), and (P x) (P x))) - ), - Π - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π (__1 : cupicef.Term cupicef.prop (P x)), - cupicef.Term cupicef.prop (P x) - ), - ex A - (unique A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x)))) - (λ - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ (H1 : cupicef.Term cupicef.prop (and (P x) (P x))), - match____and cupicef.prop (P x) (P x) - (λ (__ : cupicef.Term cupicef.prop (and (P x) (P x))), - cupicef.prod - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop))) - cupicef.prop - (cupicef.{|rule|} - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop))) - cupicef.prop) - cupicef.I - (cupicef.prod cupicef.prop - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop))) - cupicef.I - (ex A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), and (P x0) (P x0))) - (λ - ( - __0 : cupicef.Term cupicef.prop - (ex A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), and (P x0) (P x0))) - ), - cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I (P x0) - (λ (__1 : cupicef.Term cupicef.prop (P x0)), - P x0)))) - (λ - ( - __0 : Π - ( - __0 : cupicef.Term cupicef.prop - (ex A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), and (P x0) (P x0))) - ), - Π - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - Π - (__1 : cupicef.Term cupicef.prop (P x0) - ), cupicef.Term cupicef.prop (P x0) - ), - ex A - (unique A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x0)))) - (λ (Hx : cupicef.Term cupicef.prop (P x)), - λ (__tmp : cupicef.Term cupicef.prop (P x)), - λ - ( - __tmp0 : Π - ( - __ : cupicef.Term cupicef.prop - (ex A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), and (P x0) (P x0))) - ), - Π - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - Π - ( - __0 : cupicef.Term cupicef.prop - (P x0) - ), cupicef.Term cupicef.prop (P x0) - ), - ex__intro A - (unique A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), P x0)) - x - (conj (P x) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x_27_ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I (P x_27_) - (λ - (__ : cupicef.Term cupicef.prop (P x_27_)), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x_27_))) - Hx - (match____and cupicef.prop - (cupicef.prod - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.prop - (cupicef.{|rule|} - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.prop) - cupicef.I - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I (P x0) - (λ - (__ : cupicef.Term cupicef.prop (P x0)), - (λ - ( - x_27_ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x_27_) - x0))) - (λ - ( - __ : Π - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - Π - ( - __ : cupicef.Term cupicef.prop - (P x0) - ), - cupicef.Term cupicef.prop - ((λ - ( - x_27_ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x_27_) - x0) - ), - ex A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - and (P x0) - ((λ - ( - x_27_ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x_27_) - x0)))) - (cupicef.prod cupicef.prop - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop))) - cupicef.I - (ex A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - and (P x0) - ((λ - ( - x_27_ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x_27_) - x0))) - (λ - ( - __ : cupicef.Term cupicef.prop - (ex A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - and (P x0) - ((λ - ( - x_27_ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x_27_) - x0))) - ), - cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop - cupicef.prop) - cupicef.I (P x0) - (λ - ( - __0 : cupicef.Term cupicef.prop (P x0) - ), - (λ - ( - x_27_ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x_27_) - x0)))) - (λ - ( - __ : cupicef.Term cupicef.prop - (and - (cupicef.prod - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - cupicef.prop)) - cupicef.prop - (cupicef.{|rule|} - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - cupicef.prop)) - cupicef.prop) - cupicef.I - (cupicef.prod - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - cupicef.prop)) - cupicef.I A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.prod cupicef.prop - cupicef.prop - (cupicef.{|rule|} cupicef.prop - cupicef.prop) - cupicef.I (P x0) - (λ - ( - __ : cupicef.Term cupicef.prop - (P x0) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x0))) - (λ - ( - __ : Π - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - Π - ( - __ : cupicef.Term - cupicef.prop ( - P x0) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x0) - ), - ex A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - and (P x0) - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x0)))) - (cupicef.prod cupicef.prop - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - cupicef.prop)) - (cupicef.{|rule|} cupicef.prop - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - cupicef.prop))) - cupicef.I - (ex A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - and (P x0) - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x0))) - (λ - ( - __ : cupicef.Term cupicef.prop - (ex A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - and (P x0) - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x0))) - ), - cupicef.prod - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop - cupicef.prop)) - cupicef.I A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.prod cupicef.prop - cupicef.prop - (cupicef.{|rule|} cupicef.prop - cupicef.prop) - cupicef.I (P x0) - (λ - ( - __0 : cupicef.Term - cupicef.prop (P x0) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x0))))) - ), - cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} cupicef.prop cupicef.prop)) - cupicef.I A - (λ - ( - x_27_ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop - cupicef.prop) - cupicef.I (P x_27_) - (λ - ( - __0 : cupicef.Term cupicef.prop - (P x_27_) - ), - eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x_27_))) - (λ - ( - __tmp1 : Π - ( - __ : Π - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - Π - ( - __ : cupicef.Term - cupicef.prop ( - P x0) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A) - x x0) - ), - cupicef.Term cupicef.prop - (ex A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - and (P x0) - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x0))) - ), - λ - ( - Huniq : Π - ( - __ : cupicef.Term cupicef.prop - (ex A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s - cupicef.z)) - A - ), - and (P x0) - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s - cupicef.z)))) - cupicef.I) - cupicef.I A) - x x0))) - ), - Π - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - Π - ( - __0 : cupicef.Term - cupicef.prop (P x0) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x0) - ), - Huniq - (ex__intro A - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - and (P x0) - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x0)) - x - (conj (P x) - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x x) - Hx - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x)))) - (let__i0 A P H H0 x H1 Hx __tmp __tmp0)))) - H1) - (H0 - (λ - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), λ (H1 : cupicef.Term cupicef.prop (P x)), H1))) - (let__i A P H); -symbol inhabited - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - cupicef.Univ cupicef.prop; -symbol inhabits - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Term cupicef.prop (inhabited A); -rule cupicef.code_guarded? _ (inhabits _ _) ↪ cupicef.guarded; -symbol match____inhabited - : Π (s : cupicef.Sort), - Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (P : cupicef.Term cupicef.prop (inhabited A) → cupicef.Univ s), - Π - ( - case__inhabits : Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), cupicef.Term s (P (inhabits A __)) - ), - Π (x : cupicef.Term cupicef.prop (inhabited A)), - cupicef.Term s (P x); -rule match____inhabited $s $A $P $case__inhabits (inhabits $A $__) ↪ -$case__inhabits $__; -rule match____inhabited $s $A - (λ x, cupicef.uncode $_.[] (cupicef.code (cupicef.cU $s'.[]) $P.[x])) ↪ -match____inhabited $s' $A (λ x, $P.[x]); -symbol inhabited__ind - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (P : cupicef.Univ cupicef.prop), - Π - ( - f : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Term cupicef.prop P - ), - Π (i : cupicef.Term cupicef.prop (inhabited A)), - cupicef.Term cupicef.prop P - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (P : cupicef.Univ cupicef.prop), - λ - ( - f : Π - ( - __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), cupicef.Term cupicef.prop P - ), - λ (i : cupicef.Term cupicef.prop (inhabited A)), - match____inhabited cupicef.prop A - (λ (__ : cupicef.Term cupicef.prop (inhabited A)), P) f i; -opaque symbol exists__inhabited - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ cupicef.prop - ), - Π - ( - __ : cupicef.Term cupicef.prop - (ex A - (λ - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), P x)) - ), cupicef.Term cupicef.prop (inhabited A) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - λ - ( - H : cupicef.Term cupicef.prop - (ex A - (λ - ( - x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), P x)) - ), - match____ex cupicef.prop A - (λ - (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - P x) - (λ - ( - __ : cupicef.Term cupicef.prop - (ex A - (λ - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), P x)) - ), inhabited A) - (λ - (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ (H0 : cupicef.Term cupicef.prop (P x)), inhabits A x) - H; -opaque symbol inhabited__covariant - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - __ : Π - ( - __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) B - ), - Π (__0 : cupicef.Term cupicef.prop (inhabited A)), - cupicef.Term cupicef.prop (inhabited B) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (B : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - f : Π - ( - __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) B - ), - λ (H : cupicef.Term cupicef.prop (inhabited A)), - match____inhabited cupicef.prop A - (λ (__ : cupicef.Term cupicef.prop (inhabited A)), inhabited B) - (λ - (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), inhabits B (f x)) - H; -opaque symbol eq__stepl - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (z : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - __ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - Π - ( - __0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x z) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - z y) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ (z : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - H1 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - λ - ( - H2 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x z) - ), - eq__ind - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - z0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - eq - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - z0 y) - H1 z H2; -opaque symbol iff__stepl - : Π (A : cupicef.Univ cupicef.prop), - Π (B : cupicef.Univ cupicef.prop), - Π (C : cupicef.Univ cupicef.prop), - Π (__ : cupicef.Term cupicef.prop (iff A B)), - Π (__0 : cupicef.Term cupicef.prop (iff A C)), - cupicef.Term cupicef.prop (iff C B) - ≔ λ (A : cupicef.Univ cupicef.prop), - λ (B : cupicef.Univ cupicef.prop), - λ (C : cupicef.Univ cupicef.prop), - λ (H : cupicef.Term cupicef.prop (iff A B)), - match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), A)) - (λ - ( - __ : cupicef.Term cupicef.prop - (and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), A))) - ), - cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - (iff A C) - (λ (__0 : cupicef.Term cupicef.prop (iff A C)), iff C B)) - (λ - ( - H0 : Π (__ : cupicef.Term cupicef.prop A), - cupicef.Term cupicef.prop B - ), - λ - ( - H1 : Π (__ : cupicef.Term cupicef.prop B), - cupicef.Term cupicef.prop A - ), - λ (H2 : cupicef.Term cupicef.prop (iff A C)), - match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - A (λ (__ : cupicef.Term cupicef.prop A), C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) cupicef.I - C (λ (__ : cupicef.Term cupicef.prop C), A)) - (λ - ( - __ : cupicef.Term cupicef.prop - (and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I A - (λ (__ : cupicef.Term cupicef.prop A), C)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I C - (λ (__ : cupicef.Term cupicef.prop C), A))) - ), iff C B) - (λ - ( - H3 : Π (__ : cupicef.Term cupicef.prop A), - cupicef.Term cupicef.prop C - ), - λ - ( - H4 : Π (__ : cupicef.Term cupicef.prop C), - cupicef.Term cupicef.prop A - ), - conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I C - (λ (__ : cupicef.Term cupicef.prop C), B)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.{|rule|} cupicef.prop cupicef.prop) - cupicef.I B - (λ (__ : cupicef.Term cupicef.prop B), C)) - (λ (H5 : cupicef.Term cupicef.prop C), - H0 (H1 (H0 (H4 H5)))) - (λ (H5 : cupicef.Term cupicef.prop B), H3 (H1 H5))) - H2) - H; -opaque symbol eq__ex__uncurried - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ cupicef.prop - ), - Π (u1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (v1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (u2 : cupicef.Term cupicef.prop (P u1)), - Π (v2 : cupicef.Term cupicef.prop (P v1)), - Π - ( - pq : cupicef.Term cupicef.prop - (ex - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1)) - (λ - ( - p : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1) - ), - eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 v1 p) - v2)) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (ex A P)) - (ex__intro A P u1 u2) (ex__intro A P v1 v2)) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - λ (u1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - (v1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ (u2 : cupicef.Term cupicef.prop (P u1)), - λ (v2 : cupicef.Term cupicef.prop (P v1)), - λ - ( - pq : cupicef.Term cupicef.prop - (ex - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1)) - (λ - ( - p : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1) - ), - eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 v1 p) - v2)) - ), - match____ex cupicef.prop - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1)) - (λ - ( - p : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1) - ), - eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 v1 p) - v2) - (λ - ( - __ : cupicef.Term cupicef.prop - (ex - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1)) - (λ - ( - p : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1) - ), - eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 v1 p) - v2)) - ), - eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (ex A P)) - (ex__intro A P u1 u2) (ex__intro A P v1 v2)) - (λ - ( - p : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1) - ), - λ - ( - q : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 v1 p) - v2) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 v1 p) - (λ (y : cupicef.Term cupicef.prop (P v1)), - λ - ( - __ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 v1 p) - y) - ), - eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (ex A P)) - (ex__intro A P u1 u2) (ex__intro A P v1 y)) - (cupicef.cast cupicef.prop cupicef.prop - (eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (ex A P)) - (ex__intro A P u1 u2) - (ex__intro A P v1 - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 v1 p))) - (eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (ex A P)) - (ex__intro A P u1 u2) - (ex__intro A P v1 - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 v1 p))) - cupicef.I - (match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (λ - ( - y : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 y) - ), - eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (ex A P)) - (ex__intro A P u1 u2) - (ex__intro A P y - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 y e))) - (eq__refl - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (ex A P)) - (ex__intro A P u1 - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 u1 - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1)))) - v1 p)) - v2 q) - pq; -symbol eq__ex - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ cupicef.prop - ), - Π (u1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (v1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (u2 : cupicef.Term cupicef.prop (P u1)), - Π (v2 : cupicef.Term cupicef.prop (P v1)), - Π - ( - p : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1) - ), - Π - ( - q : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 v1 p) - v2) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (ex A P)) - (ex__intro A P u1 u2) (ex__intro A P v1 v2)) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - λ (u1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - (v1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ (u2 : cupicef.Term cupicef.prop (P u1)), - λ (v2 : cupicef.Term cupicef.prop (P v1)), - λ - ( - p : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1) - ), - λ - ( - q : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 v1 p) - v2) - ), - eq__ex__uncurried A P u1 v1 u2 v2 - (ex__intro - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1)) - (λ - ( - p0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1) - ), - eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 v1 p0) - v2) - p q); -symbol eq__ex__hprop - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ cupicef.prop - ), - Π - ( - P__hprop : Π - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π (p : cupicef.Term cupicef.prop (P x)), - Π (q : cupicef.Term cupicef.prop (P x)), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P x)) - p q) - ), - Π (u1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - (v1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (u2 : cupicef.Term cupicef.prop (P u1)), - Π (v2 : cupicef.Term cupicef.prop (P v1)), - Π - ( - p : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (ex A P)) - (ex__intro A P u1 u2) (ex__intro A P v1 v2)) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - λ - ( - P__hprop : Π - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π (p : cupicef.Term cupicef.prop (P x)), - Π (q : cupicef.Term cupicef.prop (P x)), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P x)) - p q) - ), - λ - (u1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - (v1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ (u2 : cupicef.Term cupicef.prop (P u1)), - λ (v2 : cupicef.Term cupicef.prop (P v1)), - λ - ( - p : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1) - ), - eq__ex A P u1 v1 u2 v2 p - (P__hprop v1 - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 v1 p) - v2); -opaque symbol rew__ex - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) - ), - Π - ( - Q : Π - ( - a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) (P a) - ), cupicef.Univ cupicef.prop - ), - Π - ( - u : cupicef.Term cupicef.prop - (ex (P x) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) (P x) - ), Q x p)) - ), - Π - (y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - H : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (ex (P y) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) ( - P y) - ), Q y p))) - (eq__rect - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - a : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I)) - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I - (λ - ( - a : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - ex (P a) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) ( - P a) - ), Q a p))) - u y H) - (match____ex cupicef.prop (P x) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) ( - P x) - ), Q x p) - (λ - ( - u0 : cupicef.Term cupicef.prop - (ex (P x) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), Q x p)) - ), - ex (P y) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) ( - P y) - ), Q y p)) - (λ - ( - u1 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) ( - P x) - ), - λ (u2 : cupicef.Term cupicef.prop (Q x u1)), - ex__intro (P y) (Q y) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P u1 y H) - (match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - a : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - x0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x a) - ), - Q a - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P u1 a x0)) - u2 y H)) - u)) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - P : Π - ( - __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) - ), - λ - ( - Q : Π - ( - a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) ( - P a) - ), cupicef.Univ cupicef.prop - ), - λ - ( - u : cupicef.Term cupicef.prop - (ex (P x) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) ( - P x) - ), Q x p)) - ), - λ - ( - y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - H : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - y0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0) - ), - eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (ex (P y0) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P y0) - ), Q y0 p))) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - a : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I - (λ - ( - a : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - ex (P a) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P a) - ), Q a p))) - u y0 e) - (match____ex cupicef.prop (P x) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), Q x p) - (λ - ( - u0 : cupicef.Term cupicef.prop - (ex (P x) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (P x) - ), Q x p)) - ), - ex (P y0) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P y0) - ), Q y0 p)) - (λ - ( - u1 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), - λ (u2 : cupicef.Term cupicef.prop (Q x u1)), - ex__intro (P y0) (Q y0) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P u1 y0 e) - (match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - a : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - λ - ( - x0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x a) - ), - Q a - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P u1 a x0)) - u2 y0 e)) - u)) - (match____ex cupicef.prop (P x) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) ( - P x) - ), Q x p) - (λ - ( - e : cupicef.Term cupicef.prop - (ex (P x) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), Q x p)) - ), - eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (ex (P x) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), Q x p))) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - a : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I - (λ - ( - a : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - ex (P a) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P a) - ), Q a p))) - e x - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x)) - (match____ex cupicef.prop (P x) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), Q x p) - (λ - ( - u0 : cupicef.Term cupicef.prop - (ex (P x) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (P x) - ), Q x p)) - ), - ex (P x) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), Q x p)) - (λ - ( - u1 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), - λ (u2 : cupicef.Term cupicef.prop (Q x u1)), - ex__intro (P x) (Q x) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P u1 x - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x)) - u2) - e)) - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) ( - P x) - ), - λ (q : cupicef.Term cupicef.prop (Q x x0)), - eq__refl - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (ex (P x) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), Q x p))) - (ex__intro (P x) (Q x) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P x0 x - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x)) - q)) - u) - y H; -opaque symbol eq__ex2__uncurried - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ cupicef.prop - ), - Π - ( - Q : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - Π (u1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - (v1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (u2 : cupicef.Term cupicef.prop (P u1)), - Π (v2 : cupicef.Term cupicef.prop (P v1)), - Π (u3 : cupicef.Term cupicef.prop (Q u1)), - Π (v3 : cupicef.Term cupicef.prop (Q v1)), - Π - ( - pq : cupicef.Term cupicef.prop - (ex2 - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1)) - (λ - ( - p : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1) - ), - eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 v1 p) - v2) - (λ - ( - p : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1) - ), - eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (Q v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I Q) - u3 v1 p) - v3)) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (ex2 A P Q)) - (ex__intro2 A P Q u1 u2 u3) - (ex__intro2 A P Q v1 v2 v3)) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - λ - ( - Q : Π - ( - __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), cupicef.Univ cupicef.prop - ), - λ - (u1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - (v1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ (u2 : cupicef.Term cupicef.prop (P u1)), - λ (v2 : cupicef.Term cupicef.prop (P v1)), - λ (u3 : cupicef.Term cupicef.prop (Q u1)), - λ (v3 : cupicef.Term cupicef.prop (Q v1)), - λ - ( - pq : cupicef.Term cupicef.prop - (ex2 - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1)) - (λ - ( - p : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1) - ), - eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 v1 p) - v2) - (λ - ( - p : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1) - ), - eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (Q v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I Q) - u3 v1 p) - v3)) - ), - match____ex2 cupicef.prop - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1)) - (λ - ( - p : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1) - ), - eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 v1 p) - v2) - (λ - ( - p : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1) - ), - eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (Q v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I Q) - u3 v1 p) - v3) - (λ - ( - __ : cupicef.Term cupicef.prop - (ex2 - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1)) - (λ - ( - p : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1) - ), - eq - (cupicef.cast - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 v1 p) - v2) - (λ - ( - p : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1) - ), - eq - (cupicef.cast - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (Q v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I Q) - u3 v1 p) - v3)) - ), - eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (ex2 A P Q)) - (ex__intro2 A P Q u1 u2 u3) - (ex__intro2 A P Q v1 v2 v3)) - (λ - ( - p : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1) - ), - λ - ( - q : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 v1 p) - v2) - ), - λ - ( - r : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (Q v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I Q) - u3 v1 p) - v3) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (Q v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I Q) - u3 v1 p) - (λ (y : cupicef.Term cupicef.prop (Q v1)), - λ - ( - __ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (Q v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I Q) - u3 v1 p) - y) - ), - eq - (cupicef.cast - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (ex2 A P Q)) - (ex__intro2 A P Q u1 u2 u3) - (ex__intro2 A P Q v1 v2 y)) - (match____eq cupicef.prop - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 v1 p) - (λ (y : cupicef.Term cupicef.prop (P v1)), - λ - ( - __ : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 v1 p) - y) - ), - eq - (cupicef.cast - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (ex2 A P Q)) - (ex__intro2 A P Q u1 u2 u3) - (ex__intro2 A P Q v1 y - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __0 : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I Q) - u3 v1 p))) - (match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (λ - ( - y : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - λ - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 y) - ), - eq - (cupicef.cast - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (ex2 A P Q)) - (ex__intro2 A P Q u1 u2 u3) - (ex__intro2 A P Q y - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 y e) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I Q) - u3 y e))) - (cupicef.cast cupicef.prop cupicef.prop - (eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (ex2 A P Q)) - (ex__intro2 A P Q u1 u2 u3) - (ex__intro2 A P Q u1 u2 u3)) - (eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (ex2 A P Q)) - (ex__intro2 A P Q u1 u2 u3) - (ex__intro2 A P Q u1 - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 u1 - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I Q) - u3 u1 - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1)))) - cupicef.I - (eq__refl - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (ex2 A P Q)) - (ex__intro2 A P Q u1 u2 u3))) - v1 p) - v2 q) - v3 r) - pq; -symbol eq__ex2 - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ cupicef.prop - ), - Π - ( - Q : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - Π (u1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - (v1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π (u2 : cupicef.Term cupicef.prop (P u1)), - Π (v2 : cupicef.Term cupicef.prop (P v1)), - Π (u3 : cupicef.Term cupicef.prop (Q u1)), - Π (v3 : cupicef.Term cupicef.prop (Q v1)), - Π - ( - p : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1) - ), - Π - ( - q : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 v1 p) - v2) - ), - Π - ( - r : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (Q v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I Q) - u3 v1 p) - v3) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (ex2 A P Q)) - (ex__intro2 A P Q u1 u2 u3) - (ex__intro2 A P Q v1 v2 v3)) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - λ - ( - Q : Π - ( - __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), cupicef.Univ cupicef.prop - ), - λ - (u1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - (v1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ (u2 : cupicef.Term cupicef.prop (P u1)), - λ (v2 : cupicef.Term cupicef.prop (P v1)), - λ (u3 : cupicef.Term cupicef.prop (Q u1)), - λ (v3 : cupicef.Term cupicef.prop (Q v1)), - λ - ( - p : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1) - ), - λ - ( - q : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 v1 p) - v2) - ), - λ - ( - r : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (Q v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I Q) - u3 v1 p) - v3) - ), - eq__ex2__uncurried A P Q u1 v1 u2 v2 u3 v3 - (ex__intro2 - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1)) - (λ - ( - p0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1) - ), - eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 v1 p0) - v2) - (λ - ( - p0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1) - ), - eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (Q v1)) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} - (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I Q) - u3 v1 p0) - v3) - p q r); -symbol eq__ex2__hprop - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - cupicef.Univ cupicef.prop - ), - Π - ( - Q : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - Π - ( - P__hprop : Π - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π (p : cupicef.Term cupicef.prop (P x)), - Π (q : cupicef.Term cupicef.prop (P x)), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P x)) - p q) - ), - Π - ( - Q__hprop : Π - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π (p : cupicef.Term cupicef.prop (Q x)), - Π (q : cupicef.Term cupicef.prop (Q x)), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (Q x)) - p q) - ), - Π - (u1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π - ( - v1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - Π (u2 : cupicef.Term cupicef.prop (P u1)), - Π (v2 : cupicef.Term cupicef.prop (P v1)), - Π (u3 : cupicef.Term cupicef.prop (Q u1)), - Π (v3 : cupicef.Term cupicef.prop (Q v1)), - Π - ( - p : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (ex2 A P Q)) - (ex__intro2 A P Q u1 u2 u3) - (ex__intro2 A P Q v1 v2 v3)) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ cupicef.prop - ), - λ - ( - Q : Π - ( - __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), cupicef.Univ cupicef.prop - ), - λ - ( - P__hprop : Π - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π (p : cupicef.Term cupicef.prop (P x)), - Π (q : cupicef.Term cupicef.prop (P x)), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (P x)) - p q) - ), - λ - ( - Q__hprop : Π - ( - x : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π (p : cupicef.Term cupicef.prop (Q x)), - Π (q : cupicef.Term cupicef.prop (Q x)), - cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I (Q x)) - p q) - ), - λ - ( - u1 : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - λ - ( - v1 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ (u2 : cupicef.Term cupicef.prop (P u1)), - λ (v2 : cupicef.Term cupicef.prop (P v1)), - λ (u3 : cupicef.Term cupicef.prop (Q u1)), - λ (v3 : cupicef.Term cupicef.prop (Q v1)), - λ - ( - p : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 v1) - ), - eq__ex2 A P Q u1 v1 u2 v2 u3 v3 p - (P__hprop v1 - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I P) - u2 v1 p) - v2) - (Q__hprop v1 - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - u1 - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I Q) - u3 v1 p) - v3); -opaque symbol rew__ex2 - : Π (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - Π (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - Π - ( - P : Π - (__ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) - ), - Π - ( - Q : Π - ( - a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) (P a) - ), cupicef.Univ cupicef.prop - ), - Π - ( - R : Π - ( - a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) ( - P a) - ), cupicef.Univ cupicef.prop - ), - Π - ( - u : cupicef.Term cupicef.prop - (ex2 (P x) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) ( - P x) - ), Q x p) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) ( - P x) - ), R x p)) - ), - Π - ( - y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π - ( - H : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - cupicef.Term cupicef.prop - (eq - (cupicef.cast (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (ex2 (P y) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) ( - P y) - ), Q y p) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) ( - P y) - ), R y p))) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - a : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) cupicef.I)) - (cupicef.prod (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I - (λ - ( - a : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - ex2 (P a) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P a) - ), Q a p) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P a) - ), R a p))) - u y H) - (match____ex2 cupicef.prop (P x) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) ( - P x) - ), Q x p) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) ( - P x) - ), R x p) - (λ - ( - u0 : cupicef.Term cupicef.prop - (ex2 (P x) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), Q x p) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), R x p)) - ), - ex2 (P y) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P y) - ), Q y p) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P y) - ), R y p)) - (λ - ( - u1 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) ( - P x) - ), - λ (u2 : cupicef.Term cupicef.prop (Q x u1)), - λ (u3 : cupicef.Term cupicef.prop (R x u1)), - ex__intro2 (P y) (Q y) (R y) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P u1 y H) - (match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - a : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - λ - ( - x0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x a) - ), - Q a - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P u1 a x0)) - u2 y H) - (match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - a : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - λ - ( - x0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x a) - ), - R a - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P u1 a x0)) - u3 y H)) - u)) - ≔ λ (A : cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z))), - λ (x : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) A), - λ - ( - P : Π - ( - __ : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), cupicef.Univ (cupicef.{|type|} (cupicef.s cupicef.z)) - ), - λ - ( - Q : Π - ( - a : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) ( - P a) - ), cupicef.Univ cupicef.prop - ), - λ - ( - R : Π - ( - a : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - Π - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) ( - P a) - ), cupicef.Univ cupicef.prop - ), - λ - ( - u : cupicef.Term cupicef.prop - (ex2 (P x) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) ( - P x) - ), Q x p) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) ( - P x) - ), R x p)) - ), - λ - ( - y : cupicef.Term (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - λ - ( - H : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y) - ), - match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - y0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - λ - ( - e : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x y0) - ), - eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (ex2 (P y0) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P y0) - ), Q y0 p) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P y0) - ), R y0 p))) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - a : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I - (λ - ( - a : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - ex2 (P a) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (P a) - ), Q a p) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (P a) - ), R a p))) - u y0 e) - (match____ex2 cupicef.prop (P x) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), Q x p) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), R x p) - (λ - ( - u0 : cupicef.Term cupicef.prop - (ex2 (P x) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (P x) - ), Q x p) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (P x) - ), R x p)) - ), - ex2 (P y0) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P y0) - ), Q y0 p) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P y0) - ), R y0 p)) - (λ - ( - u1 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), - λ (u2 : cupicef.Term cupicef.prop (Q x u1)), - λ - (u3 : cupicef.Term cupicef.prop (R x u1)), - ex__intro2 (P y0) (Q y0) (R y0) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P u1 y0 e) - (match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - a : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - λ - ( - x0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x a) - ), - Q a - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P u1 a x0)) - u2 y0 e) - (match____eq cupicef.prop - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (λ - ( - a : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - A - ), - λ - ( - x0 : cupicef.Term cupicef.prop - (eq - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x a) - ), - R a - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P u1 a x0)) - u3 y0 e)) - u)) - (match____ex2 cupicef.prop (P x) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) ( - P x) - ), Q x p) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) ( - P x) - ), R x p) - (λ - ( - e : cupicef.Term cupicef.prop - (ex2 (P x) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), Q x p) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), R x p)) - ), - eq - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (ex2 (P x) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), Q x p) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), R x p))) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x - (cupicef.cast - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I A - (λ - ( - a : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I)) - (cupicef.prod - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.{|rule|} - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z)))) - cupicef.I A - (λ - ( - __ : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - A - ), - cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I)) - cupicef.I - (λ - ( - a : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) A - ), - ex2 (P a) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P a) - ), Q a p) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P a) - ), R a p))) - e x - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x)) - (match____ex2 cupicef.prop (P x) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), Q x p) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), R x p) - (λ - ( - u0 : cupicef.Term cupicef.prop - (ex2 (P x) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (P x) - ), Q x p) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (P x) - ), R x p)) - ), - ex2 (P x) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), Q x p) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), R x p)) - (λ - ( - u1 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) - (P x) - ), - λ (u2 : cupicef.Term cupicef.prop (Q x u1)), - λ (u3 : cupicef.Term cupicef.prop (R x u1)), - ex__intro2 (P x) (Q x) (R x) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P u1 x - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x)) - u2 u3) - e)) - (λ - ( - x0 : cupicef.Term - (cupicef.{|type|} (cupicef.s cupicef.z)) ( - P x) - ), - λ (q : cupicef.Term cupicef.prop (Q x x0)), - λ (r : cupicef.Term cupicef.prop (R x x0)), - eq__refl - (cupicef.cast - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.prop - (cupicef.{|type|} (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (ex2 (P x) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (P x) - ), Q x p) - (λ - ( - p : cupicef.Term - (cupicef.{|type|} - (cupicef.s cupicef.z)) - (P x) - ), R x p))) - (ex__intro2 (P x) (Q x) (R x) - (eq__rect - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x P x0 x - (eq__refl - (cupicef.cast - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ - (cupicef.{|type|} (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.{|type|} (cupicef.s cupicef.z))) - cupicef.I) - (cupicef.univ - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.{|type|} - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I A) - x)) - q r)) - u) - y H; diff --git a/out/add_requires.sh b/out/add_requires.sh new file mode 100755 index 000000000..bdabc750e --- /dev/null +++ b/out/add_requires.sh @@ -0,0 +1,3 @@ +#!/bin/bash + +dk dep $@ | cut -d ":" -f 2 | sed "s/ /\n/g" | grep ".dko$" | sed "s/\(.*\)\.dko$/require coq.\\1 as \\1;/g" diff --git a/out/compile.sh b/out/compile.sh index 1c6eaca5c..cc73bba84 100755 --- a/out/compile.sh +++ b/out/compile.sh @@ -1,9 +1,20 @@ #!/bin/sh -set -e -set -x - -export PATH=$PWD/_build/install/default/bin:$PATH -cd out -lambdapi check --lib-root . -c coq/cupicef.lp -lambdapi check --lib-root . -c Coq__Init__Logic.lp -lambdapi check --lib-root . -c Coq__Init__Datatypes.dk \ No newline at end of file +#set -e +#set -x + +#export PATH=$PWD/_build/install/default/bin:$PATH + +# add the exports +mkdir tmp +for i in *.dk; do cat $i | sed "s/^#.*$//g" > tmp/$i; done +touch tmp/cupicef.dk +(cd tmp; for i in *.dk; do ../add_requires.sh $i > $i.imports ; done) + +cp cupicef.lp.handmade cupicef.lp + +for i in *.dk; do cat header tmp/$i.imports > `basename $i .dk`.lp && lambdapi export -o lp $i >> `basename $i .dk`.lp ; done +lambdapi check -c cupicef.lp +lambdapi check -c Coq__Init__Logic.lp +lambdapi check -c Coq__Init__Datatypes.lp + +rm -rf tmp diff --git a/out/coq/cupicef.lp b/out/cupicef.lp.handmade similarity index 94% rename from out/coq/cupicef.lp rename to out/cupicef.lp.handmade index 42d29c894..35d5d7ace 100644 --- a/out/coq/cupicef.lp +++ b/out/cupicef.lp.handmade @@ -13,6 +13,7 @@ symbol Eq : Sort → Sort → Bool; rule Eq $x $x ↪ true; associative commutative symbol and : Bool → Bool → Bool; rule and true $c ↪ $c; +rule and $c true ↪ $c; symbol BoolO : TYPE; constant symbol BoolNone : BoolO; constant symbol BoolSome : Bool → BoolO; @@ -120,13 +121,23 @@ with max $i (s (max $i $j)) ↪ s (max $i $j) with max (max $i $k) (s (max $i $j)) ↪ max $k (s (max $i $j)); rule max z $i ↪ $i; -rule s (max $i $j) ↪ max (s $i) (s $j); -rule max $i $i ↪ $i -with max (max $i $j) $i ↪ max $i $j +rule max (max $i $j) $i ↪ max $i $j +with max (max $j $i) $i ↪ max $i $j +with max $i (max $j $i) ↪ max $i $j with max (s $i) $i ↪ s $i with max (max (s $i) $j) $i ↪ max (s $i) $j +with max (max $j (s $i)) $i ↪ max (s $i) $j +with max $i (max $j (s $i)) ↪ max (s $i) $j +with max $i (s (max $j $i)) ↪ s (max $i $j) with max (s (max $i $j)) $i ↪ s (max $i $j) -with max (s (max $i $j)) (max $i $k) ↪ max $k (s (max $i $j)); +with max (s (max $j $i)) $i ↪ s (max $i $j) +with max (max $k $i) (s (max $i $j)) ↪ max $k (s (max $i $j)) +with max (max $k $i) (s (max $j $i)) ↪ max $k (s (max $i $j)) +with max (max $i $k) (s (max $j $i)) ↪ max $k (s (max $i $j)) +with max (s (max $i $j)) (max $i $k) ↪ max $k (s (max $i $j)) +with max (s (max $i $j)) (max $k $i) ↪ max $k (s (max $i $j)) +with max (s (max $j $i)) (max $i $k) ↪ max $k (s (max $i $j)) +with max (s (max $j $i)) (max $k $i) ↪ max $k (s (max $i $j)); rule sup prop $s ↪ $s with sup $s prop ↪ $s @@ -138,7 +149,9 @@ with Leq (max $i $j) $k ↪ and (Leq $i $k) (Leq $j $k) with Leq $i $i ↪ true with Leq $i (s $i) ↪ true with Leq $i (max $i $j) ↪ true -with Leq $i (max (s $i) $j) ↪ true; +with Leq $i (max $j $i) ↪ true +with Leq $i (max (s $i) $j) ↪ true +with Leq $i (max $j (s $i)) ↪ true; rule Axiom prop ({|type|} _) ↪ true with Axiom ({|type|} $i) ({|type|} $j) ↪ Leq $i $j; rule Rule _ prop prop ↪ true diff --git a/out/header b/out/header new file mode 100644 index 000000000..d390112cb --- /dev/null +++ b/out/header @@ -0,0 +1 @@ +flag "eta_equality" on; diff --git a/out/lambdapi.pkg b/out/lambdapi.pkg new file mode 100644 index 000000000..798df43c1 --- /dev/null +++ b/out/lambdapi.pkg @@ -0,0 +1,2 @@ +package_name=coq +root_path=coq From dc51dfe02e640fd69fe1c317103acfb731041e1e Mon Sep 17 00:00:00 2001 From: Claudio Sacerdoti Coen Date: Fri, 27 Jan 2023 17:36:06 +0100 Subject: [PATCH 06/25] zip everything --- out/Coq__Arith__Arith__base.dk | 4 - out/Coq__Arith__Arith__base.dk.gz | Bin 0 -> 150 bytes out/Coq__Arith__Between.dk | 2513 - out/Coq__Arith__Between.dk.gz | Bin 0 -> 7657 bytes out/Coq__Arith__Compare__dec.dk | 5066 -- out/Coq__Arith__Compare__dec.dk.gz | Bin 0 -> 9883 bytes out/Coq__Arith__EqNat.dk | 2688 - out/Coq__Arith__EqNat.dk.gz | Bin 0 -> 5081 bytes out/Coq__Arith__Factorial.dk | 342 - out/Coq__Arith__Factorial.dk.gz | Bin 0 -> 1408 bytes out/Coq__Arith__Gt.dk | 1422 - out/Coq__Arith__Gt.dk.gz | Bin 0 -> 2832 bytes out/Coq__Arith__Le.dk | 319 - 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a/out/Coq__Arith__Arith__base.dk +++ /dev/null @@ -1,4 +0,0 @@ -(; This file was automatically generated by Coqine. ;) -(; The encoding used was: "template_cic". ;) - -(; End of translation. ;) diff --git a/out/Coq__Arith__Arith__base.dk.gz b/out/Coq__Arith__Arith__base.dk.gz new file mode 100644 index 0000000000000000000000000000000000000000..c477060be45816d9af23abbf11472c998dfa26e5 GIT binary patch literal 150 zcmb2|=HQt3>~b;#vvYo7e7s{(W=RH|N=htF)l11{kUee4*KEM!_Hgg=A4gfZ*)^0O zvAsE=!luKzTh&c;dwxRQ{ngezvr}{?N!XuSa5-0fx`SqHMC4C}=N&(t^5X;ZE}E|Y zbx&f=!9_2+jKzb#nQJ}&JNc9mE9cebeA|o`-4D$s)3uiQ|6*2uR%KJqz`y_i%RWB| literal 0 HcmV?d00001 diff --git a/out/Coq__Arith__Between.dk b/out/Coq__Arith__Between.dk deleted file mode 100644 index f47ebbcb1..000000000 --- a/out/Coq__Arith__Between.dk +++ /dev/null @@ -1,2513 +0,0 @@ -(; This file was automatically generated by Coqine. ;) -(; The encoding used was: "template_cic". ;) - -def between : - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop. - -def bet__emp : - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop (between P k k). - -def bet__S : - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (between P k l) -> - __0 : cupicef.Term cupicef.prop (P l) -> - cupicef.Term cupicef.prop (between P k (Coq__Init__Datatypes.S l)). - -[ ] cupicef.code_guarded? _ (bet__emp _ _) --> cupicef.guarded. - -[ ] cupicef.code_guarded? _ (bet__S _ _ _ _ _) --> cupicef.guarded. - -def match____between : - s : cupicef.Sort -> - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - P0 : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop (between P k __) -> - cupicef.Univ s) -> - case__bet__emp : cupicef.Term s (P0 k (bet__emp P k)) -> - case__bet__S : - (l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (between P k l) -> - __0 : cupicef.Term cupicef.prop (P l) -> - cupicef.Term s (P0 (Coq__Init__Datatypes.S l) (bet__S P k l __ __0))) -> - __ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - x : cupicef.Term cupicef.prop (between P k __) -> - cupicef.Term s (P0 __ x). - -[ s, P, k, P0, case__bet__emp, case__bet__S] - match____between s P k P0 case__bet__emp case__bet__S {k} (bet__emp P k) - --> - case__bet__emp. - -[ s, P, k, P0, case__bet__emp, case__bet__S, l, __, __0] - match____between s P k P0 case__bet__emp case__bet__S - {Coq__Init__Datatypes.S l} (bet__S P k l __ __0) - --> - case__bet__S l __ __0. - -[ s, P, k, P0, s'] - match____between s P k - (__ => x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P0 __ x))) - --> - match____between s' P k (__ => x => P0 __ x). - -def between__ind : - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - P0 : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - f : cupicef.Term cupicef.prop (P0 k) -> - f0 : - (l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (between P k l) -> - __0 : cupicef.Term cupicef.prop (P0 l) -> - __1 : cupicef.Term cupicef.prop (P l) -> - cupicef.Term cupicef.prop (P0 (Coq__Init__Datatypes.S l))) -> - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - b : cupicef.Term cupicef.prop (between P k n) -> - cupicef.Term cupicef.prop (P0 n) - -:= P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - P0 : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - f : cupicef.Term cupicef.prop (P0 k) => - f0 : - (l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (between P k l) -> - __0 : cupicef.Term cupicef.prop (P0 l) -> - __1 : cupicef.Term cupicef.prop (P l) -> - cupicef.Term cupicef.prop (P0 (Coq__Init__Datatypes.S l))) => - cupicef.fixproj cupicef.1 - (c => - c - (cupicef.SA cupicef.1 cupicef.prop - (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule cupicef.set - (cupicef.rule cupicef.prop cupicef.prop)) - cupicef.I Coq__Init__Datatypes.nat - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (between P k n) - (b : cupicef.Term cupicef.prop (between P k n) => P0 n))))) - (c => - c - (F => - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - b : cupicef.Term cupicef.prop (between P k n) => - match____between cupicef.prop P k - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - __ : cupicef.Term cupicef.prop (between P k n0) => - P0 n0) - f - (l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - b0 : cupicef.Term cupicef.prop (between P k l) => - p : cupicef.Term cupicef.prop (P l) => - f0 l b0 (F l b0) p) - n b)) - cupicef.0. - -thm bet__eq : - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - l k) -> - cupicef.Term cupicef.prop (between P k l) - -:= P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - l k) => - Coq__Init__Logic.eq__ind__r - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - k - (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - between P k l0) - (bet__emp P k) l H. - -thm between__le : - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (between P k l) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) - -:= P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (between P k l) => - between__ind P k - (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Peano.le k l0) - (Coq__Init__Peano.le__n k) - (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H0 : cupicef.Term cupicef.prop (between P k l0) => - IHbetween : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l0) => - H1 : cupicef.Term cupicef.prop (P l0) => - Coq__Init__Peano.le__S k l0 IHbetween) - l H. - -thm between__Sk__l : - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (between P k l) -> - __0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) l) -> - cupicef.Term cupicef.prop (between P (Coq__Init__Datatypes.S k) l) - -:= P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (between P k l) => - between__ind P k - (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) l0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) l0) => - between P (Coq__Init__Datatypes.S k) l0)) - (H0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) k) => - bet__eq P (Coq__Init__Datatypes.S k) k - (Coq__Arith__PeanoNat.Nat__le__antisymm k (Coq__Init__Datatypes.S k) - (Coq__Arith__PeanoNat.Nat__le__succ__diag__r k) H0)) - (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H0 : cupicef.Term cupicef.prop (between P k l0) => - match____between cupicef.prop P k - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - __ : cupicef.Term cupicef.prop (between P k n) => - cupicef.prod (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop)) - (cupicef.rule (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop))) - cupicef.I - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) n) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) n) => - between P (Coq__Init__Datatypes.S k) n)) - (__0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) n) -> - cupicef.Term cupicef.prop - (between P (Coq__Init__Datatypes.S k) n)) => - cupicef.prod cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop)) - cupicef.I (P n) - (__1 : cupicef.Term cupicef.prop (P n) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) - (Coq__Init__Datatypes.S n)) - (__2 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) - (Coq__Init__Datatypes.S n)) => - between P (Coq__Init__Datatypes.S k) - (Coq__Init__Datatypes.S n))))) - (IHbetween : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) k) -> - cupicef.Term cupicef.prop - (between P (Coq__Init__Datatypes.S k) k)) => - H1 : cupicef.Term cupicef.prop (P k) => - H2 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) - (Coq__Init__Datatypes.S k)) => - bet__emp P (Coq__Init__Datatypes.S k)) - (l1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H1 : cupicef.Term cupicef.prop (between P k l1) => - H2 : cupicef.Term cupicef.prop (P l1) => - IHbetween : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) - (Coq__Init__Datatypes.S l1)) -> - cupicef.Term cupicef.prop - (between P (Coq__Init__Datatypes.S k) - (Coq__Init__Datatypes.S l1))) => - H3 : cupicef.Term cupicef.prop (P (Coq__Init__Datatypes.S l1)) => - H4 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) - (Coq__Init__Datatypes.S (Coq__Init__Datatypes.S l1))) => - bet__S P (Coq__Init__Datatypes.S k) (Coq__Init__Datatypes.S l1) - (IHbetween (Coq__Arith__Le.le__n__S k l1 (between__le P k l1 H1))) - H3) - l0 H0) - l H. - -thm between__restr : - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) -> - __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le l m) -> - __1 : cupicef.Term cupicef.prop (between P k m) -> - cupicef.Term cupicef.prop (between P l m) - -:= P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) => - Coq__Init__Peano.le__ind k - (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.prop (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule cupicef.prop (cupicef.rule cupicef.prop cupicef.prop)) - cupicef.I (Coq__Init__Peano.le l0 m) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le l0 m) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (between P k m) - (__0 : cupicef.Term cupicef.prop (between P k m) => between P l0 m))) - (H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k m) => - H1 : cupicef.Term cupicef.prop (between P k m) => - H1) - (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k m0) => - IHle : - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le m0 m) -> - __0 : cupicef.Term cupicef.prop (between P k m) -> - cupicef.Term cupicef.prop (between P m0 m)) => - H1 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S m0) m) => - H2 : cupicef.Term cupicef.prop (between P k m) => - between__Sk__l P m0 m - (IHle (Coq__Arith__PeanoNat.Nat__lt__le__incl m0 m H1) H2) H1) - l H. - -def exists__between : - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop. - -def exists__S : - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (exists__between Q k l) -> - cupicef.Term cupicef.prop (exists__between Q k (Coq__Init__Datatypes.S l)). - -def exists__le : - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) -> - __0 : cupicef.Term cupicef.prop (Q l) -> - cupicef.Term cupicef.prop (exists__between Q k (Coq__Init__Datatypes.S l)). - -[ ] cupicef.code_guarded? _ (exists__S _ _ _ _) --> cupicef.guarded. - -[ ] cupicef.code_guarded? _ (exists__le _ _ _ _ _) --> cupicef.guarded. - -def match____exists__between : - s : cupicef.Sort -> - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop (exists__between Q k __) -> - cupicef.Univ s) -> - case__exists__S : - (l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (exists__between Q k l) -> - cupicef.Term s (P (Coq__Init__Datatypes.S l) (exists__S Q k l __))) -> - case__exists__le : - (l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) -> - __0 : cupicef.Term cupicef.prop (Q l) -> - cupicef.Term s (P (Coq__Init__Datatypes.S l) (exists__le Q k l __ __0))) -> - __ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - x : cupicef.Term cupicef.prop (exists__between Q k __) -> - cupicef.Term s (P __ x). - -[ s, Q, k, P, case__exists__S, case__exists__le, l, __] - match____exists__between s Q k P case__exists__S case__exists__le - {Coq__Init__Datatypes.S l} (exists__S Q k l __) - --> - case__exists__S l __. - -[ s, Q, k, P, case__exists__S, case__exists__le, l, __, __0] - match____exists__between s Q k P case__exists__S case__exists__le - {Coq__Init__Datatypes.S l} (exists__le Q k l __ __0) - --> - case__exists__le l __ __0. - -[ s, Q, k, P, s'] - match____exists__between s Q k - (__ => x => cupicef.uncode _ (cupicef.code (cupicef.cU s') (P __ x))) - --> - match____exists__between s' Q k (__ => x => P __ x). - -def exists__between__ind : - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - f : - (l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (exists__between Q k l) -> - __0 : cupicef.Term cupicef.prop (P l) -> - cupicef.Term cupicef.prop (P (Coq__Init__Datatypes.S l))) -> - f0 : - (l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) -> - __0 : cupicef.Term cupicef.prop (Q l) -> - cupicef.Term cupicef.prop (P (Coq__Init__Datatypes.S l))) -> - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - e : cupicef.Term cupicef.prop (exists__between Q k n) -> - cupicef.Term cupicef.prop (P n) - -:= Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - f : - (l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (exists__between Q k l) -> - __0 : cupicef.Term cupicef.prop (P l) -> - cupicef.Term cupicef.prop (P (Coq__Init__Datatypes.S l))) => - f0 : - (l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) -> - __0 : cupicef.Term cupicef.prop (Q l) -> - cupicef.Term cupicef.prop (P (Coq__Init__Datatypes.S l))) => - cupicef.fixproj cupicef.1 - (c => - c - (cupicef.SA cupicef.1 cupicef.prop - (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule cupicef.set - (cupicef.rule cupicef.prop cupicef.prop)) - cupicef.I Coq__Init__Datatypes.nat - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (exists__between Q k n) - (e : cupicef.Term cupicef.prop (exists__between Q k n) => - P n))))) - (c => - c - (F => - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - e : cupicef.Term cupicef.prop (exists__between Q k n) => - match____exists__between cupicef.prop Q k - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - __ : cupicef.Term cupicef.prop (exists__between Q k n0) => - P n0) - (l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - e0 : cupicef.Term cupicef.prop (exists__between Q k l) => - f l e0 (F l e0)) - (l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - l0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) => - q : cupicef.Term cupicef.prop (Q l) => - f0 l l0 q) - n e)) - cupicef.0. - -thm exists__le__S : - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (exists__between Q k l) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) l) - -:= Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (exists__between Q k l) => - exists__between__ind Q k - (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Peano.le (Coq__Init__Datatypes.S k) l0) - (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H0 : cupicef.Term cupicef.prop (exists__between Q k l0) => - IHexists__between : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) l0) => - Coq__Init__Peano.le__S (Coq__Init__Datatypes.S k) l0 IHexists__between) - (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l0) => - H1 : cupicef.Term cupicef.prop (Q l0) => - Coq__Arith__Le.le__n__S k l0 H0) - l H. - -thm exists__lt : - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (exists__between Q k l) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt k l) - -:= Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - exists__le__S Q. - -thm exists__S__le : - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (exists__between Q k (Coq__Init__Datatypes.S l)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) - -:= Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : - cupicef.Term cupicef.prop - (exists__between Q k (Coq__Init__Datatypes.S l)) => - Coq__Arith__Le.le__S__n k l - (exists__le__S Q k (Coq__Init__Datatypes.S l) H). - -def in__int : - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop - -:= p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.and (Coq__Init__Peano.le p r) (Coq__Init__Peano.lt r q). - -thm in__int__intro : - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le p r) -> - __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt r q) -> - cupicef.Term cupicef.prop (in__int p q r) - -:= p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (Coq__Init__Peano.le p r) => - H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt r q) => - Coq__Init__Logic.conj (Coq__Init__Peano.le p r) (Coq__Init__Peano.lt r q) - H H0. - -thm in__int__lt : - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int p q r) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt p q) - -:= p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (in__int p q r) => - Coq__Init__Logic.match____and cupicef.prop (Coq__Init__Peano.le p r) - (Coq__Init__Peano.lt r q) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and (Coq__Init__Peano.le p r) - (Coq__Init__Peano.lt r q)) => - Coq__Init__Peano.lt p q) - (H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le p r) => - H1 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt r q) => - Coq__Arith__PeanoNat.Nat__le__lt__trans p r q H0 H1) - H. - -def let__o : - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H : cupicef.Term cupicef.prop (in__int p (Coq__Init__Datatypes.S q) r) -> - H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le p r) -> - H1 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt r (Coq__Init__Datatypes.S q)) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt r q) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - r q)) - -:= p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (in__int p (Coq__Init__Datatypes.S q) r) => - H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le p r) => - H1 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt r (Coq__Init__Datatypes.S q)) => - Coq__Arith__Lt.le__lt__or__eq r q (Coq__Arith__Lt.lt__n__Sm__le r q H1). - -thm in__int__p__Sq : - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int p (Coq__Init__Datatypes.S q) r) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (in__int p q r) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - r q)) - -:= p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (in__int p (Coq__Init__Datatypes.S q) r) => - Coq__Init__Logic.match____and cupicef.prop (Coq__Init__Peano.le p r) - (Coq__Init__Peano.lt r (Coq__Init__Datatypes.S q)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and (Coq__Init__Peano.le p r) - (Coq__Init__Peano.lt r (Coq__Init__Datatypes.S q))) => - Coq__Init__Logic.or (in__int p q r) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - r q)) - (H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le p r) => - H1 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt r (Coq__Init__Datatypes.S q)) => - Coq__Init__Logic.match____or cupicef.prop (Coq__Init__Peano.lt r q) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - r q) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt r q) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - r q)) => - Coq__Init__Logic.or (in__int p q r) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - r q)) - (H2 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt r q) => - Coq__Init__Logic.or__introl (in__int p q r) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - r q) - (in__int__intro p q r H0 H2)) - (H2 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - r q) => - Coq__Init__Logic.or__intror (in__int p q r) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - r q) - H2) - (let__o p q r H H0 H1)) - H. - -thm in__int__S : - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int p q r) -> - cupicef.Term cupicef.prop (in__int p (Coq__Init__Datatypes.S q) r) - -:= p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (in__int p q r) => - Coq__Init__Logic.match____and cupicef.prop (Coq__Init__Peano.le p r) - (Coq__Init__Peano.lt r q) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and (Coq__Init__Peano.le p r) - (Coq__Init__Peano.lt r q)) => - in__int p (Coq__Init__Datatypes.S q) r) - (H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le p r) => - H1 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt r q) => - in__int__intro p (Coq__Init__Datatypes.S q) r H0 - (cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S r) - (Coq__Init__Datatypes.S q)) - (Coq__Init__Peano.lt r (Coq__Init__Datatypes.S q)) cupicef.I - (Coq__Init__Peano.le__S (Coq__Init__Datatypes.S r) q H1))) - H. - -thm in__int__Sp__q : - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int (Coq__Init__Datatypes.S p) q r) -> - cupicef.Term cupicef.prop (in__int p q r) - -:= p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (in__int (Coq__Init__Datatypes.S p) q r) => - Coq__Init__Logic.match____and cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S p) r) - (Coq__Init__Peano.lt r q) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (Coq__Init__Peano.le (Coq__Init__Datatypes.S p) r) - (Coq__Init__Peano.lt r q)) => - in__int p q r) - (H0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S p) r) => - H1 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt r q) => - in__int__intro p q r (Coq__Arith__PeanoNat.Nat__lt__le__incl p r H0) H1) - H. - -def let__o0 : - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H : cupicef.Term cupicef.prop (between P k l) -> - l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H0 : cupicef.Term cupicef.prop (between P k l0) -> - IHbetween : - (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k l0 r) -> - cupicef.Term cupicef.prop (P r)) -> - H1 : cupicef.Term cupicef.prop (P l0) -> - r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H2 : cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S l0) r) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (in__int k l0 r) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - r l0)) - -:= P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (between P k l) => - l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H0 : cupicef.Term cupicef.prop (between P k l0) => - IHbetween : - (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k l0 r) -> - cupicef.Term cupicef.prop (P r)) => - H1 : cupicef.Term cupicef.prop (P l0) => - r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H2 : cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S l0) r) => - in__int__p__Sq k l0 r H2. - -thm between__in__int : - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (between P k l) -> - r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __0 : cupicef.Term cupicef.prop (in__int k l r) -> - cupicef.Term cupicef.prop (P r) - -:= P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (between P k l) => - between__ind P k - (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) - cupicef.I Coq__Init__Datatypes.nat - (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (in__int k l0 r) - (__ : cupicef.Term cupicef.prop (in__int k l0 r) => P r))) - (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H0 : cupicef.Term cupicef.prop (in__int k k r) => - Coq__Init__Logic.False__ind (P r) - (Coq__Arith__PeanoNat.Nat__lt__irrefl k (in__int__lt k k r H0))) - (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H0 : cupicef.Term cupicef.prop (between P k l0) => - IHbetween : - (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k l0 r) -> - cupicef.Term cupicef.prop (P r)) => - H1 : cupicef.Term cupicef.prop (P l0) => - r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H2 : - cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S l0) r) => - Coq__Init__Logic.match____or cupicef.prop (in__int k l0 r) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - r l0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (in__int k l0 r) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - r l0)) => - P r) - (H3 : cupicef.Term cupicef.prop (in__int k l0 r) => IHbetween r H3) - (H3 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - r l0) => - Coq__Init__Logic.eq__ind__r - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - l0 - (r0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (in__int k (Coq__Init__Datatypes.S l0) r0) - (H4 : - cupicef.Term cupicef.prop - (in__int k (Coq__Init__Datatypes.S l0) r0) => - P r0)) - (H4 : - cupicef.Term cupicef.prop - (in__int k (Coq__Init__Datatypes.S l0) l0) => - H1) - r H3 H2) - (let__o0 P k l H l0 H0 IHbetween H1 r H2)) - l H. - -thm in__int__between : - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) -> - __0 : - (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __0 : cupicef.Term cupicef.prop (in__int k l r) -> - cupicef.Term cupicef.prop (P r)) -> - cupicef.Term cupicef.prop (between P k l) - -:= P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) => - Coq__Init__Peano.le__ind k - (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod - (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) - cupicef.prop - (cupicef.rule - (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) - cupicef.prop) - cupicef.I - (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) - cupicef.I Coq__Init__Datatypes.nat - (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (in__int k l0 r) - (__ : cupicef.Term cupicef.prop (in__int k l0 r) => P r))) - (__ : - (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k l0 r) -> - cupicef.Term cupicef.prop (P r)) => - between P k l0)) - (H0 : - (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k k r) -> - cupicef.Term cupicef.prop (P r)) => - bet__emp P k) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k m) => - IHle : - (__ : - (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k m r) -> - cupicef.Term cupicef.prop (P r)) -> - cupicef.Term cupicef.prop (between P k m)) => - H1 : - (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S m) r) -> - cupicef.Term cupicef.prop (P r)) => - bet__S P k m - (IHle - (r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H2 : cupicef.Term cupicef.prop (in__int k m r) => - H1 r (in__int__S k m r H2))) - (H1 m - (in__int__intro k (Coq__Init__Datatypes.S m) m H0 - (cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S m) - (Coq__Init__Datatypes.S m)) - (Coq__Init__Peano.lt m (Coq__Init__Datatypes.S m)) cupicef.I - (Coq__Init__Peano.le__n (Coq__Init__Datatypes.S m)))))) - l H. - -thm exists__in__int : - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (exists__between Q k l) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.ex2 - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - in__int k l m) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => Q m)) - -:= Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (exists__between Q k l) => - exists__between__ind Q k - (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.ex2 - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - in__int k l0 m) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => Q m)) - (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H0 : cupicef.Term cupicef.prop (exists__between Q k l0) => - H1 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.ex2 - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - in__int k l0 m) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => Q m)) => - Coq__Init__Logic.match____ex2 cupicef.prop - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - in__int k l0 m) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => Q m) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.ex2 - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - in__int k l0 m) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => Q m)) => - Coq__Init__Logic.ex2 - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - in__int k (Coq__Init__Datatypes.S l0) m) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => Q m)) - (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H2 : cupicef.Term cupicef.prop (in__int k l0 p) => - H3 : cupicef.Term cupicef.prop (Q p) => - Coq__Init__Logic.ex__intro2 - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - in__int k (Coq__Init__Datatypes.S l0) m) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => Q m) p - (in__int__S k l0 p H2) H3) - H1) - (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l0) => - H1 : cupicef.Term cupicef.prop (Q l0) => - Coq__Init__Logic.ex__intro2 - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - in__int k (Coq__Init__Datatypes.S l0) m) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => Q m) l0 - (in__int__intro k (Coq__Init__Datatypes.S l0) l0 H0 - (cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S l0) - (Coq__Init__Datatypes.S l0)) - (Coq__Init__Peano.lt l0 (Coq__Init__Datatypes.S l0)) cupicef.I - (Coq__Init__Peano.le__n (Coq__Init__Datatypes.S l0)))) - H1) - l H. - -thm in__int__exists : - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k l r) -> - __0 : cupicef.Term cupicef.prop (Q r) -> - cupicef.Term cupicef.prop (exists__between Q k l) - -:= Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - r : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (in__int k l r) => - Coq__Init__Logic.match____and cupicef.prop (Coq__Init__Peano.le k r) - (Coq__Init__Peano.lt r l) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and (Coq__Init__Peano.le k r) - (Coq__Init__Peano.lt r l)) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (Q r) - (__0 : cupicef.Term cupicef.prop (Q r) => exists__between Q k l)) - (H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k r) => - lt__r__l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt r l) => - H1 : cupicef.Term cupicef.prop (Q r) => - Coq__Init__Peano.le__ind (Coq__Init__Datatypes.S r) - (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - exists__between Q k l0) - (exists__le Q k r H0 H1) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - lt__r__l0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S r) m) => - IHlt__r__l : cupicef.Term cupicef.prop (exists__between Q k m) => - exists__S Q k m IHlt__r__l) - l lt__r__l) - H. - -def let__o1 : - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k m) -> - IHle : - (__ : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k m n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.or (P n) (Q n))) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (between P k m) (exists__between Q k m))) -> - P__or__Q : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S m) n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.or (P n) (Q n))) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (between P k m) (exists__between Q k m)) - -:= P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k m) => - IHle : - (__ : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k m n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.or (P n) (Q n))) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (between P k m) (exists__between Q k m))) => - P__or__Q : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S m) n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.or (P n) (Q n))) => - IHle - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H1 : cupicef.Term cupicef.prop (in__int k m n) => - P__or__Q n (in__int__S k m n H1)). - -def let__o2 : - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k m) -> - IHle : - (__ : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k m n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.or (P n) (Q n))) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (between P k m) (exists__between Q k m))) -> - P__or__Q : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S m) n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.or (P n) (Q n))) -> - H1 : cupicef.Term cupicef.prop (between P k m) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.or (P m) (Q m)) - -:= P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k m) => - IHle : - (__ : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k m n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.or (P n) (Q n))) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (between P k m) (exists__between Q k m))) => - P__or__Q : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S m) n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.or (P n) (Q n))) => - H1 : cupicef.Term cupicef.prop (between P k m) => - P__or__Q m - (in__int__intro k (Coq__Init__Datatypes.S m) m H0 - (cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S m) - (Coq__Init__Datatypes.S m)) - (Coq__Init__Peano.lt m (Coq__Init__Datatypes.S m)) cupicef.I - (Coq__Init__Peano.le__n (Coq__Init__Datatypes.S m)))). - -thm between__or__exists : - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) -> - __0 : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __0 : cupicef.Term cupicef.prop (in__int k l n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.or (P n) (Q n))) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (between P k l) (exists__between Q k l)) - -:= P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l) => - Coq__Init__Peano.le__ind k - (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod - (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) - cupicef.prop - (cupicef.rule - (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) - cupicef.prop) - cupicef.I - (cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) - cupicef.I Coq__Init__Datatypes.nat - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (in__int k l0 n) - (__ : cupicef.Term cupicef.prop (in__int k l0 n) => - Coq__Init__Logic.or (P n) (Q n)))) - (__ : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k l0 n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.or (P n) (Q n))) => - Coq__Init__Logic.or (between P k l0) (exists__between Q k l0))) - (H0 : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k k n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.or (P n) (Q n))) => - Coq__Init__Logic.or__introl (between P k k) (exists__between Q k k) - (bet__emp P k)) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k m) => - IHle : - (__ : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k m n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.or (P n) (Q n))) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (between P k m) (exists__between Q k m))) => - P__or__Q : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S m) n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.or (P n) (Q n))) => - Coq__Init__Logic.match____or cupicef.prop (between P k m) - (exists__between Q k m) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (between P k m) (exists__between Q k m)) => - Coq__Init__Logic.or (between P k (Coq__Init__Datatypes.S m)) - (exists__between Q k (Coq__Init__Datatypes.S m))) - (H1 : cupicef.Term cupicef.prop (between P k m) => - Coq__Init__Logic.match____or cupicef.prop (P m) (Q m) - (__ : cupicef.Term cupicef.prop (Coq__Init__Logic.or (P m) (Q m)) => - Coq__Init__Logic.or (between P k (Coq__Init__Datatypes.S m)) - (exists__between Q k (Coq__Init__Datatypes.S m))) - (H2 : cupicef.Term cupicef.prop (P m) => - Coq__Init__Logic.or__introl - (between P k (Coq__Init__Datatypes.S m)) - (exists__between Q k (Coq__Init__Datatypes.S m)) - (bet__S P k m H1 H2)) - (H2 : cupicef.Term cupicef.prop (Q m) => - Coq__Init__Logic.or__intror - (between P k (Coq__Init__Datatypes.S m)) - (exists__between Q k (Coq__Init__Datatypes.S m)) - (exists__le Q k m H0 H2)) - (let__o2 P Q k l H m H0 IHle P__or__Q H1)) - (H1 : cupicef.Term cupicef.prop (exists__between Q k m) => - Coq__Init__Logic.or__intror (between P k (Coq__Init__Datatypes.S m)) - (exists__between Q k (Coq__Init__Datatypes.S m)) - (exists__S Q k m H1)) - (let__o1 P Q k l H m H0 IHle P__or__Q)) - l H. - -def let__e : - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H : cupicef.Term cupicef.prop (between P k l) -> - l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H0 : cupicef.Term cupicef.prop (between P k l0) -> - IHbetween : - (__ : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k l0 n) -> - __0 : cupicef.Term cupicef.prop (P n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (exists__between Q k l0))) -> - H1 : cupicef.Term cupicef.prop (P l0) -> - H2 : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S l0) n) -> - __0 : cupicef.Term cupicef.prop (P n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) -> - H3 : - cupicef.Term cupicef.prop - (exists__between Q k (Coq__Init__Datatypes.S l0)) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.ex2 - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - in__int k (Coq__Init__Datatypes.S l0) m) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => Q m)) - -:= P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (between P k l) => - l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H0 : cupicef.Term cupicef.prop (between P k l0) => - IHbetween : - (__ : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k l0 n) -> - __0 : cupicef.Term cupicef.prop (P n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (exists__between Q k l0))) => - H1 : cupicef.Term cupicef.prop (P l0) => - H2 : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S l0) n) -> - __0 : cupicef.Term cupicef.prop (P n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) => - H3 : - cupicef.Term cupicef.prop - (exists__between Q k (Coq__Init__Datatypes.S l0)) => - exists__in__int Q k (Coq__Init__Datatypes.S l0) H3. - -def let__o3 : - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H : cupicef.Term cupicef.prop (between P k l) -> - l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H0 : cupicef.Term cupicef.prop (between P k l0) -> - IHbetween : - (__ : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k l0 n) -> - __0 : cupicef.Term cupicef.prop (P n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (exists__between Q k l0))) -> - H1 : cupicef.Term cupicef.prop (P l0) -> - H2 : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S l0) n) -> - __0 : cupicef.Term cupicef.prop (P n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) -> - H3 : - cupicef.Term cupicef.prop - (exists__between Q k (Coq__Init__Datatypes.S l0)) -> - l_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H4 : - cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S l0) l_27_) -> - H5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l_27_) -> - H6 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt l_27_ (Coq__Init__Datatypes.S l0)) -> - H7 : cupicef.Term cupicef.prop (Q l_27_) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt l_27_ l0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - l_27_ l0)) - -:= P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (between P k l) => - l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H0 : cupicef.Term cupicef.prop (between P k l0) => - IHbetween : - (__ : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k l0 n) -> - __0 : cupicef.Term cupicef.prop (P n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (exists__between Q k l0))) => - H1 : cupicef.Term cupicef.prop (P l0) => - H2 : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S l0) n) -> - __0 : cupicef.Term cupicef.prop (P n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) => - H3 : - cupicef.Term cupicef.prop - (exists__between Q k (Coq__Init__Datatypes.S l0)) => - l_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H4 : - cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S l0) l_27_) => - H5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l_27_) => - H6 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt l_27_ (Coq__Init__Datatypes.S l0)) => - H7 : cupicef.Term cupicef.prop (Q l_27_) => - Coq__Arith__Lt.le__lt__or__eq l_27_ l0 - (Coq__Arith__Lt.lt__n__Sm__le l_27_ l0 H6). - -def let__H6 : - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H : cupicef.Term cupicef.prop (between P k l) -> - l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H0 : cupicef.Term cupicef.prop (between P k l0) -> - IHbetween : - (__ : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k l0 n) -> - __0 : cupicef.Term cupicef.prop (P n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (exists__between Q k l0))) -> - H1 : cupicef.Term cupicef.prop (P l0) -> - H2 : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S l0) n) -> - __0 : cupicef.Term cupicef.prop (P n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) -> - H3 : - cupicef.Term cupicef.prop - (exists__between Q k (Coq__Init__Datatypes.S l0)) -> - l_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H4 : - cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S l0) l_27_) -> - H5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l_27_) -> - H6 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt l_27_ (Coq__Init__Datatypes.S l0)) -> - H7 : cupicef.Term cupicef.prop (Q l_27_) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - l_27_ l0) - -:= P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (between P k l) => - l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H0 : cupicef.Term cupicef.prop (between P k l0) => - IHbetween : - (__ : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k l0 n) -> - __0 : cupicef.Term cupicef.prop (P n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (exists__between Q k l0))) => - H1 : cupicef.Term cupicef.prop (P l0) => - H2 : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S l0) n) -> - __0 : cupicef.Term cupicef.prop (P n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) => - H3 : - cupicef.Term cupicef.prop - (exists__between Q k (Coq__Init__Datatypes.S l0)) => - l_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H4 : - cupicef.Term cupicef.prop (in__int k (Coq__Init__Datatypes.S l0) l_27_) => - H5 : cupicef.Term cupicef.prop (Coq__Init__Peano.le k l_27_) => - H6 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt l_27_ (Coq__Init__Datatypes.S l0)) => - H7 : cupicef.Term cupicef.prop (Q l_27_) => - Coq__Init__Logic.match____or cupicef.prop (Coq__Init__Peano.lt l_27_ l0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - l_27_ l0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt l_27_ l0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - l_27_ l0)) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - l_27_ l0) - (H8 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt l_27_ l0) => - Coq__Init__Logic.False__ind - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - l_27_ l0) - (IHbetween - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H9 : cupicef.Term cupicef.prop (in__int k l0 n) => - H10 : cupicef.Term cupicef.prop (P n) => - H2 n (in__int__S k l0 n H9) H10) - (in__int__exists Q k l0 l_27_ (in__int__intro k l0 l_27_ H5 H8) H7))) - (H8 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - l_27_ l0) => - H8) - (let__o3 P Q k l H l0 H0 IHbetween H1 H2 H3 l_27_ H4 H5 H6 H7). - -thm between__not__exists : - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (between P k l) -> - __0 : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __0 : cupicef.Term cupicef.prop (in__int k l n) -> - __1 : cupicef.Term cupicef.prop (P n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (exists__between Q k l)) - -:= P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (between P k l) => - between__ind P k - (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod - (cupicef.rule cupicef.set - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop))) - cupicef.prop - (cupicef.rule - (cupicef.rule cupicef.set - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop))) - cupicef.prop) - cupicef.I - (cupicef.prod cupicef.set - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop)) - (cupicef.rule cupicef.set - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop))) - cupicef.I Coq__Init__Datatypes.nat - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop)) - cupicef.I (in__int k l0 n) - (__ : cupicef.Term cupicef.prop (in__int k l0 n) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( - P n) - (__0 : cupicef.Term cupicef.prop (P n) => - Coq__Init__Logic.not (Q n))))) - (__ : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k l0 n) -> - __0 : cupicef.Term cupicef.prop (P n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) => - Coq__Init__Logic.not (exists__between Q k l0))) - (cupicef.cast - (cupicef.rule - (cupicef.rule cupicef.set - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop))) - (cupicef.rule cupicef.prop cupicef.prop)) - (cupicef.rule - (cupicef.rule cupicef.set - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop))) - cupicef.prop) - (cupicef.prod - (cupicef.rule cupicef.set - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop))) - (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule - (cupicef.rule cupicef.set - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop))) - (cupicef.rule cupicef.prop cupicef.prop)) - cupicef.I - (cupicef.prod cupicef.set - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop)) - (cupicef.rule cupicef.set - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop))) - cupicef.I Coq__Init__Datatypes.nat - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop)) - cupicef.I (in__int k k n) - (__ : cupicef.Term cupicef.prop (in__int k k n) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( - P n) - (__0 : cupicef.Term cupicef.prop (P n) => - Coq__Init__Logic.not (Q n))))) - (H0 : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k k n) -> - __0 : cupicef.Term cupicef.prop (P n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (exists__between Q k k) - (H1 : cupicef.Term cupicef.prop (exists__between Q k k) => - Coq__Init__Logic.False))) - (cupicef.prod - (cupicef.rule cupicef.set - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop))) - cupicef.prop - (cupicef.rule - (cupicef.rule cupicef.set - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop))) - cupicef.prop) - cupicef.I - (cupicef.prod cupicef.set - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop)) - (cupicef.rule cupicef.set - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop))) - cupicef.I Coq__Init__Datatypes.nat - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop)) - cupicef.I (in__int k k n) - (__ : cupicef.Term cupicef.prop (in__int k k n) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( - P n) - (__0 : cupicef.Term cupicef.prop (P n) => - Coq__Init__Logic.not (Q n))))) - (__ : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k k n) -> - __0 : cupicef.Term cupicef.prop (P n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) => - Coq__Init__Logic.not (exists__between Q k k))) - cupicef.I - (H0 : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k k n) -> - __0 : cupicef.Term cupicef.prop (P n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) => - H1 : cupicef.Term cupicef.prop (exists__between Q k k) => - Coq__Init__Logic.False__ind Coq__Init__Logic.False - (Coq__Arith__PeanoNat.Nat__lt__irrefl k - (cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S k) k) - (Coq__Init__Peano.lt k k) cupicef.I (exists__le__S Q k k H1))))) - (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H0 : cupicef.Term cupicef.prop (between P k l0) => - IHbetween : - (__ : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (in__int k l0 n) -> - __0 : cupicef.Term cupicef.prop (P n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (exists__between Q k l0))) => - H1 : cupicef.Term cupicef.prop (P l0) => - cupicef.cast - (cupicef.rule - (cupicef.rule cupicef.set - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop))) - (cupicef.rule cupicef.prop cupicef.prop)) - (cupicef.rule - (cupicef.rule cupicef.set - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop))) - cupicef.prop) - (cupicef.prod - (cupicef.rule cupicef.set - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop))) - (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule - (cupicef.rule cupicef.set - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop))) - (cupicef.rule cupicef.prop cupicef.prop)) - cupicef.I - (cupicef.prod cupicef.set - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop)) - (cupicef.rule cupicef.set - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop))) - cupicef.I Coq__Init__Datatypes.nat - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop)) - cupicef.I (in__int k (Coq__Init__Datatypes.S l0) n) - (__ : - cupicef.Term cupicef.prop - (in__int k (Coq__Init__Datatypes.S l0) n) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( - P n) - (__0 : cupicef.Term cupicef.prop (P n) => - Coq__Init__Logic.not (Q n))))) - (H2 : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (in__int k (Coq__Init__Datatypes.S l0) n) -> - __0 : cupicef.Term cupicef.prop (P n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (exists__between Q k (Coq__Init__Datatypes.S l0)) - (H3 : - cupicef.Term cupicef.prop - (exists__between Q k (Coq__Init__Datatypes.S l0)) => - Coq__Init__Logic.False))) - (cupicef.prod - (cupicef.rule cupicef.set - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop))) - cupicef.prop - (cupicef.rule - (cupicef.rule cupicef.set - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop))) - cupicef.prop) - cupicef.I - (cupicef.prod cupicef.set - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop)) - (cupicef.rule cupicef.set - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop))) - cupicef.I Coq__Init__Datatypes.nat - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop)) - cupicef.I (in__int k (Coq__Init__Datatypes.S l0) n) - (__ : - cupicef.Term cupicef.prop - (in__int k (Coq__Init__Datatypes.S l0) n) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I ( - P n) - (__0 : cupicef.Term cupicef.prop (P n) => - Coq__Init__Logic.not (Q n))))) - (__ : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (in__int k (Coq__Init__Datatypes.S l0) n) -> - __0 : cupicef.Term cupicef.prop (P n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) => - Coq__Init__Logic.not - (exists__between Q k (Coq__Init__Datatypes.S l0)))) - cupicef.I - (H2 : - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (in__int k (Coq__Init__Datatypes.S l0) n) -> - __0 : cupicef.Term cupicef.prop (P n) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Q n))) => - H3 : - cupicef.Term cupicef.prop - (exists__between Q k (Coq__Init__Datatypes.S l0)) => - Coq__Init__Logic.False__ind Coq__Init__Logic.False - (H2 l0 - (in__int__intro k (Coq__Init__Datatypes.S l0) l0 - (exists__S__le Q k l0 H3) - (cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S l0) - (Coq__Init__Datatypes.S l0)) - (Coq__Init__Peano.lt l0 (Coq__Init__Datatypes.S l0)) - cupicef.I - (Coq__Init__Peano.le__n (Coq__Init__Datatypes.S l0)))) - H1 - (Coq__Init__Logic.match____ex2 cupicef.prop - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - in__int k (Coq__Init__Datatypes.S l0) m) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Q m) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.ex2 - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (m : - cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - in__int k (Coq__Init__Datatypes.S l0) m) - (m : - cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Q m)) => - Q l0) - (l_27_ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H4 : - cupicef.Term cupicef.prop - (in__int k (Coq__Init__Datatypes.S l0) l_27_) => - Coq__Init__Logic.match____and cupicef.prop - (Coq__Init__Peano.le k l_27_) - (Coq__Init__Peano.lt l_27_ (Coq__Init__Datatypes.S l0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and (Coq__Init__Peano.le k l_27_) - (Coq__Init__Peano.lt l_27_ - (Coq__Init__Datatypes.S l0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Q l_27_) - (__0 : cupicef.Term cupicef.prop (Q l_27_) => Q l0)) - (H5 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le k l_27_) => - H6 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt l_27_ - (Coq__Init__Datatypes.S l0)) => - H7 : cupicef.Term cupicef.prop (Q l_27_) => - Coq__Init__Logic.eq__ind - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - l_27_ - (l1 : - cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Q l1) - H7 l0 - (let__H6 P Q k l H l0 H0 IHbetween H1 H2 H3 l_27_ H4 - H5 H6 H7)) - H4) - (let__e P Q k l H l0 H0 IHbetween H1 H2 H3))))) - l H. - -def P__nth : - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - init : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop. - -def nth__O : - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - init : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop (P__nth P Q init init Coq__Init__Datatypes.O). - -def nth__S : - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - init : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (P__nth P Q init k n) -> - __0 : cupicef.Term cupicef.prop (between P (Coq__Init__Datatypes.S k) l) -> - __1 : cupicef.Term cupicef.prop (Q l) -> - cupicef.Term cupicef.prop (P__nth P Q init l (Coq__Init__Datatypes.S n)). - -[ ] cupicef.code_guarded? _ (nth__O _ _ _) --> cupicef.guarded. - -[ ] cupicef.code_guarded? _ (nth__S _ _ _ _ _ _ _ _ _) --> cupicef.guarded. - -def match____P__nth : - s : cupicef.Sort -> - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - init : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - P0 : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop (P__nth P Q init __ __0) -> - cupicef.Univ s) -> - case__nth__O : - cupicef.Term s (P0 init Coq__Init__Datatypes.O (nth__O P Q init)) -> - case__nth__S : - (k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (P__nth P Q init k n) -> - __0 : cupicef.Term cupicef.prop (between P (Coq__Init__Datatypes.S k) l) -> - __1 : cupicef.Term cupicef.prop (Q l) -> - cupicef.Term s - (P0 l (Coq__Init__Datatypes.S n) (nth__S P Q init k l n __ __0 __1))) -> - __ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - x : cupicef.Term cupicef.prop (P__nth P Q init __ __0) -> - cupicef.Term s (P0 __ __0 x). - -[ s, P, Q, init, P0, case__nth__O, case__nth__S] - match____P__nth s P Q init P0 case__nth__O case__nth__S {init} - {Coq__Init__Datatypes.O} (nth__O P Q init) - --> - case__nth__O. - -[ s, P, Q, init, P0, case__nth__O, case__nth__S, k, l, n, __, __0, __1] - match____P__nth s P Q init P0 case__nth__O case__nth__S {l} - {Coq__Init__Datatypes.S n} (nth__S P Q init k l n __ __0 __1) - --> - case__nth__S k l n __ __0 __1. - -[ s, P, Q, init, P0, s'] - match____P__nth s P Q init - (__ => - __0 => - x => - cupicef.uncode _ (cupicef.code (cupicef.cU s') (P0 __ __0 x))) - --> - match____P__nth s' P Q init (__ => __0 => x => P0 __ __0 x). - -def P__nth__ind : - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - init : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - P0 : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - f : cupicef.Term cupicef.prop (P0 init Coq__Init__Datatypes.O) -> - f0 : - (k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (P__nth P Q init k n) -> - __0 : cupicef.Term cupicef.prop (P0 k n) -> - __1 : cupicef.Term cupicef.prop (between P (Coq__Init__Datatypes.S k) l) -> - __2 : cupicef.Term cupicef.prop (Q l) -> - cupicef.Term cupicef.prop (P0 l (Coq__Init__Datatypes.S n))) -> - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.prop (P__nth P Q init n n0) -> - cupicef.Term cupicef.prop (P0 n n0) - -:= P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - init : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - P0 : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - f : cupicef.Term cupicef.prop (P0 init Coq__Init__Datatypes.O) => - f0 : - (k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (P__nth P Q init k n) -> - __0 : cupicef.Term cupicef.prop (P0 k n) -> - __1 : - cupicef.Term cupicef.prop (between P (Coq__Init__Datatypes.S k) l) -> - __2 : cupicef.Term cupicef.prop (Q l) -> - cupicef.Term cupicef.prop (P0 l (Coq__Init__Datatypes.S n))) => - cupicef.fixproj cupicef.1 - (c => - c - (cupicef.SA cupicef.2 cupicef.prop - (cupicef.prod cupicef.set - (cupicef.rule cupicef.set - (cupicef.rule cupicef.prop cupicef.prop)) - (cupicef.rule cupicef.set - (cupicef.rule cupicef.set - (cupicef.rule cupicef.prop cupicef.prop))) - cupicef.I Coq__Init__Datatypes.nat - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.set - (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule cupicef.set - (cupicef.rule cupicef.prop cupicef.prop)) - cupicef.I Coq__Init__Datatypes.nat - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (P__nth P Q init n n0) - (p : cupicef.Term cupicef.prop (P__nth P Q init n n0) => - P0 n n0)))))) - (c => - c - (F => - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.prop (P__nth P Q init n n0) => - match____P__nth cupicef.prop P Q init - (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n2 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - __ : cupicef.Term cupicef.prop (P__nth P Q init n1 n2) => - P0 n1 n2) - f - (k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p0 : cupicef.Term cupicef.prop (P__nth P Q init k n1) => - b : - cupicef.Term cupicef.prop - (between P (Coq__Init__Datatypes.S k) l) => - q : cupicef.Term cupicef.prop (Q l) => - f0 k l n1 p0 (F k n1 p0) b q) - n n0 p)) - cupicef.0. - -thm nth__le : - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - init : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (P__nth P Q init l n) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le init l) - -:= P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - init : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - l : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (P__nth P Q init l n) => - P__nth__ind P Q init - (l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Peano.le init l0) - (Coq__Init__Peano.le__n init) - (k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - l0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H0 : cupicef.Term cupicef.prop (P__nth P Q init k n0) => - IHP__nth : cupicef.Term cupicef.prop (Coq__Init__Peano.le init k) => - H1 : - cupicef.Term cupicef.prop (between P (Coq__Init__Datatypes.S k) l0) => - H2 : cupicef.Term cupicef.prop (Q l0) => - Coq__Arith__PeanoNat.Nat__le__trans init k l0 IHP__nth - (Coq__Arith__PeanoNat.Nat__le__trans k k l0 - (Coq__Init__Peano.le__n k) - (Coq__Arith__PeanoNat.Nat__le__trans k k l0 - (Coq__Init__Peano.le__n k) - (Coq__Arith__PeanoNat.Nat__le__trans k k l0 - (Coq__Init__Peano.le__n k) - (Coq__Arith__PeanoNat.Nat__le__trans k - (Coq__Init__Datatypes.S k) l0 - (Coq__Arith__PeanoNat.Nat__le__succ__diag__r k) - (between__le P (Coq__Init__Datatypes.S k) l0 H1)))))) - l n H. - -def eventually : - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop - -:= Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.ex2 - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Peano.le k n) - (k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => Q k). - -def let__H1 : - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - H : cupicef.Term cupicef.prop (eventually Q Coq__Init__Datatypes.O) -> - x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H0 : - cupicef.Term cupicef.prop (Coq__Init__Peano.le x Coq__Init__Datatypes.O) -> - H1 : cupicef.Term cupicef.prop (Q x) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - x Coq__Init__Datatypes.O) - -:= Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - H : cupicef.Term cupicef.prop (eventually Q Coq__Init__Datatypes.O) => - x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H0 : - cupicef.Term cupicef.prop (Coq__Init__Peano.le x Coq__Init__Datatypes.O) => - H1 : cupicef.Term cupicef.prop (Q x) => - Coq__Arith__PeanoNat.Nat__le__antisymm x Coq__Init__Datatypes.O H0 - (Coq__Arith__PeanoNat.Nat__le__0__l x). - -thm event__O : - Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - __ : cupicef.Term cupicef.prop (eventually Q Coq__Init__Datatypes.O) -> - cupicef.Term cupicef.prop (Q Coq__Init__Datatypes.O) - -:= Q : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - H : cupicef.Term cupicef.prop (eventually Q Coq__Init__Datatypes.O) => - Coq__Init__Logic.match____ex2 cupicef.prop - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Peano.le k Coq__Init__Datatypes.O) - (k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => Q k) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.ex2 - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Peano.le k Coq__Init__Datatypes.O) - (k : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => Q k)) => - Q Coq__Init__Datatypes.O) - (x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le x Coq__Init__Datatypes.O) => - H1 : cupicef.Term cupicef.prop (Q x) => - Coq__Init__Logic.eq__ind - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - x (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => Q n) H1 - Coq__Init__Datatypes.O (let__H1 Q H x H0 H1)) - H. - -(; End of translation. ;) diff --git a/out/Coq__Arith__Between.dk.gz b/out/Coq__Arith__Between.dk.gz new file mode 100644 index 0000000000000000000000000000000000000000..32a04f860a726da916d74d93dce93e4bbafe6dd5 GIT binary patch literal 7657 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cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n Coq__Init__Datatypes.O) - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.match____nat cupicef.set - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Specif.sumbool - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O) - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) - (Coq__Init__Specif.left - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O Coq__Init__Datatypes.O) - (Coq__Init__Peano.lt Coq__Init__Datatypes.O Coq__Init__Datatypes.O) - (Coq__Init__Logic.eq__refl - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O)) - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Specif.right - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) - (Coq__Init__Peano.lt Coq__Init__Datatypes.O - (Coq__Init__Datatypes.S n0)) - (cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.le - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O) - (Coq__Init__Datatypes.S n0)) - (Coq__Init__Peano.lt Coq__Init__Datatypes.O - (Coq__Init__Datatypes.S n0)) - cupicef.I - (Coq__Arith__Gt.gt__le__S Coq__Init__Datatypes.O - (Coq__Init__Datatypes.S n0) - (cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O - (Coq__Init__Datatypes.S n0)) - (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n0) - Coq__Init__Datatypes.O) - cupicef.I (Coq__Arith__PeanoNat.Nat__lt__0__succ n0))))) - n. - -def let__s : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - IHn : - (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.set - (Coq__Init__Specif.sumor cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (Coq__Init__Peano.lt m0 n0))) -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.set - (Coq__Init__Specif.sumor cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n0 m1) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m1)) - (Coq__Init__Peano.lt m1 n0)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - IHn : - (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.set - (Coq__Init__Specif.sumor cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (Coq__Init__Peano.lt m0 n0))) => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - IHn m1. - -def lt__eq__lt__dec : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.set - (Coq__Init__Specif.sumor cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) - (Coq__Init__Peano.lt m n)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.nat__rec - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.set cupicef.set - (cupicef.rule cupicef.set cupicef.set) cupicef.I - Coq__Init__Datatypes.nat - (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Specif.sumor cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (Coq__Init__Peano.lt m0 n0))) - (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.match____nat cupicef.set - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Specif.sumor cupicef.set - (Coq__Init__Specif.sumbool - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O n0)) - (Coq__Init__Peano.lt n0 Coq__Init__Datatypes.O)) - (Coq__Init__Specif.inleft - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I - (Coq__Init__Specif.sumbool - (Coq__Init__Peano.lt Coq__Init__Datatypes.O - Coq__Init__Datatypes.O) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O Coq__Init__Datatypes.O))) - (Coq__Init__Peano.lt Coq__Init__Datatypes.O Coq__Init__Datatypes.O) - (Coq__Init__Specif.right - (Coq__Init__Peano.lt Coq__Init__Datatypes.O - Coq__Init__Datatypes.O) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O Coq__Init__Datatypes.O) - (Coq__Init__Logic.eq__refl - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O))) - (m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Specif.inleft - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I - (Coq__Init__Specif.sumbool - (Coq__Init__Peano.lt Coq__Init__Datatypes.O - (Coq__Init__Datatypes.S m1)) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1)))) - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S m1) - Coq__Init__Datatypes.O) - (Coq__Init__Specif.left - (Coq__Init__Peano.lt Coq__Init__Datatypes.O - (Coq__Init__Datatypes.S m1)) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1)) - (cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.le - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O) - (Coq__Init__Datatypes.S m1)) - (Coq__Init__Peano.lt Coq__Init__Datatypes.O - (Coq__Init__Datatypes.S m1)) - cupicef.I - (Coq__Arith__Gt.gt__le__S Coq__Init__Datatypes.O - (Coq__Init__Datatypes.S m1) (Coq__Arith__Gt.gt__Sn__O m1))))) - m0) - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - IHn : - (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.set - (Coq__Init__Specif.sumor cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (Coq__Init__Peano.lt m0 n0))) => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.match____nat cupicef.set - (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Specif.sumor cupicef.set - (Coq__Init__Specif.sumbool - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) n1) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S n0) n1)) - (Coq__Init__Peano.lt n1 (Coq__Init__Datatypes.S n0))) - (Coq__Init__Specif.inright - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I - (Coq__Init__Specif.sumbool - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - Coq__Init__Datatypes.O) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O))) - (Coq__Init__Peano.lt Coq__Init__Datatypes.O - (Coq__Init__Datatypes.S n0)) - (cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.le - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O) - (Coq__Init__Datatypes.S n0)) - (Coq__Init__Peano.lt Coq__Init__Datatypes.O - (Coq__Init__Datatypes.S n0)) - cupicef.I - (Coq__Arith__Gt.gt__le__S Coq__Init__Datatypes.O - (Coq__Init__Datatypes.S n0) - (cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O - (Coq__Init__Datatypes.S n0)) - (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n0) - Coq__Init__Datatypes.O) - cupicef.I (Coq__Arith__PeanoNat.Nat__lt__0__succ n0))))) - (m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Specif.match____sumor cupicef.set - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n0 m1) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m1))) - (Coq__Init__Peano.lt m1 n0) - (s0 : - cupicef.Term cupicef.set - (Coq__Init__Specif.sumor cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n0 m1) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m1)) - (Coq__Init__Peano.lt m1 n0)) => - Coq__Init__Specif.sumor cupicef.set - (Coq__Init__Specif.sumbool - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m1)) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S n0) (Coq__Init__Datatypes.S m1))) - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S m1) - (Coq__Init__Datatypes.S n0))) - (H : - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n0 m1) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m1)) => - Coq__Init__Specif.match____sumbool cupicef.set - (Coq__Init__Peano.lt n0 m1) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m1) - (s0 : - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n0 m1) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m1)) => - Coq__Init__Specif.sumor cupicef.set - (Coq__Init__Specif.sumbool - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m1)) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m1))) - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S m1) - (Coq__Init__Datatypes.S n0))) - (l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m1) => - Coq__Init__Specif.inleft - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I - (Coq__Init__Specif.sumbool - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m1)) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m1)))) - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S m1) - (Coq__Init__Datatypes.S n0)) - (Coq__Init__Specif.left - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m1)) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m1)) - (cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.le - (Coq__Init__Datatypes.S (Coq__Init__Datatypes.S n0)) - (Coq__Init__Datatypes.S m1)) - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m1)) - cupicef.I - (Coq__Arith__Le.le__n__S (Coq__Init__Datatypes.S n0) - m1 l)))) - (e : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m1) => - Coq__Init__Specif.inleft - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I - (Coq__Init__Specif.sumbool - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m1)) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m1)))) - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S m1) - (Coq__Init__Datatypes.S n0)) - (Coq__Init__Specif.right - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m1)) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m1)) - (Coq__Init__Peano.f__equal__nat - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.S n0 m1 e))) - H) - (H : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m1 n0) => - Coq__Init__Specif.inright - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I - (Coq__Init__Specif.sumbool - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m1)) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m1)))) - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S m1) - (Coq__Init__Datatypes.S n0)) - (cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.le - (Coq__Init__Datatypes.S (Coq__Init__Datatypes.S m1)) - (Coq__Init__Datatypes.S n0)) - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S m1) - (Coq__Init__Datatypes.S n0)) - cupicef.I - (Coq__Arith__Gt.gt__le__S (Coq__Init__Datatypes.S m1) - (Coq__Init__Datatypes.S n0) - (cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S m1) - (Coq__Init__Datatypes.S n0)) - (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m1)) - cupicef.I - (Coq__Arith__Lt.le__lt__n__Sm - (Coq__Init__Datatypes.S m1) n0 H))))) - (let__s n m n0 IHn m0 m1)) - m0) - n m. - -def gt__eq__gt__dec : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.set - (Coq__Init__Specif.sumor cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.gt m n) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) - (Coq__Init__Peano.gt n m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - lt__eq__lt__dec n m. - -def le__lt__dec : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.le n m) - (Coq__Init__Peano.lt m n)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.nat__rec - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.set cupicef.set - (cupicef.rule cupicef.set cupicef.set) cupicef.I - Coq__Init__Datatypes.nat - (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Specif.sumbool (Coq__Init__Peano.le n0 m0) - (Coq__Init__Peano.lt m0 n0))) - (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Specif.left (Coq__Init__Peano.le Coq__Init__Datatypes.O m0) - (Coq__Init__Peano.lt m0 Coq__Init__Datatypes.O) - (Coq__Arith__PeanoNat.Nat__le__0__l m0)) - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - IHn : - (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.le n0 m0) - (Coq__Init__Peano.lt m0 n0))) => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.match____nat cupicef.set - (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Specif.sumbool - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) n1) - (Coq__Init__Peano.lt n1 (Coq__Init__Datatypes.S n0))) - (Coq__Init__Specif.right - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - Coq__Init__Datatypes.O) - (Coq__Init__Peano.lt Coq__Init__Datatypes.O - (Coq__Init__Datatypes.S n0)) - (cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.le - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O) - (Coq__Init__Datatypes.S n0)) - (Coq__Init__Peano.lt Coq__Init__Datatypes.O - (Coq__Init__Datatypes.S n0)) - cupicef.I - (Coq__Arith__Gt.gt__le__S Coq__Init__Datatypes.O - (Coq__Init__Datatypes.S n0) - (cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O - (Coq__Init__Datatypes.S n0)) - (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n0) - Coq__Init__Datatypes.O) - cupicef.I - (cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.le - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O) - (Coq__Init__Datatypes.S n0)) - (Coq__Init__Peano.lt Coq__Init__Datatypes.O - (Coq__Init__Datatypes.S n0)) - cupicef.I - (Coq__Arith__Lt.lt__le__S Coq__Init__Datatypes.O - (Coq__Init__Datatypes.S n0) - (Coq__Arith__PeanoNat.Nat__lt__0__succ n0))))))) - (m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Specif.sumbool__rec (Coq__Init__Peano.le n0 m1) - (Coq__Init__Peano.lt m1 n0) - (s : - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.le n0 m1) - (Coq__Init__Peano.lt m1 n0)) => - Coq__Init__Specif.sumbool - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m1)) - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S m1) - (Coq__Init__Datatypes.S n0))) - (a : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m1) => - Coq__Init__Specif.left - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m1)) - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S m1) - (Coq__Init__Datatypes.S n0)) - (Coq__Arith__Gt.gt__le__S n0 (Coq__Init__Datatypes.S m1) - (cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m1)) - (Coq__Init__Peano.gt (Coq__Init__Datatypes.S m1) n0) - cupicef.I - (cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m1)) - (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m1)) - cupicef.I (Coq__Arith__Le.le__n__S n0 m1 a))))) - (b : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m1 n0) => - Coq__Init__Specif.right - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m1)) - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S m1) - (Coq__Init__Datatypes.S n0)) - (cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.le - (Coq__Init__Datatypes.S (Coq__Init__Datatypes.S m1)) - (Coq__Init__Datatypes.S n0)) - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S m1) - (Coq__Init__Datatypes.S n0)) - cupicef.I - (Coq__Arith__Gt.gt__le__S (Coq__Init__Datatypes.S m1) - (Coq__Init__Datatypes.S n0) - (cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S m1) - (Coq__Init__Datatypes.S n0)) - (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m1)) - cupicef.I (Coq__Arith__Lt.lt__n__S m1 n0 b))))) - (IHn m1)) - m0) - n m. - -def le__le__S__dec : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.le n m) - (Coq__Init__Peano.le (Coq__Init__Datatypes.S m) n)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - le__lt__dec n m. - -def le__ge__dec : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.le n m) - (Coq__Init__Peano.ge n m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Specif.sumbool__rec (Coq__Init__Peano.le n m) - (Coq__Init__Peano.lt m n) - (s : - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.le n m) - (Coq__Init__Peano.lt m n)) => - Coq__Init__Specif.sumbool (Coq__Init__Peano.le n m) - (Coq__Init__Peano.ge n m)) - (a : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) => - Coq__Init__Specif.left (Coq__Init__Peano.le n m) - (Coq__Init__Peano.ge n m) a) - (b : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) => - Coq__Init__Specif.right (Coq__Init__Peano.le n m) - (Coq__Init__Peano.ge n m) - (cupicef.cast cupicef.prop cupicef.prop (Coq__Init__Peano.le m n) - (Coq__Init__Peano.ge n m) cupicef.I - (Coq__Arith__PeanoNat.Nat__lt__le__incl m n b))) - (le__lt__dec n m). - -def le__gt__dec : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.le n m) - (Coq__Init__Peano.gt n m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - le__lt__dec n m. - -def let__s0 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> - cupicef.Term cupicef.set - (Coq__Init__Specif.sumor cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) - (Coq__Init__Peano.lt m n)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) => - lt__eq__lt__dec n m. - -def le__lt__eq__dec : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) => - Coq__Init__Specif.match____sumor cupicef.set - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) - cupicef.I - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m))) - (Coq__Init__Peano.lt m n) - (s0 : - cupicef.Term cupicef.set - (Coq__Init__Specif.sumor cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) - (Coq__Init__Peano.lt m n)) => - Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) - (s0 : - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) => - s0) - (l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) => - Coq__Init__Logic.False__rec - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) - (Coq__Arith__Lt.le__not__lt n m H l)) - (let__s0 n m H). - -def let__s1 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.le n m) - (Coq__Init__Peano.gt n m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - le__gt__dec n m. - -def le__dec : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.le n m) - (Coq__Init__Logic.not (Coq__Init__Peano.le n m))) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Specif.match____sumbool cupicef.set (Coq__Init__Peano.le n m) - (Coq__Init__Peano.gt n m) - (s0 : - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.le n m) - (Coq__Init__Peano.gt n m)) => - Coq__Init__Specif.sumbool (Coq__Init__Peano.le n m) - (Coq__Init__Logic.not (Coq__Init__Peano.le n m))) - (l : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) => - Coq__Init__Specif.left (Coq__Init__Peano.le n m) - (Coq__Init__Logic.not (Coq__Init__Peano.le n m)) l) - (g : cupicef.Term cupicef.prop (Coq__Init__Peano.gt n m) => - Coq__Init__Specif.right (Coq__Init__Peano.le n m) - (Coq__Init__Logic.not (Coq__Init__Peano.le n m)) - (Coq__Arith__Gt.gt__not__le n m g)) - (let__s1 n m). - -def lt__dec : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.not (Coq__Init__Peano.lt n m))) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - le__dec (Coq__Init__Datatypes.S n) m. - -def gt__dec : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.gt n m) - (Coq__Init__Logic.not (Coq__Init__Peano.gt n m))) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - lt__dec m n. - -def ge__dec : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.ge n m) - (Coq__Init__Logic.not (Coq__Init__Peano.ge n m))) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - le__dec m n. - -thm dec__le : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Logic__Decidable.decidable (Coq__Init__Peano.le n m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Arith__PeanoNat.Nat__le__decidable n m. - -thm dec__lt : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Logic__Decidable.decidable (Coq__Init__Peano.lt n m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Arith__PeanoNat.Nat__lt__decidable n m. - -thm dec__gt : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Logic__Decidable.decidable (Coq__Init__Peano.gt n m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Arith__PeanoNat.Nat__lt__decidable m n. - -thm dec__ge : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Logic__Decidable.decidable (Coq__Init__Peano.ge n m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Arith__PeanoNat.Nat__le__decidable m n. - -def let__H : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Peano.lt m0 n0)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) => - Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Peano.lt m0 n0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Peano.lt m0 n0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Peano.lt m0 n0)) => - Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0))) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) => - Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Peano.lt m0 n0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Peano.lt m0 n0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Peano.lt m0 n0)) => - Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) => - Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Peano.lt m0 n0))) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Peano.lt m0 n0))) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Peano.lt m0 n0)) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0))) => - __) - (Coq__Arith__PeanoNat.Nat__lt__gt__cases n0 m0). - -thm not__eq : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt n m) (Coq__Init__Peano.lt m n)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H n m n m. - -def let__H0 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) => - Coq__Init__Peano.lt m0 n0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt m0 n0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) => - Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0))) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) => - Coq__Init__Peano.lt m0 n0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt m0 n0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) => - Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) => - Coq__Init__Peano.lt m0 n0)) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0)) => - __0 : - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0))) => - __) - (Coq__Arith__PeanoNat.Nat__nle__gt n0 m0). - -thm not__le : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le n m)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.gt n m) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H0 n m n m. - -def let__H1 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le m0 n0) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) => - Coq__Init__Peano.le m0 n0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le m0 n0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le m0 n0) => - Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0))) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) => - Coq__Init__Peano.le m0 n0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le m0 n0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le m0 n0) => - Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) => - Coq__Init__Peano.le m0 n0)) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le m0 n0)) => - __0 : - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m0 n0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0))) => - __) - (Coq__Arith__PeanoNat.Nat__nlt__ge n0 m0). - -thm not__gt : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.gt n m)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H1 n m m n. - -def let__H2 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) => - Coq__Init__Peano.lt m0 n0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt m0 n0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) => - Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0))) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) => - Coq__Init__Peano.lt m0 n0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt m0 n0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) => - Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) => - Coq__Init__Peano.lt m0 n0)) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0)) => - __0 : - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le n0 m0))) => - __) - (Coq__Arith__PeanoNat.Nat__nle__gt n0 m0). - -thm not__ge : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.ge n m)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H2 n m m n. - -def let__H3 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le m0 n0) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) => - Coq__Init__Peano.le m0 n0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le m0 n0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le m0 n0) => - Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0))) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) => - Coq__Init__Peano.le m0 n0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le m0 n0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le m0 n0) => - Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) => - Coq__Init__Peano.le m0 n0)) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le m0 n0)) => - __0 : - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le m0 n0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.lt n0 m0))) => - __) - (Coq__Arith__PeanoNat.Nat__nlt__ge n0 m0). - -thm not__lt : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.lt n m)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.ge n m) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H3 n m n m. - -thm nat__compare__lt : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.iff (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n m) Coq__Init__Datatypes.Lt)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Classes__RelationClasses.symmetry - (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) - cupicef.I - (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) - cupicef.I)) - Coq__Init__Logic.iff Coq__Classes__RelationClasses.iff__Symmetric - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n m) Coq__Init__Datatypes.Lt) - (Coq__Init__Peano.lt n m) - (Coq__Arith__PeanoNat.Nat__compare__lt__iff n m). - -thm nat__compare__gt : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.iff (Coq__Init__Peano.gt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n m) Coq__Init__Datatypes.Gt)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Classes__RelationClasses.symmetry - (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) - cupicef.I - (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) - cupicef.I)) - Coq__Init__Logic.iff Coq__Classes__RelationClasses.iff__Symmetric - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n m) Coq__Init__Datatypes.Gt) - (Coq__Init__Peano.gt n m) - (Coq__Arith__PeanoNat.Nat__compare__gt__iff n m). - -thm nat__compare__le : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.iff (Coq__Init__Peano.le n m) - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n m) Coq__Init__Datatypes.Gt))) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Classes__RelationClasses.symmetry - (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) - cupicef.I - (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) - cupicef.I)) - Coq__Init__Logic.iff Coq__Classes__RelationClasses.iff__Symmetric - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n m) Coq__Init__Datatypes.Gt)) - (Coq__Init__Peano.le n m) - (Coq__Arith__PeanoNat.Nat__compare__le__iff n m). - -thm nat__compare__ge : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.iff (Coq__Init__Peano.ge n m) - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n m) Coq__Init__Datatypes.Lt))) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Classes__RelationClasses.symmetry - (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) - cupicef.I - (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) - cupicef.I)) - Coq__Init__Logic.iff Coq__Classes__RelationClasses.iff__Symmetric - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n m) Coq__Init__Datatypes.Lt)) - (Coq__Init__Peano.ge n m) - (Coq__Arith__PeanoNat.Nat__compare__ge__iff n m). - -def let__H4 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Eq) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Eq) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) - Coq__Init__Datatypes.Eq) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Eq)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) - Coq__Init__Datatypes.Eq) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) - Coq__Init__Datatypes.Eq) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) - Coq__Init__Datatypes.Eq))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Eq) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) - Coq__Init__Datatypes.Eq) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) - Coq__Init__Datatypes.Eq) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) - Coq__Init__Datatypes.Eq)) => - __) - (Coq__Arith__PeanoNat.Nat__compare__eq__iff n0 m0). - -thm nat__compare__eq : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n m) Coq__Init__Datatypes.Eq) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H4 n m n m. - -def let__H5 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Lt) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Lt) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) - Coq__Init__Datatypes.Lt) => - Coq__Init__Peano.lt n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Lt)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) - Coq__Init__Datatypes.Lt) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) - Coq__Init__Datatypes.Lt) => - Coq__Init__Peano.lt n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) - Coq__Init__Datatypes.Lt))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Lt) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) - Coq__Init__Datatypes.Lt) => - Coq__Init__Peano.lt n0 m0)) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) - Coq__Init__Datatypes.Lt) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => - __0 : - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) - Coq__Init__Datatypes.Lt)) => - __) - (Coq__Arith__PeanoNat.Nat__compare__lt__iff n0 m0). - -thm nat__compare__Lt__lt : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n m) Coq__Init__Datatypes.Lt) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H5 n m n m. - -def let__H6 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare x y) Coq__Init__Datatypes.Gt) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt y x) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare x y) Coq__Init__Datatypes.Gt) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare x y) - Coq__Init__Datatypes.Gt) => - Coq__Init__Peano.lt y x)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt y x) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt y x) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare x y) Coq__Init__Datatypes.Gt)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare x y) - Coq__Init__Datatypes.Gt) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare x y) - Coq__Init__Datatypes.Gt) => - Coq__Init__Peano.lt y x)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt y x) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt y x) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare x y) - Coq__Init__Datatypes.Gt))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare x y) Coq__Init__Datatypes.Gt) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare x y) - Coq__Init__Datatypes.Gt) => - Coq__Init__Peano.lt y x)) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare x y) - Coq__Init__Datatypes.Gt) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt y x)) => - __0 : - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt y x) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare x y) Coq__Init__Datatypes.Gt)) => - __) - (Coq__Arith__PeanoNat.Nat__compare__gt__iff x y). - -thm nat__compare__Gt__gt : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n m) Coq__Init__Datatypes.Gt) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.gt n m) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H6 n m n m. - -def nat__compare__alt : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.set Coq__Init__Datatypes.comparison - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Specif.match____sumor cupicef.set - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) - cupicef.I - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m))) - (Coq__Init__Peano.lt m n) - (x : - cupicef.Term cupicef.set - (Coq__Init__Specif.sumor cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) - (Coq__Init__Peano.lt m n)) => - Coq__Init__Datatypes.comparison) - (s : - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) => - Coq__Init__Specif.match____sumbool cupicef.set - (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m) - (s0 : - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) => - Coq__Init__Datatypes.comparison) - (l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) => - Coq__Init__Datatypes.Lt) - (e : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m) => - Coq__Init__Datatypes.Eq) - s) - (l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) => - Coq__Init__Datatypes.Gt) - (lt__eq__lt__dec n m). - -def let__s2 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.set - (Coq__Init__Specif.sumor cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) - (Coq__Init__Peano.lt m n)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - lt__eq__lt__dec n m. - -def let__H7 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - s0 : - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) -> - l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Lt) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - s0 : - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) => - l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Lt) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) - Coq__Init__Datatypes.Lt) => - Coq__Init__Peano.lt n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Lt)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) - Coq__Init__Datatypes.Lt) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) - Coq__Init__Datatypes.Lt) => - Coq__Init__Peano.lt n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) - Coq__Init__Datatypes.Lt))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Lt)) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) - Coq__Init__Datatypes.Lt) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => - __0 : - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) - Coq__Init__Datatypes.Lt)) => - __0) - (Coq__Arith__PeanoNat.Nat__compare__lt__iff n0 m0). - -def let__H8 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - s0 : - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) -> - e : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m) -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Eq) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - s0 : - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) => - e : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m) => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Eq) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) - Coq__Init__Datatypes.Eq) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Eq)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) - Coq__Init__Datatypes.Eq) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) - Coq__Init__Datatypes.Eq) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) - Coq__Init__Datatypes.Eq))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) Coq__Init__Datatypes.Eq)) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) - Coq__Init__Datatypes.Eq) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n0 m0) - Coq__Init__Datatypes.Eq)) => - __0) - (Coq__Arith__PeanoNat.Nat__compare__eq__iff n0 m0). - -def let__H9 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) -> - x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt y x) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare x y) Coq__Init__Datatypes.Gt) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) => - x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare x y) Coq__Init__Datatypes.Gt) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare x y) - Coq__Init__Datatypes.Gt) => - Coq__Init__Peano.lt y x)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt y x) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt y x) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare x y) Coq__Init__Datatypes.Gt)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare x y) - Coq__Init__Datatypes.Gt) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare x y) - Coq__Init__Datatypes.Gt) => - Coq__Init__Peano.lt y x)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt y x) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt y x) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare x y) - Coq__Init__Datatypes.Gt))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt y x) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt y x) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare x y) Coq__Init__Datatypes.Gt)) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare x y) - Coq__Init__Datatypes.Gt) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt y x)) => - __0 : - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt y x) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare x y) Coq__Init__Datatypes.Gt)) => - __0) - (Coq__Arith__PeanoNat.Nat__compare__gt__iff x y). - -thm nat__compare__equiv : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n m) (nat__compare__alt n m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n m) - (Coq__Init__Specif.match____sumor cupicef.set - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m))) - (Coq__Init__Peano.lt m n) - (x : - cupicef.Term cupicef.set - (Coq__Init__Specif.sumor cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) - (Coq__Init__Peano.lt m n)) => - Coq__Init__Datatypes.comparison) - (s : - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) => - Coq__Init__Specif.match____sumbool cupicef.set - (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m) - (s0 : - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) => - Coq__Init__Datatypes.comparison) - (l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) => - Coq__Init__Datatypes.Lt) - (e : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m) => - Coq__Init__Datatypes.Eq) - s) - (l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) => - Coq__Init__Datatypes.Gt) - (lt__eq__lt__dec n m))) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n m) (nat__compare__alt n m)) - cupicef.I - (Coq__Init__Specif.match____sumor cupicef.prop - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) - cupicef.I - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m))) - (Coq__Init__Peano.lt m n) - (s0 : - cupicef.Term cupicef.set - (Coq__Init__Specif.sumor cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) - (Coq__Init__Peano.lt m n)) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n m) - (Coq__Init__Specif.match____sumor cupicef.set - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m))) - (Coq__Init__Peano.lt m n) - (x : - cupicef.Term cupicef.set - (Coq__Init__Specif.sumor cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast - (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) - (Coq__Init__Peano.lt m n)) => - Coq__Init__Datatypes.comparison) - (s1 : - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) => - Coq__Init__Specif.match____sumbool cupicef.set - (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m) - (s2 : - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast - (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) => - Coq__Init__Datatypes.comparison) - (l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) => - Coq__Init__Datatypes.Lt) - (e : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m) => - Coq__Init__Datatypes.Eq) - s1) - (l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) => - Coq__Init__Datatypes.Gt) - s0)) - (s0 : - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) => - Coq__Init__Specif.match____sumbool cupicef.prop - (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m) - (s1 : - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n m) - (Coq__Init__Specif.match____sumbool cupicef.set - (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m) - (s2 : - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast - (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) => - Coq__Init__Datatypes.comparison) - (l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) => - Coq__Init__Datatypes.Lt) - (e : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m) => - Coq__Init__Datatypes.Eq) - s1)) - (l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) => - let__H7 n m s0 l n m l) - (e : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m) => - let__H8 n m s0 e n m e) - s0) - (l : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) => - let__H9 n m l n m l) - (let__s2 n m)). - -thm leb__iff__conv : - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.iff - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n m) Coq__Init__Datatypes.false) - (Coq__Init__Peano.lt m n)) - -:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - (lemma : - cupicef.Term cupicef.prop - (Coq__Init__Logic.iff - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n m) Coq__Init__Datatypes.false) - (Coq__Init__Logic.not (Coq__Init__Peano.le n m))) => - Coq__Classes__Morphisms.trans__co__eq__inv__impl__morphism - (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) - cupicef.I - (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) - cupicef.I)) - Coq__Init__Logic.iff Coq__Classes__RelationClasses.iff__Transitive - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n m) Coq__Init__Datatypes.false) - (Coq__Init__Logic.not (Coq__Init__Peano.le n m)) lemma - (Coq__Init__Peano.lt m n) (Coq__Init__Peano.lt m n) - (Coq__Classes__Morphisms.eq__proper__proxy - (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) - cupicef.I - (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) - cupicef.I)) - (Coq__Init__Peano.lt m n))) - (Coq__Arith__PeanoNat.Nat__leb__nle n m) - (Coq__Arith__PeanoNat.Nat__nle__gt n m). - -def let__H10 : - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.true) - -:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.true) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) - Coq__Init__Datatypes.true) => - Coq__Init__Peano.le n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.true)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) - Coq__Init__Datatypes.true) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) - Coq__Init__Datatypes.true) => - Coq__Init__Peano.le n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) - Coq__Init__Datatypes.true))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.true)) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) - Coq__Init__Datatypes.true) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => - __0 : - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.true)) => - __0) - (Coq__Arith__PeanoNat.Nat__leb__le n0 m0). - -thm leb__correct : - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb m n) Coq__Init__Datatypes.true) - -:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H10 m n m n. - -def let__H11 : - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.true) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) - -:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.true) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) - Coq__Init__Datatypes.true) => - Coq__Init__Peano.le n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.true)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) - Coq__Init__Datatypes.true) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) - Coq__Init__Datatypes.true) => - Coq__Init__Peano.le n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) - Coq__Init__Datatypes.true))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.true) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) - Coq__Init__Datatypes.true) => - Coq__Init__Peano.le n0 m0)) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) - Coq__Init__Datatypes.true) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => - __0 : - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.true)) => - __) - (Coq__Arith__PeanoNat.Nat__leb__le n0 m0). - -thm leb__complete : - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb m n) Coq__Init__Datatypes.true) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le m n) - -:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H11 m n m n. - -def let__H12 : - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.false) - -:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.false) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) - Coq__Init__Datatypes.false) => - Coq__Init__Peano.lt m0 n0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt m0 n0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.false)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) - Coq__Init__Datatypes.false) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) - Coq__Init__Datatypes.false) => - Coq__Init__Peano.lt m0 n0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt m0 n0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) - Coq__Init__Datatypes.false))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt m0 n0) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.false)) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) - Coq__Init__Datatypes.false) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0)) => - __0 : - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) - Coq__Init__Datatypes.false)) => - __0) - (leb__iff__conv m0 n0). - -thm leb__correct__conv : - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n m) Coq__Init__Datatypes.false) - -:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H12 m n m n. - -def let__H13 : - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.false) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) - -:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.false) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) - Coq__Init__Datatypes.false) => - Coq__Init__Peano.lt m0 n0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt m0 n0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.false)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) - Coq__Init__Datatypes.false) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) - Coq__Init__Datatypes.false) => - Coq__Init__Peano.lt m0 n0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt m0 n0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) - Coq__Init__Datatypes.false))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) Coq__Init__Datatypes.false) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) - Coq__Init__Datatypes.false) => - Coq__Init__Peano.lt m0 n0)) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) - Coq__Init__Datatypes.false) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0)) => - __0 : - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m0 n0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n0 m0) - Coq__Init__Datatypes.false)) => - __) - (leb__iff__conv m0 n0). - -thm leb__complete__conv : - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n m) Coq__Init__Datatypes.false) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) - -:= m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H13 m n m n. - -thm leb__compare : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.iff - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n m) Coq__Init__Datatypes.true) - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n m) Coq__Init__Datatypes.Gt))) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - (lemma : - cupicef.Term cupicef.prop - (Coq__Init__Logic.iff - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n m) - Coq__Init__Datatypes.Gt)) - (Coq__Init__Peano.le n m)) => - Coq__Classes__Morphisms.trans__sym__co__inv__impl__morphism - (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) - cupicef.I - (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) - cupicef.I)) - Coq__Init__Logic.iff - (Coq__Classes__RelationClasses.Equivalence__PER - (cupicef.cast (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) - cupicef.I - (cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) - cupicef.I)) - Coq__Init__Logic.iff Coq__Classes__RelationClasses.iff__equivalence) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__leb n m) Coq__Init__Datatypes.true) - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.comparison) - (Coq__Arith__PeanoNat.Nat__compare n m) Coq__Init__Datatypes.Gt)) - (Coq__Init__Peano.le n m) lemma) - (Coq__Arith__PeanoNat.Nat__compare__le__iff n m) - (Coq__Arith__PeanoNat.Nat__leb__le n m). - -(; End of translation. ;) diff --git a/out/Coq__Arith__Compare__dec.dk.gz b/out/Coq__Arith__Compare__dec.dk.gz new file mode 100644 index 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Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop - -:= cupicef.fixproj cupicef.1 - (c => - c - (cupicef.SA cupicef.0 (cupicef.type (cupicef.s cupicef.z)) - (cupicef.prod cupicef.set - (cupicef.rule cupicef.set (cupicef.type (cupicef.s cupicef.z))) - (cupicef.rule cupicef.set - (cupicef.rule cupicef.set - (cupicef.type (cupicef.s cupicef.z)))) - cupicef.I Coq__Init__Datatypes.nat - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.set (cupicef.type (cupicef.s cupicef.z)) - (cupicef.rule cupicef.set - (cupicef.type (cupicef.s cupicef.z))) - cupicef.I Coq__Init__Datatypes.nat - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.univ cupicef.prop - (cupicef.type (cupicef.s cupicef.z)) cupicef.I))))) - (c => - c - (eq__nat0 => - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.match____nat - (cupicef.type (cupicef.s cupicef.z)) - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (Coq__Init__Datatypes.match____nat - (cupicef.type (cupicef.s cupicef.z)) - (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - Coq__Init__Logic.True - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.False) - m) - (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.match____nat - (cupicef.type (cupicef.s cupicef.z)) - (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - Coq__Init__Logic.False - (m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - eq__nat0 n1 m1) - m) - n)) - cupicef.0. - -thm eq__nat__refl : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop (eq__nat n n) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.nat__ind - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => eq__nat n0 n0) - (cupicef.cast cupicef.prop cupicef.prop Coq__Init__Logic.True - (eq__nat Coq__Init__Datatypes.O Coq__Init__Datatypes.O) cupicef.I - Coq__Init__Logic.I) - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - IHn : cupicef.Term cupicef.prop (eq__nat n0 n0) => - cupicef.cast cupicef.prop cupicef.prop (eq__nat n0 n0) - (eq__nat (Coq__Init__Datatypes.S n0) (Coq__Init__Datatypes.S n0)) - cupicef.I IHn) - n. - -thm eq__nat__is__eq : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.iff (eq__nat n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.conj - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eq__nat n m) - (__ : cupicef.Term cupicef.prop (eq__nat n m) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m) => - eq__nat n m)) - (Coq__Init__Datatypes.nat__ind - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) - cupicef.I Coq__Init__Datatypes.nat - (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (eq__nat n0 m0) - (__ : cupicef.Term cupicef.prop (eq__nat n0 m0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0))) - (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.match____nat cupicef.prop - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (eq__nat Coq__Init__Datatypes.O n0) - (__ : - cupicef.Term cupicef.prop - (eq__nat Coq__Init__Datatypes.O n0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O n0)) - (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - Coq__Init__Logic.True - (H : cupicef.Term cupicef.prop Coq__Init__Logic.True => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O Coq__Init__Datatypes.O)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (eq__nat Coq__Init__Datatypes.O Coq__Init__Datatypes.O) - (__ : - cupicef.Term cupicef.prop - (eq__nat Coq__Init__Datatypes.O Coq__Init__Datatypes.O) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O Coq__Init__Datatypes.O)) - cupicef.I - (H : cupicef.Term cupicef.prop Coq__Init__Logic.True => - Coq__Init__Logic.eq__refl - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O)) - (m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - Coq__Init__Logic.False - (H : cupicef.Term cupicef.prop Coq__Init__Logic.False => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (eq__nat Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1)) - (__ : - cupicef.Term cupicef.prop - (eq__nat Coq__Init__Datatypes.O - (Coq__Init__Datatypes.S m1)) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1))) - cupicef.I - (H : cupicef.Term cupicef.prop Coq__Init__Logic.False => - Coq__Init__Logic.False__ind - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m1)) - H)) - m0) - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - IHn : - (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (eq__nat n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.match____nat cupicef.prop - (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (eq__nat (Coq__Init__Datatypes.S n0) n1) - (__ : - cupicef.Term cupicef.prop - (eq__nat (Coq__Init__Datatypes.S n0) n1) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S n0) n1)) - (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - Coq__Init__Logic.False - (H : cupicef.Term cupicef.prop Coq__Init__Logic.False => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (eq__nat (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) - (__ : - cupicef.Term cupicef.prop - (eq__nat (Coq__Init__Datatypes.S n0) - Coq__Init__Datatypes.O) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O)) - cupicef.I - (H : cupicef.Term cupicef.prop Coq__Init__Logic.False => - Coq__Init__Logic.False__ind - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) - H)) - (m1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (eq__nat n0 m1) - (H : cupicef.Term cupicef.prop (eq__nat n0 m1) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S n0) (Coq__Init__Datatypes.S m1))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (eq__nat (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m1)) - (__ : - cupicef.Term cupicef.prop - (eq__nat (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m1)) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S n0) (Coq__Init__Datatypes.S m1))) - cupicef.I - (H : cupicef.Term cupicef.prop (eq__nat n0 m1) => - Coq__Init__Peano.f__equal__nat - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.S n0 m1 (IHn m1 H))) - m0) - n m) - (H : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m) => - Coq__Init__Logic.eq__ind - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n - (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - eq__nat n m0) - (eq__nat__refl n) m H). - -def let__H : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) -> - cupicef.Term cupicef.prop (eq__nat n0 m0) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eq__nat n0 m0) - (__ : cupicef.Term cupicef.prop (eq__nat n0 m0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) => - eq__nat n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (eq__nat n0 m0) - (__ : cupicef.Term cupicef.prop (eq__nat n0 m0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) => - eq__nat n0 m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) => - eq__nat n0 m0)) - (__ : - (__ : cupicef.Term cupicef.prop (eq__nat n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) -> - cupicef.Term cupicef.prop (eq__nat n0 m0)) => - __0) - (eq__nat__is__eq n0 m0). - -thm eq__eq__nat : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m) -> - cupicef.Term cupicef.prop (eq__nat n m) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H n m n m. - -def let__H0 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (eq__nat n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eq__nat n0 m0) - (__ : cupicef.Term cupicef.prop (eq__nat n0 m0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) => - eq__nat n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (eq__nat n0 m0) - (__ : cupicef.Term cupicef.prop (eq__nat n0 m0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) => - eq__nat n0 m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I (eq__nat n0 m0) - (__0 : cupicef.Term cupicef.prop (eq__nat n0 m0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (__ : - (__ : cupicef.Term cupicef.prop (eq__nat n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) -> - cupicef.Term cupicef.prop (eq__nat n0 m0)) => - __) - (eq__nat__is__eq n0 m0). - -thm eq__nat__eq : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (eq__nat n m) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H0 n m n m. - -def let__H1 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - H : cupicef.Term cupicef.prop (P n) -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H0 : cupicef.Term cupicef.prop (eq__nat n m) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - H : cupicef.Term cupicef.prop (P n) => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H0 : cupicef.Term cupicef.prop (eq__nat n m) => - eq__nat__eq n m H0. - -thm eq__nat__elim : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - __ : cupicef.Term cupicef.prop (P n) -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __0 : cupicef.Term cupicef.prop (eq__nat n m) -> - cupicef.Term cupicef.prop (P m) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - H : cupicef.Term cupicef.prop (P n) => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H0 : cupicef.Term cupicef.prop (eq__nat n m) => - Coq__Init__Logic.eq__ind - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => P m0) H m - (let__H1 n P H m H0). - -def eq__nat__decide : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (eq__nat n m) - (Coq__Init__Logic.not (eq__nat n m))) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.nat__rec - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.set cupicef.set - (cupicef.rule cupicef.set cupicef.set) cupicef.I - Coq__Init__Datatypes.nat - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Specif.sumbool (eq__nat n0 m) - (Coq__Init__Logic.not (eq__nat n0 m)))) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.match____nat cupicef.set - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Specif.sumbool (eq__nat Coq__Init__Datatypes.O n0) - (Coq__Init__Logic.not (eq__nat Coq__Init__Datatypes.O n0))) - (cupicef.cast cupicef.set cupicef.set - (Coq__Init__Specif.sumbool Coq__Init__Logic.True - (Coq__Init__Logic.not Coq__Init__Logic.True)) - (Coq__Init__Specif.sumbool - (eq__nat Coq__Init__Datatypes.O Coq__Init__Datatypes.O) - (Coq__Init__Logic.not - (eq__nat Coq__Init__Datatypes.O Coq__Init__Datatypes.O))) - cupicef.I - (Coq__Init__Specif.left Coq__Init__Logic.True - (Coq__Init__Logic.not Coq__Init__Logic.True) Coq__Init__Logic.I)) - (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.cast cupicef.set cupicef.set - (Coq__Init__Specif.sumbool Coq__Init__Logic.False - (Coq__Init__Logic.not Coq__Init__Logic.False)) - (Coq__Init__Specif.sumbool - (eq__nat Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m0)) - (Coq__Init__Logic.not - (eq__nat Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m0)))) - cupicef.I - (Coq__Init__Specif.right Coq__Init__Logic.False - (Coq__Init__Logic.not Coq__Init__Logic.False) - (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) - cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - Coq__Init__Logic.False - (H : cupicef.Term cupicef.prop Coq__Init__Logic.False => - Coq__Init__Logic.False)) - (Coq__Init__Logic.not Coq__Init__Logic.False) cupicef.I - (H : cupicef.Term cupicef.prop Coq__Init__Logic.False => H)))) - m) - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - IHn : - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.set - (Coq__Init__Specif.sumbool (eq__nat n0 m) - (Coq__Init__Logic.not (eq__nat n0 m)))) => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.match____nat cupicef.set - (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Specif.sumbool (eq__nat (Coq__Init__Datatypes.S n0) n1) - (Coq__Init__Logic.not (eq__nat (Coq__Init__Datatypes.S n0) n1))) - (cupicef.cast cupicef.set cupicef.set - (Coq__Init__Specif.sumbool Coq__Init__Logic.False - (Coq__Init__Logic.not Coq__Init__Logic.False)) - (Coq__Init__Specif.sumbool - (eq__nat (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) - (Coq__Init__Logic.not - (eq__nat (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O))) - cupicef.I - (Coq__Init__Specif.right Coq__Init__Logic.False - (Coq__Init__Logic.not Coq__Init__Logic.False) - (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) - cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - Coq__Init__Logic.False - (H : cupicef.Term cupicef.prop Coq__Init__Logic.False => - Coq__Init__Logic.False)) - (Coq__Init__Logic.not Coq__Init__Logic.False) cupicef.I - (H : cupicef.Term cupicef.prop Coq__Init__Logic.False => H)))) - (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.cast cupicef.set cupicef.set - (Coq__Init__Specif.sumbool (eq__nat n0 m0) - (Coq__Init__Logic.not (eq__nat n0 m0))) - (Coq__Init__Specif.sumbool - (eq__nat (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) - (Coq__Init__Logic.not - (eq__nat (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)))) - cupicef.I (IHn m0)) - m) - n. - -thm beq__nat__refl : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true (Coq__Arith__PeanoNat.Nat__eqb n n)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq__sym - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__eqb n n) Coq__Init__Datatypes.true - (Coq__Arith__PeanoNat.Nat__eqb__refl n). - -def let__H2 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__eqb n0 m0) Coq__Init__Datatypes.true) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__eqb n0 m0) Coq__Init__Datatypes.true) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__eqb n0 m0) - Coq__Init__Datatypes.true) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__eqb n0 m0) Coq__Init__Datatypes.true)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__eqb n0 m0) - Coq__Init__Datatypes.true) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__eqb n0 m0) - Coq__Init__Datatypes.true) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__eqb n0 m0) - Coq__Init__Datatypes.true))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__eqb n0 m0) Coq__Init__Datatypes.true) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__eqb n0 m0) - Coq__Init__Datatypes.true) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__eqb n0 m0) - Coq__Init__Datatypes.true) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__eqb n0 m0) Coq__Init__Datatypes.true)) => - __) - (Coq__Arith__PeanoNat.Nat__eqb__eq n0 m0). - -thm beq__nat__true : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__eqb n m) Coq__Init__Datatypes.true) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H2 n m n m. - -def let__H3 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__eqb x y) Coq__Init__Datatypes.false) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - x y)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - x : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__eqb x y) Coq__Init__Datatypes.false) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__eqb x y) - Coq__Init__Datatypes.false) => - Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - x y))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - x y)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - x y)) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__eqb x y) Coq__Init__Datatypes.false)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__eqb x y) - Coq__Init__Datatypes.false) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__eqb x y) - Coq__Init__Datatypes.false) => - Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - x y))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - x y)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - x y)) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__eqb x y) - Coq__Init__Datatypes.false))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__eqb x y) Coq__Init__Datatypes.false) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__eqb x y) - Coq__Init__Datatypes.false) => - Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - x y))) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__eqb x y) - Coq__Init__Datatypes.false) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - x y))) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - x y)) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__eqb x y) Coq__Init__Datatypes.false)) => - __) - (Coq__Arith__PeanoNat.Nat__eqb__neq x y). - -thm beq__nat__false : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - (Coq__Arith__PeanoNat.Nat__eqb n m) Coq__Init__Datatypes.false) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H3 n m n m. - -def let__H4 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true Coq__Init__Datatypes.false) -> - cupicef.Term cupicef.prop Coq__Init__Logic.False - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => - Coq__Init__Logic.eq__ind - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true - (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => - Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) - (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => - cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - Coq__Init__Logic.True Coq__Init__Logic.False e) - Coq__Init__Logic.I Coq__Init__Datatypes.false H. - -def let__H5 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - IHn : - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true (Coq__Arith__PeanoNat.Nat__eqb n0 m)) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m)) -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true Coq__Init__Datatypes.false) -> - cupicef.Term cupicef.prop Coq__Init__Logic.False - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - IHn : - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true (Coq__Arith__PeanoNat.Nat__eqb n0 m)) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m)) => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => - Coq__Init__Logic.eq__ind - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true - (e : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => - Coq__Init__Datatypes.match____bool (cupicef.type (cupicef.s cupicef.z)) - (b : cupicef.Term cupicef.set Coq__Init__Datatypes.bool => - cupicef.univ cupicef.prop (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - Coq__Init__Logic.True Coq__Init__Logic.False e) - Coq__Init__Logic.I Coq__Init__Datatypes.false H. - -def beq__nat__eq : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true (Coq__Arith__PeanoNat.Nat__eqb n m)) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.nat__ind - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) - cupicef.I Coq__Init__Datatypes.nat - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true (Coq__Arith__PeanoNat.Nat__eqb n0 m)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true - (Coq__Arith__PeanoNat.Nat__eqb n0 m)) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m))) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.match____nat cupicef.prop - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true - (Coq__Arith__PeanoNat.Nat__eqb Coq__Init__Datatypes.O n0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true - (Coq__Arith__PeanoNat.Nat__eqb Coq__Init__Datatypes.O n0)) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O n0)) - (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true Coq__Init__Datatypes.true) - (H : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O Coq__Init__Datatypes.O)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true - (Coq__Arith__PeanoNat.Nat__eqb Coq__Init__Datatypes.O - Coq__Init__Datatypes.O)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true - (Coq__Arith__PeanoNat.Nat__eqb Coq__Init__Datatypes.O - Coq__Init__Datatypes.O)) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O Coq__Init__Datatypes.O)) - cupicef.I - (H : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true Coq__Init__Datatypes.true) => - Coq__Init__Logic.eq__refl - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O)) - (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true Coq__Init__Datatypes.false) - (H : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true - (Coq__Arith__PeanoNat.Nat__eqb Coq__Init__Datatypes.O - (Coq__Init__Datatypes.S m0))) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true - (Coq__Arith__PeanoNat.Nat__eqb Coq__Init__Datatypes.O - (Coq__Init__Datatypes.S m0))) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m0))) - cupicef.I - (H : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => - Coq__Init__Logic.False__ind - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O (Coq__Init__Datatypes.S m0)) - (let__H4 n m m0 H))) - m) - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - IHn : - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true - (Coq__Arith__PeanoNat.Nat__eqb n0 m)) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m)) => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.match____nat cupicef.prop - (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true - (Coq__Arith__PeanoNat.Nat__eqb (Coq__Init__Datatypes.S n0) n1)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true - (Coq__Arith__PeanoNat.Nat__eqb (Coq__Init__Datatypes.S n0) - n1)) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S n0) n1)) - (cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true Coq__Init__Datatypes.false) - (H : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true - (Coq__Arith__PeanoNat.Nat__eqb (Coq__Init__Datatypes.S n0) - Coq__Init__Datatypes.O)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true - (Coq__Arith__PeanoNat.Nat__eqb - (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O)) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O)) - cupicef.I - (H : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true Coq__Init__Datatypes.false) => - Coq__Init__Logic.False__ind - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S n0) Coq__Init__Datatypes.O) - (let__H5 n n0 IHn m H))) - (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true - (Coq__Arith__PeanoNat.Nat__eqb n0 m0)) - (H : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true - (Coq__Arith__PeanoNat.Nat__eqb n0 m0)) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S n0) (Coq__Init__Datatypes.S m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true - (Coq__Arith__PeanoNat.Nat__eqb (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true - (Coq__Arith__PeanoNat.Nat__eqb - (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S n0) (Coq__Init__Datatypes.S m0))) - cupicef.I - (H : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.bool) - Coq__Init__Datatypes.true - (Coq__Arith__PeanoNat.Nat__eqb n0 m0)) => - Coq__Init__Logic.match____eq cupicef.prop - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 - (y : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 y) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S n0) (Coq__Init__Datatypes.S y)) - (Coq__Init__Logic.eq__refl - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S n0)) - m0 (IHn m0 H))) - m) - n. - -(; End of translation. ;) diff --git a/out/Coq__Arith__EqNat.dk.gz b/out/Coq__Arith__EqNat.dk.gz new file mode 100644 index 0000000000000000000000000000000000000000..77abe4a9d3c0ac66a3ba8c90e88d85e717f2ed78 GIT binary patch literal 5081 zcmV;~6DI5*iwFpw=F?*U14D0dUtd9TX>@2`Uqx|FVRSBJYXI$CQE%fm5`NFGAo`R% zG;rGM%Pv}==w0Cj4($TFeKx#~vg;yNHi>h$xnF;hELkEsB!}e0lATY$;kpjR;gB=m 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- (cupicef.rule cupicef.set cupicef.set) cupicef.I - Coq__Init__Datatypes.nat - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.nat)))) - (c => - c - (fact0 => - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.match____nat cupicef.set - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O) - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Nat.mul (Coq__Init__Datatypes.S n0) (fact0 n0)) - n)) - cupicef.0. - -thm lt__O__fact : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O (fact n)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.nat__ind - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Peano.lt Coq__Init__Datatypes.O (fact n0)) - (cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - (Coq__Init__Peano.lt Coq__Init__Datatypes.O - (fact Coq__Init__Datatypes.O)) - cupicef.I - (cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.le - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O) - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - (Coq__Init__Peano.lt Coq__Init__Datatypes.O - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - cupicef.I - (Coq__Init__Peano.le__n - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)))) - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - IHn : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O (fact n0)) => - cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O - (Coq__Init__Nat.add (fact n0) (Coq__Init__Nat.mul n0 (fact n0)))) - (Coq__Init__Peano.lt Coq__Init__Datatypes.O - (fact (Coq__Init__Datatypes.S n0))) - cupicef.I - (cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.le - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O) - (Coq__Init__Nat.add (fact n0) (Coq__Init__Nat.mul n0 (fact n0)))) - (Coq__Init__Peano.lt Coq__Init__Datatypes.O - (Coq__Init__Nat.add (fact n0) (Coq__Init__Nat.mul n0 (fact n0)))) - cupicef.I - (Coq__Arith__Plus.le__plus__trans - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O) (fact n0) - (Coq__Init__Nat.mul n0 (fact n0)) IHn))) - n. - -def let__H : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) => - Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) => - Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O))) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) => - Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) => - Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) => - Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O))) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O))) => - __0) - (Coq__Arith__PeanoNat.Nat__neq__0__lt__0 n0). - -thm fact__neq__0 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (fact n) Coq__Init__Datatypes.O)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H n (fact n) (lt__O__fact n). - -thm fact__le : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le (fact n) (fact m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) => - Coq__Init__Peano.le__ind n - (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Peano.le (fact n) (fact m0)) - (Coq__Init__Peano.le__n (fact n)) - (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m0) => - IHle : - cupicef.Term cupicef.prop (Coq__Init__Peano.le (fact n) (fact m0)) => - cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.le (fact n) - (Coq__Init__Nat.add (fact m0) (Coq__Init__Nat.mul m0 (fact m0)))) - (Coq__Init__Peano.le (fact n) (fact (Coq__Init__Datatypes.S m0))) - cupicef.I - (Coq__Classes__RelationClasses.transitivity - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Peano.le - (Coq__Classes__RelationClasses.PreOrder__Transitive - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Peano.le Coq__Arith__PeanoNat.Nat__le__preorder) - (fact n) (fact m0) - (Coq__Init__Nat.add (fact m0) (Coq__Init__Nat.mul m0 (fact m0))) - IHle - (Coq__Arith__PeanoNat.Nat__le__add__r (fact m0) - (Coq__Init__Nat.mul m0 (fact m0))))) - m H. - -(; End of translation. ;) diff --git a/out/Coq__Arith__Factorial.dk.gz b/out/Coq__Arith__Factorial.dk.gz new file mode 100644 index 0000000000000000000000000000000000000000..1d691cb7f05ff62b62fb532f8ab607133147012f GIT binary patch literal 1408 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+0,0 @@ -(; This file was automatically generated by Coqine. ;) -(; The encoding used was: "template_cic". ;) - -thm gt__Sn__O : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n) Coq__Init__Datatypes.O) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Arith__PeanoNat.Nat__lt__0__succ n. - -thm gt__Sn__n : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n) n) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Arith__PeanoNat.Nat__lt__succ__diag__r n. - -def let__H : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) => - Coq__Init__Peano.lt n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) => - Coq__Init__Peano.lt n0 m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) - (__ : - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => - __) - (Coq__Arith__PeanoNat.Nat__succ__lt__mono n0 m0). - -thm gt__n__S : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.gt n m) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n) - (Coq__Init__Datatypes.S m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H n m m n. - -def let__H0 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) => - Coq__Init__Peano.lt n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) => - Coq__Init__Peano.lt n0 m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) => - Coq__Init__Peano.lt n0 m0)) - (__ : - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => - __0) - (Coq__Arith__PeanoNat.Nat__succ__lt__mono n0 m0). - -thm gt__S__n : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.gt (Coq__Init__Datatypes.S m) - (Coq__Init__Datatypes.S n)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.gt m n) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H0 n m n m. - -def let__H1 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H : - cupicef.Term cupicef.prop - (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n) m) -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : - cupicef.Term cupicef.prop - (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n) m) => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) => - Coq__Init__Peano.le n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) => - Coq__Init__Peano.le n0 m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0))) - (__ : - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0))) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => - __) - (Coq__Arith__PeanoNat.Nat__lt__eq__cases n0 m0). - -def let__H2 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H : - cupicef.Term cupicef.prop - (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n) m) -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : - cupicef.Term cupicef.prop - (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n) m) => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) => - Coq__Init__Peano.le n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) => - Coq__Init__Peano.le n0 m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) => - Coq__Init__Peano.le n0 m0)) - (__ : - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => - __0) - (Coq__Arith__PeanoNat.Nat__succ__le__mono n0 m0). - -thm gt__S : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n) m) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.gt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - m n)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : - cupicef.Term cupicef.prop - (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n) m) => - let__H1 n m H m n (let__H2 n m H m n H). - -def let__H3 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) => - Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0)) => - Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) => - Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 - (Coq__Arith__PeanoNat.Nat__pred m0)) => - Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) => - Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0))) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0))) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0)) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0)) => - __) - (Coq__Arith__PeanoNat.Nat__lt__succ__lt__pred n0 m0). - -thm gt__pred : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.gt m (Coq__Init__Datatypes.S n)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.gt (Coq__Init__Nat.pred m) n) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H3 n m n m. - -thm gt__irrefl : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Coq__Init__Peano.gt n n)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Arith__PeanoNat.Nat__lt__irrefl n. - -thm gt__asym : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.gt n m) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Coq__Init__Peano.gt m n)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Arith__PeanoNat.Nat__lt__asymm m n. - -def let__H4 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0)) => - Coq__Init__Peano.le n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0)) => - Coq__Init__Peano.le n0 m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0))) - (__ : - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0))) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => - __) - (Coq__Arith__PeanoNat.Nat__le__ngt n0 m0). - -thm le__not__gt : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Coq__Init__Peano.gt n m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H4 n m n m. - -def let__H5 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) => - Coq__Init__Peano.lt n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) => - Coq__Init__Peano.lt n0 m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0))) - (__ : - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0))) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => - __) - (Coq__Arith__PeanoNat.Nat__lt__nge n0 m0). - -thm gt__not__le : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.gt n m) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Coq__Init__Peano.le n m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H5 n m m n. - -def let__H6 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) => - Coq__Init__Peano.lt n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) => - Coq__Init__Peano.lt n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0)) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => - __0 : - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0)) => - __0) - (Coq__Arith__PeanoNat.Nat__le__succ__l n0 m0). - -thm le__S__gt : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n) m) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.gt m n) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H6 n m n m. - -def let__H7 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) => - Coq__Init__Peano.le n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) => - Coq__Init__Peano.le n0 m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) => - Coq__Init__Peano.le n0 m0)) - (__ : - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => - __0) - (Coq__Arith__PeanoNat.Nat__succ__le__mono n0 m0). - -thm gt__S__le : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.gt (Coq__Init__Datatypes.S m) n) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H7 n m n m. - -def let__H8 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) => - Coq__Init__Peano.lt n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) => - Coq__Init__Peano.lt n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0)) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => - __0 : - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0)) => - __0) - (Coq__Arith__PeanoNat.Nat__le__succ__l n0 m0). - -thm gt__le__S : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.gt m n) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n) m) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H8 n m n m. - -def let__H9 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) => - Coq__Init__Peano.le n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) => - Coq__Init__Peano.le n0 m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) - (__ : - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => - __) - (Coq__Arith__PeanoNat.Nat__succ__le__mono n0 m0). - -thm le__gt__S : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.gt (Coq__Init__Datatypes.S m) n) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H9 n m n m. - -thm le__gt__trans : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n) -> - __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.gt m p) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.gt n p) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n) => - H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.gt m p) => - Coq__Arith__PeanoNat.Nat__lt__le__trans p m n H0 H. - -thm gt__le__trans : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.gt n m) -> - __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le p m) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.gt n p) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (Coq__Init__Peano.gt n m) => - H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le p m) => - Coq__Arith__PeanoNat.Nat__le__lt__trans p m n H0 H. - -thm gt__trans : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.gt n m) -> - __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.gt m p) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.gt n p) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (Coq__Init__Peano.gt n m) => - H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.gt m p) => - Coq__Arith__PeanoNat.Nat__lt__trans p m n H0 H. - -def let__H10 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H : - cupicef.Term cupicef.prop - (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n) m) -> - H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.gt m p) -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : - cupicef.Term cupicef.prop - (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n) m) => - H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.gt m p) => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) => - Coq__Init__Peano.le n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) => - Coq__Init__Peano.le n0 m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) => - Coq__Init__Peano.le n0 m0)) - (__ : - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => - __0) - (Coq__Arith__PeanoNat.Nat__succ__le__mono n0 m0). - -thm gt__trans__S : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n) m) -> - __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.gt m p) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.gt n p) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : - cupicef.Term cupicef.prop - (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n) m) => - H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.gt m p) => - Coq__Arith__PeanoNat.Nat__lt__le__trans p m n H0 - (let__H10 n m p H H0 m n H). - -thm gt__0__eq : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.gt n Coq__Init__Datatypes.O) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O n)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.match____nat cupicef.prop - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.or (Coq__Init__Peano.gt n0 Coq__Init__Datatypes.O) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O n0)) - (Coq__Init__Logic.or__intror - (Coq__Init__Peano.gt Coq__Init__Datatypes.O Coq__Init__Datatypes.O) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O Coq__Init__Datatypes.O) - (Coq__Init__Logic.eq__refl - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O)) - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.or__introl - (Coq__Init__Peano.gt (Coq__Init__Datatypes.S n0) - Coq__Init__Datatypes.O) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O (Coq__Init__Datatypes.S n0)) - (Coq__Arith__PeanoNat.Nat__lt__0__succ n0)) - n. - -def let__H11 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) - (Coq__Arith__PeanoNat.Nat__add p0 m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) - (Coq__Arith__PeanoNat.Nat__add p0 m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) - (Coq__Arith__PeanoNat.Nat__add p0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) - (Coq__Arith__PeanoNat.Nat__add p0 m0)) => - Coq__Init__Peano.lt n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) - (Coq__Arith__PeanoNat.Nat__add p0 m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) - (Coq__Arith__PeanoNat.Nat__add p0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt - (Coq__Arith__PeanoNat.Nat__add p0 n0) - (Coq__Arith__PeanoNat.Nat__add p0 m0)) => - Coq__Init__Peano.lt n0 m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) - (Coq__Arith__PeanoNat.Nat__add p0 m0)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) - (Coq__Arith__PeanoNat.Nat__add p0 m0)) => - Coq__Init__Peano.lt n0 m0)) - (__ : - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) - (Coq__Arith__PeanoNat.Nat__add p0 m0))) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) - (Coq__Arith__PeanoNat.Nat__add p0 m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => - __0) - (Coq__Arith__PeanoNat.Nat__add__lt__mono__l n0 m0 p0). - -thm plus__gt__reg__l : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.gt (Coq__Init__Nat.add p n) (Coq__Init__Nat.add p m)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.gt n m) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H11 n m p m n p. - -def let__H12 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) - (Coq__Arith__PeanoNat.Nat__add p0 m0)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) - (Coq__Arith__PeanoNat.Nat__add p0 m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) - (Coq__Arith__PeanoNat.Nat__add p0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) - (Coq__Arith__PeanoNat.Nat__add p0 m0)) => - Coq__Init__Peano.lt n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) - (Coq__Arith__PeanoNat.Nat__add p0 m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) - (Coq__Arith__PeanoNat.Nat__add p0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt - (Coq__Arith__PeanoNat.Nat__add p0 n0) - (Coq__Arith__PeanoNat.Nat__add p0 m0)) => - Coq__Init__Peano.lt n0 m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) - (Coq__Arith__PeanoNat.Nat__add p0 m0))) - (__ : - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) - (Coq__Arith__PeanoNat.Nat__add p0 m0))) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add p0 n0) - (Coq__Arith__PeanoNat.Nat__add p0 m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => - __) - (Coq__Arith__PeanoNat.Nat__add__lt__mono__l n0 m0 p0). - -thm plus__gt__compat__l : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.gt n m) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.gt (Coq__Init__Nat.add p n) (Coq__Init__Nat.add p m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H12 n m p m n p. - -(; End of translation. ;) diff --git a/out/Coq__Arith__Gt.dk.gz b/out/Coq__Arith__Gt.dk.gz new file mode 100644 index 0000000000000000000000000000000000000000..5182b69575aec5545895ae2e97e6b42a4e145e28 GIT binary patch literal 2832 zcmV+r3-9zFiwFpw=F?*U14D0dUtd9TX>@2`Uq^Hv;{Ii(GHp!U`yQIMt<*+kJCH!?R#=S zPss+Of8qm*pR;ruXUQ_&Y+leB6TFWzyh4AypkLA-3Bhx8H=9gZF^n)-rmKXk(er`6 zLW|#`R~g>!Hnd2zOqQ?mmnYfxEm~($^hl!U0o_gzBIp(^pLfX;Kh3}4{r33ffHT%k z6n!NLE%9rd#o5aaALb;^(CeRQQEsu_KHbr02u*8+|HLs#=j*KTnzbk!MO|eX3pwoDWUQ~-#mtQR8v*whv%Jt*x%w-#S$p{~ z<5E%9kWFR7Hic`Ce^1xRa!xJgl1g|K#bni{Jet;iYCh+@Tz`N6OwDrV`?KS>ucxBj zLUMY?LeL5V6dyx@k_pu;45B$0Nt5FBQesu*R<7GIY44=GELdK#qcH@P{!d$KLbc;; ze40FVz-u9C&lIJ>J_iaTBDM}!C5)YgO=O27<#UQxi=x80?6|b|%Cb=^Vy!@M6H4cF 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zh;7DcWo8E*#qWLee2((!Q&d*bVEAt;^;I7G*HY^0BT(*8b^#1xmz{-~!!^v_u8Omw zjG_%2Vh2IbtaY5|N4~3oHeNtiwyEHlGKco33W@GE&xhg!-X5>n3!?-KQ z0G|i=JizAxJ`eDDgr5Vq_Zben5LfGkaO#9urrTW}Z67C?v`ZzS3(JgL7o$a8M>j*$ i12Mfrzft&-KB4?fz$VU;l+4lHZ1O*S8ri - H : - cupicef.Term cupicef.prop (Coq__Init__Peano.le n Coq__Init__Datatypes.O) -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 Coq__Init__Datatypes.O) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : - cupicef.Term cupicef.prop (Coq__Init__Peano.le n Coq__Init__Datatypes.O) => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 Coq__Init__Datatypes.O) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le n0 Coq__Init__Datatypes.O) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O) => - Coq__Init__Peano.le n0 Coq__Init__Datatypes.O)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 Coq__Init__Datatypes.O) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le n0 Coq__Init__Datatypes.O) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O) => - Coq__Init__Peano.le n0 Coq__Init__Datatypes.O))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 Coq__Init__Datatypes.O) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le n0 Coq__Init__Datatypes.O) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le n0 Coq__Init__Datatypes.O) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.le n0 Coq__Init__Datatypes.O)) => - __) - (Coq__Arith__PeanoNat.Nat__le__0__r n0). - -thm le__n__0__eq : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop (Coq__Init__Peano.le n Coq__Init__Datatypes.O) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O n) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : - cupicef.Term cupicef.prop (Coq__Init__Peano.le n Coq__Init__Datatypes.O) => - Coq__Init__Logic.eq__sym - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n Coq__Init__Datatypes.O (let__H0 n H n H). - -thm le__n__S : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n) - (Coq__Init__Datatypes.S m)) - -:= Coq__Init__Peano.le__n__S. - -thm le__S__n : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n) - (Coq__Init__Datatypes.S m)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) - -:= Coq__Init__Peano.le__S__n. - -thm le__Sn__le : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n) m) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) - -:= Coq__Arith__PeanoNat.Nat__lt__le__incl. - -thm le__elim__rel : - P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) -> - __ : - (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop (P Coq__Init__Datatypes.O p)) -> - __0 : - (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le p q) -> - __1 : cupicef.Term cupicef.prop (P p q) -> - cupicef.Term cupicef.prop - (P (Coq__Init__Datatypes.S p) (Coq__Init__Datatypes.S q))) -> - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __1 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> - cupicef.Term cupicef.prop (P n m) - -:= P : - (__ : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Univ cupicef.prop) => - H0 : - (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop (P Coq__Init__Datatypes.O p)) => - HS : - (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le p q) -> - __0 : cupicef.Term cupicef.prop (P p q) -> - cupicef.Term cupicef.prop - (P (Coq__Init__Datatypes.S p) (Coq__Init__Datatypes.S q))) => - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.nat__ind - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.set (cupicef.rule cupicef.prop cupicef.prop) - (cupicef.rule cupicef.set (cupicef.rule cupicef.prop cupicef.prop)) - cupicef.I Coq__Init__Datatypes.nat - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m) => - P n0 m))) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le Coq__Init__Datatypes.O m) => - H0 m) - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - IHn : - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m) -> - cupicef.Term cupicef.prop (P n0 m)) => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Le : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m) => - Coq__Init__Peano.le__ind (Coq__Init__Datatypes.S n0) - (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - P (Coq__Init__Datatypes.S n0) m0) - (HS n0 n0 (Coq__Init__Peano.le__n n0) - (IHn n0 (Coq__Init__Peano.le__n n0))) - (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) => - H1 : cupicef.Term cupicef.prop (P (Coq__Init__Datatypes.S n0) m0) => - HS n0 m0 (le__Sn__le n0 m0 H) (IHn m0 (le__Sn__le n0 m0 H))) - m Le) - n. - -(; End of translation. ;) diff --git a/out/Coq__Arith__Le.dk.gz b/out/Coq__Arith__Le.dk.gz new file mode 100644 index 0000000000000000000000000000000000000000..7767122f78ca7bb2055d736d5c41335fc84a7c98 GIT binary patch literal 1270 zcmV@2`Urc2#WNQHJTT5@-FbuxuR}ekfhX&cT zMF$MqfD|aU0PW6(=c=s%TTWtUTlecH`62I?3jp!-?Y`JTzzEo&7#B;jGAtJRJ#3`nmAyq#WNFe$ZIe`7t&A0U zS3IZ567A{-^jSALiOXvEMa2)l%3kK*FzjZsljA>Wk;`X`1&Ku+aQcfw)X*jd+*`Q< zQ&V{b9@i~1x4$WksD89OI2Mbm8%qM+(6H(m?X8CNx;;YXNL7MN;5Uut^O5N#Zp@NLe>l zOLtz_sx|n-GT@jY1N*PUZ zXO2~RTHBNu6^_231Vt3{PxV5%zRc^jMycN6{myyvHI!Z z>md~DBOmjLkNL!h zgk|E$0Q`Ordh~xB$&wGQBJ*g z&8N9rDmD-7+&JU*lLMhW_mrreJ0&CXu;H>mY6ByK>OHx0W>nP$ZZTe~0YplWCR`); zsT^GjCyOqoEy90@?{r}6peJsr+O{giNyb2lvNR`+7Ru@cZQ5l8a#~J<1MORq!~AaW z4>s>N`*S@0!Pbi&0n8g>nx~5i37d?A{7%dBAL87cvq)s3mOp@slqIWP&Wli3eqm zc43F?xkx<$gJmtEaP$Q280JhdXO5{z);6B~-iBjW6oujoISeLchX~P;E-FAn3_)XeguR_jw!Z!_)wp0Qb|2Vq44tk}jthP11NB@ad zOy4zG+vo%uP3qyH-^Hx`eKDLsg;zV-x*AEVVJ2t<%xh+IXB{@6N**$3<|K~09U1bb g6&+#RQnj0SzM(BuK!= - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) => - Coq__Init__Peano.lt n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) => - Coq__Init__Peano.lt n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0)) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => - __0 : - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n0) m0)) => - __0) - (Coq__Arith__PeanoNat.Nat__le__succ__l n0 m0). - -thm lt__le__S : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Datatypes.S n) m) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H n m n m. - -def let__H0 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) => - Coq__Init__Peano.le n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0))) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) => - Coq__Init__Peano.le n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) => - Coq__Init__Peano.le n0 m0)) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => - __0 : - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0))) => - __) - (Coq__Arith__PeanoNat.Nat__lt__succ__r n0 m0). - -thm lt__n__Sm__le : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n (Coq__Init__Datatypes.S m)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H0 n m n m. - -def let__H1 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) => - Coq__Init__Peano.le n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0))) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) => - Coq__Init__Peano.le n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0))) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => - __0 : - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 (Coq__Init__Datatypes.S m0))) => - __0) - (Coq__Arith__PeanoNat.Nat__lt__succ__r n0 m0). - -thm le__lt__n__Sm : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n (Coq__Init__Datatypes.S m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H1 n m n m. - -def let__H2 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0)) => - Coq__Init__Peano.le n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0)) => - Coq__Init__Peano.le n0 m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0))) - (__ : - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0))) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.lt m0 n0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => - __) - (Coq__Arith__PeanoNat.Nat__le__ngt n0 m0). - -thm le__not__lt : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Coq__Init__Peano.lt m n)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H2 n m n m. - -def let__H3 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) => - Coq__Init__Peano.lt n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) => - Coq__Init__Peano.lt n0 m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0))) - (__ : - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0))) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => - __) - (Coq__Arith__PeanoNat.Nat__lt__nge n0 m0). - -thm lt__not__le : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) -> - cupicef.Term cupicef.prop (Coq__Init__Logic.not (Coq__Init__Peano.le m n)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H3 n m n m. - -def let__H4 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O n)) -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O n)) => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) => - Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) => - Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O))) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) => - Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) => - Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) => - Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O))) => - __) - (Coq__Arith__PeanoNat.Nat__neq__0__lt__0 n0). - -thm neq__0__lt : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O n)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O n)) => - let__H4 n H n - (Coq__Arith__PeanoNat.Nat__neq__sym Coq__Init__Datatypes.O n H). - -def let__H5 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H : - cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n) -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : - cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n) => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) => - Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) => - Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O))) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) => - Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) => - Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) => - Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O))) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O))) => - __0) - (Coq__Arith__PeanoNat.Nat__neq__0__lt__0 n0). - -thm lt__0__neq : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O n)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : - cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n) => - Coq__Arith__PeanoNat.Nat__neq__sym n Coq__Init__Datatypes.O - (let__H5 n H n H). - -def let__H6 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) => - Coq__Init__Peano.lt n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) => - Coq__Init__Peano.lt n0 m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) - (__ : - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => - __) - (Coq__Arith__PeanoNat.Nat__succ__lt__mono n0 m0). - -thm lt__n__S : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n) - (Coq__Init__Datatypes.S m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H6 n m n m. - -def let__H7 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) => - Coq__Init__Peano.lt n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) => - Coq__Init__Peano.lt n0 m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) => - Coq__Init__Peano.lt n0 m0)) - (__ : - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0))) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) - (Coq__Init__Datatypes.S m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => - __0) - (Coq__Arith__PeanoNat.Nat__succ__lt__mono n0 m0). - -thm lt__S__n : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n) - (Coq__Init__Datatypes.S m)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H7 n m n m. - -thm S__pred : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n (Coq__Init__Datatypes.S (Coq__Init__Nat.pred n))) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) => - Coq__Init__Logic.eq__sym - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S (Coq__Init__Nat.pred n)) n - (Coq__Arith__PeanoNat.Nat__lt__succ__pred m n H). - -thm S__pred__pos : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n (Coq__Init__Datatypes.S (Coq__Init__Nat.pred n))) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - S__pred n Coq__Init__Datatypes.O. - -def let__H8 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) => - Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0)) => - Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) => - Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 - (Coq__Arith__PeanoNat.Nat__pred m0)) => - Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) => - Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0))) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0))) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__pred m0)) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n0) m0)) => - __) - (Coq__Arith__PeanoNat.Nat__lt__succ__lt__pred n0 m0). - -thm lt__pred : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Datatypes.S n) m) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt n (Coq__Init__Nat.pred m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H8 n m n m. - -def let__H9 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H : - cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n) -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : - cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n) => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) => - Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) => - Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O))) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) => - Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) => - Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) => - Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O))) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O)) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0)) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O n0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O))) => - __0) - (Coq__Arith__PeanoNat.Nat__neq__0__lt__0 n0). - -thm lt__pred__n__n : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt (Coq__Init__Nat.pred n) n) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : - cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O n) => - Coq__Arith__PeanoNat.Nat__lt__pred__l n (let__H9 n H n H). - -def let__H10 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) => - Coq__Init__Peano.le n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) => - Coq__Init__Peano.le n0 m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0))) - (__ : - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0))) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => - __) - (Coq__Arith__PeanoNat.Nat__lt__eq__cases n0 m0). - -thm le__lt__or__eq : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt n m) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H10 n m n m. - -def let__H11 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Peano.lt m0 n0)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) => - Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Peano.lt m0 n0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Peano.lt m0 n0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Peano.lt m0 n0)) => - Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0))) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) => - Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Peano.lt m0 n0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Peano.lt m0 n0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Peano.lt m0 n0)) => - Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) => - Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Peano.lt m0 n0))) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0)) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Peano.lt m0 n0))) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt n0 m0) - (Coq__Init__Peano.lt m0 n0)) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m0))) => - __) - (Coq__Arith__PeanoNat.Nat__lt__gt__cases n0 m0). - -thm nat__total__order : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n m)) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or (Coq__Init__Peano.lt n m) (Coq__Init__Peano.lt m n)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H11 n m n m. - -(; End of translation. ;) diff --git a/out/Coq__Arith__Lt.dk.gz b/out/Coq__Arith__Lt.dk.gz new file mode 100644 index 0000000000000000000000000000000000000000..11c83dee2a67be6df4a8028e17c5deebbcb03da8 GIT binary patch literal 3218 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zINr^a=Q!Tojg)vfsV*hu{7(hf-kL9jCkj?Y57u*L2Ve?<%uaw7WZ^c@NQyz?aKbZsii? zO);=co1?=q5Qwo+#n=&WNpVh7t0^$RXkFWGuHFv@7*3w#VEgzPX?wVS5)z1vxy#hcs7|8ktkR{~Q3 E06rf@ssI20 literal 0 HcmV?d00001 diff --git a/out/Coq__Arith__Minus.dk b/out/Coq__Arith__Minus.dk deleted file mode 100644 index c5dee9532..000000000 --- a/out/Coq__Arith__Minus.dk +++ /dev/null @@ -1,796 +0,0 @@ -(; This file was automatically generated by Coqine. ;) -(; The encoding used was: "template_cic". ;) - -thm minus__n__O : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n (Coq__Init__Nat.sub n Coq__Init__Datatypes.O)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq__sym - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.sub n Coq__Init__Datatypes.O) n - (Coq__Arith__PeanoNat.Nat__sub__0__r n). - -thm minus__Sn__m : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S (Coq__Init__Nat.sub n m)) - (Coq__Init__Nat.sub (Coq__Init__Datatypes.S n) m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (Coq__Init__Peano.le m n) => - Coq__Init__Logic.eq__sym - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.sub (Coq__Init__Datatypes.S n) m) - (Coq__Init__Datatypes.S (Coq__Init__Nat.sub n m)) - (Coq__Arith__PeanoNat.Nat__sub__succ__l m n H). - -thm pred__of__minus : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.pred n) - (Coq__Init__Nat.sub n (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq__sym - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.sub n (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - (Coq__Init__Nat.pred n) (Coq__Arith__PeanoNat.Nat__sub__1__r n). - -thm minus__diag__reverse : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O (Coq__Init__Nat.sub n n)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq__sym - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.sub n n) Coq__Init__Datatypes.O - (Coq__Arith__PeanoNat.Nat__sub__diag n). - -thm minus__plus__simpl__l__reverse : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.sub n m) - (Coq__Init__Nat.sub (Coq__Init__Nat.add p n) (Coq__Init__Nat.add p m))) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq__ind__r - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__sub - (Coq__Arith__PeanoNat.Nat__sub (Coq__Init__Nat.add p n) p) m) - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.sub n m) n0) - (Coq__Init__Logic.eq__ind__r - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__add n p) - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.sub n m) - (Coq__Arith__PeanoNat.Nat__sub - (Coq__Arith__PeanoNat.Nat__sub n0 p) m)) - (Coq__Init__Logic.eq__ind__r - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.sub n m) (Coq__Arith__PeanoNat.Nat__sub n0 m)) - (Coq__Init__Logic.eq__refl - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__sub n m)) - (Coq__Arith__PeanoNat.Nat__sub (Coq__Arith__PeanoNat.Nat__add n p) - p) - (Coq__Arith__PeanoNat.Nat__add__sub n p)) - (Coq__Arith__PeanoNat.Nat__add p n) - (Coq__Arith__PeanoNat.Nat__add__comm p n)) - (Coq__Arith__PeanoNat.Nat__sub (Coq__Init__Nat.add p n) - (Coq__Arith__PeanoNat.Nat__add p m)) - (Coq__Arith__PeanoNat.Nat__sub__add__distr (Coq__Init__Nat.add p n) p m). - -thm plus__minus : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n (Coq__Init__Nat.add m p)) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - p (Coq__Init__Nat.sub n m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n (Coq__Init__Nat.add m p)) => - Coq__Init__Logic.eq__sym - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.sub n m) p - (Coq__Arith__PeanoNat.Nat__add__sub__eq__l n m p - (Coq__Init__Logic.eq__sym - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n (Coq__Arith__PeanoNat.Nat__add m p) H)). - -thm minus__plus : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.sub (Coq__Init__Nat.add n m) n) m) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq__ind__r - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__add m n) - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.sub n0 n) m) - (Coq__Arith__PeanoNat.Nat__add__sub m n) - (Coq__Arith__PeanoNat.Nat__add n m) - (Coq__Arith__PeanoNat.Nat__add__comm n m). - -thm le__plus__minus__r : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.add n (Coq__Init__Nat.sub m n)) m) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq__ind__r - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__add (Coq__Init__Nat.sub m n) n) - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n m) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m)) - (Coq__Arith__PeanoNat.Nat__sub__add n m) - (Coq__Arith__PeanoNat.Nat__add n (Coq__Init__Nat.sub m n)) - (Coq__Arith__PeanoNat.Nat__add__comm n (Coq__Init__Nat.sub m n)). - -thm le__plus__minus : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - m (Coq__Init__Nat.add n (Coq__Init__Nat.sub m n))) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) => - Coq__Init__Logic.eq__sym - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.add n (Coq__Init__Nat.sub m n)) m - (Coq__Init__Logic.eq__ind__r - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__add (Coq__Init__Nat.sub m n) n) - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 m) - (Coq__Arith__PeanoNat.Nat__sub__add n m H) - (Coq__Arith__PeanoNat.Nat__add n (Coq__Init__Nat.sub m n)) - (Coq__Arith__PeanoNat.Nat__add__comm n (Coq__Init__Nat.sub m n))). - -def let__H : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__sub m0 p)) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p) m0) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p) m0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p) m0) => - Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__sub m0 p))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__sub m0 p)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__sub m0 p)) => - Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p) m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p) m0) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt - (Coq__Arith__PeanoNat.Nat__add n0 p) m0) => - Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__sub m0 p))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__sub m0 p)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 - (Coq__Arith__PeanoNat.Nat__sub m0 p)) => - Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p) m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__sub m0 p)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__sub m0 p)) => - Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p) m0)) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p) m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__sub m0 p))) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt n0 (Coq__Arith__PeanoNat.Nat__sub m0 p)) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__add n0 p) m0)) => - __0) - (Coq__Arith__PeanoNat.Nat__lt__add__lt__sub__r n0 m0 p). - -thm lt__O__minus__lt : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O (Coq__Init__Nat.sub n m)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt m n) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H n m Coq__Init__Datatypes.O n m. - -def let__H0 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le m n)) -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__sub n0 m0) Coq__Init__Datatypes.O) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le m n)) => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__sub n0 m0) Coq__Init__Datatypes.O) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__sub n0 m0) Coq__Init__Datatypes.O) => - Coq__Init__Peano.le n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__sub n0 m0) Coq__Init__Datatypes.O)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__sub n0 m0) - Coq__Init__Datatypes.O) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__sub n0 m0) - Coq__Init__Datatypes.O) => - Coq__Init__Peano.le n0 m0)) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__sub n0 m0) - Coq__Init__Datatypes.O))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__sub n0 m0) Coq__Init__Datatypes.O)) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__sub n0 m0) Coq__Init__Datatypes.O) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => - __0 : - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__sub n0 m0) Coq__Init__Datatypes.O)) => - __0) - (Coq__Arith__PeanoNat.Nat__sub__0__le n0 m0). - -def let__H1 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le m n)) -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le m n)) => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) => - Coq__Init__Peano.lt n0 m0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) => - Coq__Init__Peano.lt n0 m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) => - Coq__Init__Peano.lt n0 m0)) - (__ : - (__ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0))) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le m0 n0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => - __0) - (Coq__Arith__PeanoNat.Nat__lt__nge n0 m0). - -thm not__le__minus__0 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le m n)) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.sub n m) Coq__Init__Datatypes.O) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : - cupicef.Term cupicef.prop - (Coq__Init__Logic.not (Coq__Init__Peano.le m n)) => - let__H0 n m H n m - (Coq__Arith__PeanoNat.Nat__lt__le__incl n m (let__H1 n m H n m H)). - -(; End of translation. ;) diff --git a/out/Coq__Arith__Minus.dk.gz b/out/Coq__Arith__Minus.dk.gz new file mode 100644 index 0000000000000000000000000000000000000000..d2b3dd585f9c46d4f32b6c906e44fc444144da3d GIT binary patch literal 2150 zcmV-s2$}aEiwFpw=F?*U14D0dUtd9TX>@2`UrlLlb#pFcYXI$CO^@3)5WV|X5INb0 z7|L}QDNwWp9H8hHNbZCpYj*`mlp@J#x4*umY{`=FJ;&q+LFe zZ?8#3o|7~t|AZ9@cXghHbrOYX`bJ)27MEciZ^`#J@>~8RVey(gt`Li z$gbi=c=ZSJp^md6Mi~2PqS;BS|rHs-i@~|L;kp~j(iX@7k*WcnY>wc-? zTJ%Ke7nanN{vOt0{Z_=)nuRs__zSNptyYwI!M`J9*{l3#{*pxN_($`F_^vG-g;jn0 zO|*aft?JtRZ?$S`_A7_~CV2+emNz7CYR8hFgPB{7i5yqs3*3S2P=_5L&KnwT_aV6R zU1>Jh;J?CpUG2W}Po))~c<+mc&yHaGEWk!2aH6z&%NA1+g|@DEXp(FW9+EI|BpIkN zhOF>ISijP*Wm3OV`ga_%TvX-%I2Lx_DGews30tiPWcQV`T{e~NGWV*KLUjdO4i=)B 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zZ4-KZD+KN~e&zy~f3ulU*E(A1zS{|PFP#{9&?>c#8+tuprj4KEoivF#bH+u7 zcw8J902Ah44Py6T#vR=_KOxBSTDAl6;ShJ$qXDO@w%|yX<}tg!Y&{bANZ*G2T24TGFhnL#K*fwd>&G0g0ip$|~xlVAsJJt}s*1$C33XCDi!86JTd>dUHcDHZxPhF*Mb5?Tu`jhq27o zY7(>pN;_j1a2n?^+8tcqr-49)7_)){`Pvve;G#zOrSrMD}1c&vHLwg?PzkoFaerQz?7ynEnL8 z`N?C7IUxjBT4Ouw0~$8!OV^8PrCc7gU@tx0E*`Dm!xUn^d$?lGO#HKG-J^t8x9U^c zM;%Oyt8tI3rNgy|m7zy$?lX(Med?xpr-%68h&%*MBrMwh35R7H@po+;-o(FrB!BQ= cC4VM$8L}!3>m+Au^0-?34?3{DoOYN10O-mtiU0rr literal 0 HcmV?d00001 diff --git a/out/Coq__Arith__Mult.dk b/out/Coq__Arith__Mult.dk deleted file mode 100644 index 16146079c..000000000 --- a/out/Coq__Arith__Mult.dk +++ /dev/null @@ -1,2942 +0,0 @@ -(; This file was automatically generated by Coqine. ;) -(; The encoding used was: "template_cic". ;) - -thm mult__assoc__reverse : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.mul (Coq__Init__Nat.mul n m) p) - (Coq__Init__Nat.mul n (Coq__Init__Nat.mul m p))) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq__sym - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.mul n (Coq__Init__Nat.mul m p)) - (Coq__Init__Nat.mul (Coq__Init__Nat.mul n m) p) - (Coq__Arith__PeanoNat.Nat__mul__assoc n m p). - -def let__H : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__mul n0 m0) Coq__Init__Datatypes.O) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - m0 Coq__Init__Datatypes.O)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__mul n0 m0) Coq__Init__Datatypes.O) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__mul n0 m0) Coq__Init__Datatypes.O) => - Coq__Init__Logic.or - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - m0 Coq__Init__Datatypes.O))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.or - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - m0 Coq__Init__Datatypes.O)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.or - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - m0 Coq__Init__Datatypes.O)) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__mul n0 m0) Coq__Init__Datatypes.O)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__mul n0 m0) - Coq__Init__Datatypes.O) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__mul n0 m0) - Coq__Init__Datatypes.O) => - Coq__Init__Logic.or - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - m0 Coq__Init__Datatypes.O))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.or - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - m0 Coq__Init__Datatypes.O)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.or - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - m0 Coq__Init__Datatypes.O)) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__mul n0 m0) - Coq__Init__Datatypes.O))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__mul n0 m0) Coq__Init__Datatypes.O) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__mul n0 m0) Coq__Init__Datatypes.O) => - Coq__Init__Logic.or - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - m0 Coq__Init__Datatypes.O))) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__mul n0 m0) Coq__Init__Datatypes.O) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - m0 Coq__Init__Datatypes.O))) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.or - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - m0 Coq__Init__Datatypes.O)) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__mul n0 m0) Coq__Init__Datatypes.O)) => - __) - (Coq__Arith__PeanoNat.Nat__eq__mul__0 n0 m0). - -thm mult__is__O : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.mul n m) Coq__Init__Datatypes.O) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n Coq__Init__Datatypes.O) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - m Coq__Init__Datatypes.O)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H n m n m. - -def let__H0 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__mul n0 m0) - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - m0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__mul n0 m0) - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__mul n0 m0) - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) => - Coq__Init__Logic.and - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - m0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.and - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - m0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - m0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__mul n0 m0) - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__mul n0 m0) - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__mul n0 m0) - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) => - Coq__Init__Logic.and - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - m0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.and - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - m0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - m0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__mul n0 m0) - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__mul n0 m0) - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__mul n0 m0) - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) => - Coq__Init__Logic.and - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - m0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)))) - (__ : - (__ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__mul n0 m0) - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - m0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)))) => - __0 : - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - m0 (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__mul n0 m0) - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) => - __) - (Coq__Arith__PeanoNat.Nat__eq__mul__1 n0 m0). - -thm mult__is__one : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.mul n m) - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - m (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H0 n m n m. - -thm mult__O__le : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.or - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - m Coq__Init__Datatypes.O) - (Coq__Init__Peano.le n (Coq__Init__Nat.mul m n))) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.match____nat cupicef.prop - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.or - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 Coq__Init__Datatypes.O) - (Coq__Init__Peano.le n (Coq__Init__Nat.mul n0 n))) - (Coq__Init__Logic.or__introl - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O Coq__Init__Datatypes.O) - (Coq__Init__Peano.le n (Coq__Init__Nat.mul Coq__Init__Datatypes.O n)) - (cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O Coq__Init__Datatypes.O) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O Coq__Init__Datatypes.O) - cupicef.I - (Coq__Init__Logic.eq__refl - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - Coq__Init__Datatypes.O))) - (m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.or__intror - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Datatypes.S m0) Coq__Init__Datatypes.O) - (Coq__Init__Peano.le n - (Coq__Init__Nat.mul (Coq__Init__Datatypes.S m0) n)) - (cupicef.cast cupicef.prop cupicef.prop - (Coq__Init__Peano.le n - (Coq__Arith__PeanoNat.Nat__add n (Coq__Init__Nat.mul m0 n))) - (Coq__Init__Peano.le n - (Coq__Init__Nat.mul (Coq__Init__Datatypes.S m0) n)) - cupicef.I - (Coq__Arith__PeanoNat.Nat__le__add__r n (Coq__Init__Nat.mul m0 n)))) - m. - -thm mult__le__compat__l : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Nat.mul p n) (Coq__Init__Nat.mul p m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Arith__PeanoNat.Nat__mul__le__mono__nonneg__l n m p - (Coq__Arith__PeanoNat.Nat__le__0__l p). - -thm mult__le__compat__r : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Nat.mul n p) (Coq__Init__Nat.mul m p)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Arith__PeanoNat.Nat__mul__le__mono__nonneg__r n m p - (Coq__Arith__PeanoNat.Nat__le__0__l p). - -thm mult__le__compat : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) -> - __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le p q) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Nat.mul n p) (Coq__Init__Nat.mul m q)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (Coq__Init__Peano.le n m) => - H0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le p q) => - Coq__Arith__PeanoNat.Nat__mul__le__mono__nonneg n m p q - (Coq__Arith__PeanoNat.Nat__le__0__l n) H - (Coq__Arith__PeanoNat.Nat__le__0__l p) H0. - -def let__H1 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O p0) -> - __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O p0) => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0)) => - Coq__Init__Peano.lt n0 m0)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt - (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0)) => - Coq__Init__Peano.lt n0 m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__1 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0))) - (__0 : - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0))) => - __1 : - (__1 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => - __0) - (Coq__Arith__PeanoNat.Nat__mul__lt__mono__pos__l p0 n0 m0 __). - -thm mult__S__lt__compat__l : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt m p) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Nat.mul (Coq__Init__Datatypes.S n) m) - (Coq__Init__Nat.mul (Coq__Init__Datatypes.S n) p)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H1 n m p (Coq__Init__Datatypes.S n) m p - (Coq__Arith__PeanoNat.Nat__lt__0__succ n). - -def let__H2 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) -> - H0 : - cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O p) -> - p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O p0) -> - __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) => - H0 : - cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O p) => - p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O p0) => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0)) => - Coq__Init__Peano.lt n0 m0)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt - (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0)) => - Coq__Init__Peano.lt n0 m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__1 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0))) - (__0 : - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0))) => - __1 : - (__1 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => - __0) - (Coq__Arith__PeanoNat.Nat__mul__lt__mono__pos__l p0 n0 m0 __). - -thm mult__lt__compat__l : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) -> - __0 : - cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O p) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Nat.mul p n) (Coq__Init__Nat.mul p m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) => - H0 : - cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O p) => - let__H2 n m p H H0 p n m H0 H. - -def let__H3 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - H : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) -> - H0 : - cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O p) -> - p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O p0) -> - __0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul n0 p0) - (Coq__Arith__PeanoNat.Nat__mul m0 p0)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) => - H0 : - cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O p) => - p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O p0) => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul n0 p0) - (Coq__Arith__PeanoNat.Nat__mul m0 p0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul n0 p0) - (Coq__Arith__PeanoNat.Nat__mul m0 p0)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul n0 p0) - (Coq__Arith__PeanoNat.Nat__mul m0 p0)) => - Coq__Init__Peano.lt n0 m0)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul n0 p0) - (Coq__Arith__PeanoNat.Nat__mul m0 p0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul n0 p0) - (Coq__Arith__PeanoNat.Nat__mul m0 p0)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt - (Coq__Arith__PeanoNat.Nat__mul n0 p0) - (Coq__Arith__PeanoNat.Nat__mul m0 p0)) => - Coq__Init__Peano.lt n0 m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.lt n0 m0) - (__1 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) => - Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul n0 p0) - (Coq__Arith__PeanoNat.Nat__mul m0 p0))) - (__0 : - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul n0 p0) - (Coq__Arith__PeanoNat.Nat__mul m0 p0))) => - __1 : - (__1 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Arith__PeanoNat.Nat__mul n0 p0) - (Coq__Arith__PeanoNat.Nat__mul m0 p0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.lt n0 m0)) => - __0) - (Coq__Arith__PeanoNat.Nat__mul__lt__mono__pos__r p0 n0 m0 __). - -thm mult__lt__compat__r : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) -> - __0 : - cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O p) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt (Coq__Init__Nat.mul n p) (Coq__Init__Nat.mul m p)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - H : cupicef.Term cupicef.prop (Coq__Init__Peano.lt n m) => - H0 : - cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O p) => - let__H3 n m p H H0 p n m H0 H. - -def let__H4 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop (Coq__Init__Peano.lt Coq__Init__Datatypes.O p0) -> - __0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.lt Coq__Init__Datatypes.O p0) => - Coq__Init__Logic.match____and cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0)) => - Coq__Init__Peano.le n0 m0)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.and - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le n0 m0) - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) => - Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0))) - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0)) - (__0 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le - (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0)) => - Coq__Init__Peano.le n0 m0))) => - cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0)) - (__1 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0)) => - Coq__Init__Peano.le n0 m0)) - (__0 : - (__0 : cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0) -> - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0))) => - __1 : - (__1 : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Arith__PeanoNat.Nat__mul p0 n0) - (Coq__Arith__PeanoNat.Nat__mul p0 m0)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le n0 m0)) => - __1) - (Coq__Arith__PeanoNat.Nat__mul__le__mono__pos__l n0 m0 p0 __). - -thm mult__S__le__reg__l : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - __ : - cupicef.Term cupicef.prop - (Coq__Init__Peano.le (Coq__Init__Nat.mul (Coq__Init__Datatypes.S n) m) - (Coq__Init__Nat.mul (Coq__Init__Datatypes.S n) p)) -> - cupicef.Term cupicef.prop (Coq__Init__Peano.le m p) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - let__H4 n m p m p (Coq__Init__Datatypes.S n) - (Coq__Arith__PeanoNat.Nat__lt__0__succ n). - -thm odd__even__lem : - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.add - (Coq__Init__Nat.mul - (Coq__Init__Datatypes.S - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - p) - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - (Coq__Init__Nat.mul - (Coq__Init__Datatypes.S - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - q))) - -:= p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - q : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.cast (cupicef.rule cupicef.prop cupicef.prop) cupicef.prop - (cupicef.prod cupicef.prop cupicef.prop - (cupicef.rule cupicef.prop cupicef.prop) cupicef.I - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.add - (Coq__Init__Nat.mul - (Coq__Init__Datatypes.S - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - p) - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - (Coq__Init__Nat.mul - (Coq__Init__Datatypes.S - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - q)) - (H : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.add - (Coq__Init__Nat.mul - (Coq__Init__Datatypes.S - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - p) - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - (Coq__Init__Nat.mul - (Coq__Init__Datatypes.S - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - q)) => - Coq__Init__Logic.False)) - (Coq__Init__Logic.not - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.add - (Coq__Init__Nat.mul - (Coq__Init__Datatypes.S - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - p) - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - (Coq__Init__Nat.mul - (Coq__Init__Datatypes.S - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - q))) - cupicef.I - (H : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.add - (Coq__Init__Nat.mul - (Coq__Init__Datatypes.S - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - p) - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - (Coq__Init__Nat.mul - (Coq__Init__Datatypes.S - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - q)) => - Coq__Arith__PeanoNat.Nat__Even__Odd__False - (Coq__Init__Nat.mul - (Coq__Init__Datatypes.S - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - q) - (Coq__Init__Logic.ex__intro - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.mul - (Coq__Init__Datatypes.S - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - q) - (Coq__Init__Nat.mul - (Coq__Init__Datatypes.S - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - m)) - q - (Coq__Init__Logic.eq__refl - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.mul - (Coq__Init__Datatypes.S - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - q))) - (Coq__Init__Logic.ex__intro - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.mul - (Coq__Init__Datatypes.S - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - q) - (Coq__Init__Nat.add - (Coq__Init__Nat.mul - (Coq__Init__Datatypes.S - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - m) - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O))) - p - (Coq__Init__Logic.eq__sym - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.add - (Coq__Init__Nat.mul - (Coq__Init__Datatypes.S - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - p) - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - (Coq__Init__Nat.mul - (Coq__Init__Datatypes.S - (Coq__Init__Datatypes.S Coq__Init__Datatypes.O)) - q) - H))). - -def mult__acc : - s : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.set Coq__Init__Datatypes.nat - -:= cupicef.fixproj cupicef.1 - (c => - c - (cupicef.SA cupicef.2 cupicef.set - (cupicef.prod cupicef.set - (cupicef.rule cupicef.set - (cupicef.rule cupicef.set cupicef.set)) - (cupicef.rule cupicef.set - (cupicef.rule cupicef.set - (cupicef.rule cupicef.set cupicef.set))) - cupicef.I Coq__Init__Datatypes.nat - (s : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.set - (cupicef.rule cupicef.set cupicef.set) - (cupicef.rule cupicef.set - (cupicef.rule cupicef.set cupicef.set)) - cupicef.I Coq__Init__Datatypes.nat - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.set cupicef.set - (cupicef.rule cupicef.set cupicef.set) cupicef.I - Coq__Init__Datatypes.nat - (n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.nat)))))) - (c => - c - (mult__acc0 => - s : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.match____nat cupicef.set - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.nat) - s - (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - mult__acc0 (Coq__Arith__Plus.tail__plus m s) m p) - n)) - cupicef.0. - -def let__H5 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - IHn : - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.add m (Coq__Init__Nat.mul n0 p)) (mult__acc m p n0))) -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 n0) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - IHn : - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.add m (Coq__Init__Nat.mul n0 p)) - (mult__acc m p n0))) => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq__refl - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0. - -def let__H6 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - IHn : - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.add m (Coq__Init__Nat.mul n0 p)) (mult__acc m p n0))) -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - p p) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - IHn : - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.add m (Coq__Init__Nat.mul n0 p)) - (mult__acc m p n0))) => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq__refl - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - p. - -def let__H7 : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - IHn : - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.add m (Coq__Init__Nat.mul n0 p)) (mult__acc m p n0))) -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__add m p) (Coq__Arith__Plus.tail__plus p m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - IHn : - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.add m (Coq__Init__Nat.mul n0 p)) - (mult__acc m p n0))) => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq__ind__r - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__add p m) - (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n1 (Coq__Arith__Plus.tail__plus p m)) - (Coq__Arith__Plus.plus__tail__plus p m) - (Coq__Arith__PeanoNat.Nat__add m p) - (Coq__Arith__PeanoNat.Nat__add__comm m p). - -thm mult__acc__aux : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.add m (Coq__Init__Nat.mul n p)) (mult__acc m p n)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.nat__ind - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.set (cupicef.rule cupicef.set cupicef.prop) - (cupicef.rule cupicef.set (cupicef.rule cupicef.set cupicef.prop)) - cupicef.I Coq__Init__Datatypes.nat - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.set cupicef.prop - (cupicef.rule cupicef.set cupicef.prop) cupicef.I - Coq__Init__Datatypes.nat - (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.add m (Coq__Init__Nat.mul n0 p)) - (mult__acc m p n0)))) - (cupicef.cast - (cupicef.rule cupicef.set (cupicef.rule cupicef.set cupicef.prop)) - (cupicef.rule cupicef.set (cupicef.rule cupicef.set cupicef.prop)) - (cupicef.prod cupicef.set (cupicef.rule cupicef.set cupicef.prop) - (cupicef.rule cupicef.set (cupicef.rule cupicef.set cupicef.prop)) - cupicef.I Coq__Init__Datatypes.nat - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.set cupicef.prop - (cupicef.rule cupicef.set cupicef.prop) cupicef.I - Coq__Init__Datatypes.nat - (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.add m Coq__Init__Datatypes.O) m))) - (cupicef.prod cupicef.set (cupicef.rule cupicef.set cupicef.prop) - (cupicef.rule cupicef.set (cupicef.rule cupicef.set cupicef.prop)) - cupicef.I Coq__Init__Datatypes.nat - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.set cupicef.prop - (cupicef.rule cupicef.set cupicef.prop) cupicef.I - Coq__Init__Datatypes.nat - (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.add m - (Coq__Init__Nat.mul Coq__Init__Datatypes.O p)) - (mult__acc m p Coq__Init__Datatypes.O)))) - cupicef.I - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq__sym - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - m (Coq__Init__Nat.add m Coq__Init__Datatypes.O) - (Coq__Init__Peano.plus__n__O m))) - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - IHn : - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.add m (Coq__Init__Nat.mul n0 p)) - (mult__acc m p n0))) => - cupicef.cast - (cupicef.rule cupicef.set (cupicef.rule cupicef.set cupicef.prop)) - (cupicef.rule cupicef.set (cupicef.rule cupicef.set cupicef.prop)) - (cupicef.prod cupicef.set (cupicef.rule cupicef.set cupicef.prop) - (cupicef.rule cupicef.set (cupicef.rule cupicef.set cupicef.prop)) - cupicef.I Coq__Init__Datatypes.nat - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.set cupicef.prop - (cupicef.rule cupicef.set cupicef.prop) cupicef.I - Coq__Init__Datatypes.nat - (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n1 (mult__acc (Coq__Arith__Plus.tail__plus p m) p n0)) - (Coq__Arith__PeanoNat.Nat__add m - (Coq__Arith__PeanoNat.Nat__add p - (Coq__Init__Nat.mul n0 p)))))) - (cupicef.prod cupicef.set (cupicef.rule cupicef.set cupicef.prop) - (cupicef.rule cupicef.set (cupicef.rule cupicef.set cupicef.prop)) - cupicef.I Coq__Init__Datatypes.nat - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.prod cupicef.set cupicef.prop - (cupicef.rule cupicef.set cupicef.prop) cupicef.I - Coq__Init__Datatypes.nat - (p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.add m - (Coq__Init__Nat.mul (Coq__Init__Datatypes.S n0) p)) - (mult__acc m p (Coq__Init__Datatypes.S n0))))) - cupicef.I - (m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - p : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq__ind__r - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__add (Coq__Arith__PeanoNat.Nat__add m p) - (Coq__Init__Nat.mul n0 p)) - (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n1 (mult__acc (Coq__Arith__Plus.tail__plus p m) p n0)) - (Coq__Init__Logic.eq__ind__r - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s cupicef.z))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (mult__acc (Coq__Arith__PeanoNat.Nat__add m p) p n0) - (n1 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n1 (mult__acc (Coq__Arith__Plus.tail__plus p m) p n0)) - ((H0 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Arith__PeanoNat.Nat__add m p) - (Coq__Arith__Plus.tail__plus p m)) => - H1 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - p p) => - H2 : - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0 n0) => - Coq__Init__Logic.eq__trans - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (mult__acc (Coq__Init__Nat.add m p) p n0) - (mult__acc (Coq__Arith__Plus.tail__plus p m) p n0) - (mult__acc (Coq__Arith__Plus.tail__plus p m) p n0) - (Coq__Init__Logic.f__equal - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (cupicef.prod cupicef.set cupicef.set - (cupicef.rule cupicef.set cupicef.set) cupicef.I - Coq__Init__Datatypes.nat - (n1 : - cupicef.Term cupicef.set - Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.nat))) - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (f : - (n1 : - cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.set Coq__Init__Datatypes.nat) => - f n0) - (mult__acc (Coq__Init__Nat.add m p) p) - (mult__acc (Coq__Arith__Plus.tail__plus p m) p) - (Coq__Init__Logic.eq__trans - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (cupicef.prod cupicef.set cupicef.set - (cupicef.rule cupicef.set cupicef.set) - cupicef.I Coq__Init__Datatypes.nat - (n1 : - cupicef.Term cupicef.set - Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.nat))) - (mult__acc (Coq__Init__Nat.add m p) p) - (mult__acc (Coq__Arith__Plus.tail__plus p m) p) - (mult__acc (Coq__Arith__Plus.tail__plus p m) p) - (Coq__Init__Logic.f__equal - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (cupicef.prod cupicef.set - (cupicef.rule cupicef.set cupicef.set) - (cupicef.rule cupicef.set - (cupicef.rule cupicef.set cupicef.set)) - cupicef.I Coq__Init__Datatypes.nat - (m0 : - cupicef.Term cupicef.set - Coq__Init__Datatypes.nat => - cupicef.prod cupicef.set cupicef.set - (cupicef.rule cupicef.set cupicef.set) - cupicef.I Coq__Init__Datatypes.nat - (n1 : - cupicef.Term cupicef.set - Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.nat)))) - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (cupicef.prod cupicef.set cupicef.set - (cupicef.rule cupicef.set cupicef.set) - cupicef.I Coq__Init__Datatypes.nat - (n1 : - cupicef.Term cupicef.set - Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.nat))) - (f : - (m0 : - cupicef.Term cupicef.set - Coq__Init__Datatypes.nat -> - n1 : - cupicef.Term cupicef.set - Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.set - Coq__Init__Datatypes.nat) => - f p) - (mult__acc (Coq__Init__Nat.add m p)) - (mult__acc (Coq__Arith__Plus.tail__plus p m)) - (Coq__Init__Logic.eq__trans - (cupicef.cast - (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (cupicef.prod cupicef.set - (cupicef.rule cupicef.set cupicef.set) - (cupicef.rule cupicef.set - (cupicef.rule cupicef.set cupicef.set)) - cupicef.I Coq__Init__Datatypes.nat - (m0 : - cupicef.Term cupicef.set - Coq__Init__Datatypes.nat => - cupicef.prod cupicef.set cupicef.set - (cupicef.rule cupicef.set cupicef.set) - cupicef.I Coq__Init__Datatypes.nat - (n1 : - cupicef.Term cupicef.set - Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.nat)))) - (mult__acc (Coq__Init__Nat.add m p)) - (mult__acc (Coq__Init__Nat.add m p)) - (mult__acc (Coq__Arith__Plus.tail__plus p m)) - (Coq__Init__Logic.f__equal - (cupicef.cast - (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (cupicef.prod cupicef.set - (cupicef.rule cupicef.set - (cupicef.rule cupicef.set - cupicef.set)) - (cupicef.rule cupicef.set - (cupicef.rule cupicef.set - (cupicef.rule cupicef.set - cupicef.set))) - cupicef.I Coq__Init__Datatypes.nat - (s : - cupicef.Term cupicef.set - Coq__Init__Datatypes.nat => - cupicef.prod cupicef.set - (cupicef.rule cupicef.set - cupicef.set) - (cupicef.rule cupicef.set - (cupicef.rule cupicef.set - cupicef.set)) - cupicef.I Coq__Init__Datatypes.nat - (m0 : - cupicef.Term cupicef.set - Coq__Init__Datatypes.nat => - cupicef.prod cupicef.set - cupicef.set - (cupicef.rule cupicef.set - cupicef.set) - cupicef.I - Coq__Init__Datatypes.nat - (n1 : - cupicef.Term cupicef.set - Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.nat))))) - (cupicef.cast - (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (cupicef.prod cupicef.set - (cupicef.rule cupicef.set cupicef.set) - (cupicef.rule cupicef.set - (cupicef.rule cupicef.set - cupicef.set)) - cupicef.I Coq__Init__Datatypes.nat - (m0 : - cupicef.Term cupicef.set - Coq__Init__Datatypes.nat => - cupicef.prod cupicef.set cupicef.set - (cupicef.rule cupicef.set - cupicef.set) - cupicef.I Coq__Init__Datatypes.nat - (n1 : - cupicef.Term cupicef.set - Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.nat)))) - (f : - (s : - cupicef.Term cupicef.set - Coq__Init__Datatypes.nat -> - m0 : - cupicef.Term cupicef.set - Coq__Init__Datatypes.nat -> - n1 : - cupicef.Term cupicef.set - Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.set - Coq__Init__Datatypes.nat) => - f (Coq__Init__Nat.add m p)) - mult__acc mult__acc - (Coq__Init__Logic.eq__refl - (cupicef.cast - (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s - (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (cupicef.prod cupicef.set - (cupicef.rule cupicef.set - (cupicef.rule cupicef.set - cupicef.set)) - (cupicef.rule cupicef.set - (cupicef.rule cupicef.set - (cupicef.rule cupicef.set - cupicef.set))) - cupicef.I Coq__Init__Datatypes.nat - (s : - cupicef.Term cupicef.set - Coq__Init__Datatypes.nat => - cupicef.prod cupicef.set - (cupicef.rule cupicef.set - cupicef.set) - (cupicef.rule cupicef.set - (cupicef.rule cupicef.set - cupicef.set)) - cupicef.I - Coq__Init__Datatypes.nat - (m0 : - cupicef.Term cupicef.set - Coq__Init__Datatypes.nat => - cupicef.prod cupicef.set - cupicef.set - (cupicef.rule cupicef.set - cupicef.set) - cupicef.I - Coq__Init__Datatypes.nat - (n1 : - cupicef.Term cupicef.set - Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.nat))))) - mult__acc)) - (Coq__Init__Logic.f__equal - (cupicef.cast - (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (cupicef.cast - (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (cupicef.prod cupicef.set - (cupicef.rule cupicef.set cupicef.set) - (cupicef.rule cupicef.set - (cupicef.rule cupicef.set - cupicef.set)) - cupicef.I Coq__Init__Datatypes.nat - (m0 : - cupicef.Term cupicef.set - Coq__Init__Datatypes.nat => - cupicef.prod cupicef.set cupicef.set - (cupicef.rule cupicef.set - cupicef.set) - cupicef.I Coq__Init__Datatypes.nat - (n1 : - cupicef.Term cupicef.set - Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.nat)))) - mult__acc (Coq__Init__Nat.add m p) - (Coq__Arith__Plus.tail__plus p m) H0))) - (Coq__Init__Logic.f__equal - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I - (cupicef.prod cupicef.set cupicef.set - (cupicef.rule cupicef.set cupicef.set) - cupicef.I Coq__Init__Datatypes.nat - (n1 : - cupicef.Term cupicef.set - Coq__Init__Datatypes.nat => - Coq__Init__Datatypes.nat))) - (mult__acc (Coq__Arith__Plus.tail__plus p m)) p p - (Coq__Init__Logic.eq__refl - (cupicef.cast - (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ - (cupicef.type - (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - p)))) - (Coq__Init__Logic.f__equal - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (mult__acc (Coq__Arith__Plus.tail__plus p m) p) n0 n0 - (Coq__Init__Logic.eq__refl - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set - (cupicef.type (cupicef.s cupicef.z)) cupicef.I) - (cupicef.univ - (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type - (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - n0))) - (let__H7 n n0 IHn m p) (let__H6 n n0 IHn m p) - (let__H5 n n0 IHn m p)) - (Coq__Init__Nat.add (Coq__Arith__PeanoNat.Nat__add m p) - (Coq__Init__Nat.mul n0 p)) - (IHn (Coq__Arith__PeanoNat.Nat__add m p) p)) - (Coq__Arith__PeanoNat.Nat__add m - (Coq__Arith__PeanoNat.Nat__add p (Coq__Init__Nat.mul n0 p))) - (Coq__Arith__PeanoNat.Nat__add__assoc m p - (Coq__Init__Nat.mul n0 p)))) - n. - -def tail__mult : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.set Coq__Init__Datatypes.nat - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - mult__acc Coq__Init__Datatypes.O m n. - -thm mult__tail__mult : - n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat -> - cupicef.Term cupicef.prop - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.mul n m) (tail__mult n m)) - -:= n : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - m : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - cupicef.cast cupicef.prop cupicef.prop - ((n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.mul n m) n0) - (mult__acc Coq__Init__Datatypes.O m n)) - (Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.mul n m) (tail__mult n m)) - cupicef.I - (Coq__Init__Logic.eq__ind - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.add Coq__Init__Datatypes.O (Coq__Init__Nat.mul n m)) - (n0 : cupicef.Term cupicef.set Coq__Init__Datatypes.nat => - Coq__Init__Logic.eq - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.mul n m) n0) - (Coq__Init__Logic.eq__refl - (cupicef.cast (cupicef.type (cupicef.s cupicef.z)) - (cupicef.axiom (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - (cupicef.univ cupicef.set (cupicef.type (cupicef.s cupicef.z)) - cupicef.I) - (cupicef.univ (cupicef.type (cupicef.s (cupicef.s cupicef.z))) - (cupicef.axiom - (cupicef.type (cupicef.s (cupicef.s cupicef.z)))) - cupicef.I) - cupicef.I Coq__Init__Datatypes.nat) - (Coq__Init__Nat.add Coq__Init__Datatypes.O - (Coq__Init__Nat.mul n m))) - (mult__acc Coq__Init__Datatypes.O m n) - (mult__acc__aux n Coq__Init__Datatypes.O m)). - -(; End of translation. ;) diff --git a/out/Coq__Arith__Mult.dk.gz b/out/Coq__Arith__Mult.dk.gz new file mode 100644 index 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